Monte Carlo Integration Explanation in 1D

Let’s say that you have a function y=\sin(x)^2 and you want to know what the area is under the curve between 0 and pi.

We could solve this specific problem by doing some algebra and calculus to get the exact answer analytically (which is \frac{\pi}{2}), but let’s pretend like we can’t, or don’t want to solve it that way.

Another way to solve this problem is to use Monte Carlo integration, which lets you solve it numerically and get an approximated answer.

How you would do that is like this:

  1. Pick a random number between 0 and pi.
  2. Plug that value into the function y=\sin(x)^2 as x to get a y value.
  3. Do this multiple times and take the average to get the average y value of the function.
  4. Pretending that the function is a rectangle, you can use the average y as the height of the rectangle, and use pi as the width because we are looking between 0 and pi.
  5. Multiply that width and height to get the area of a rectangle, which is the estimated area under the curve.

That’s all you need to do!

Monte Carlo integration is pretty powerful in how simple it is, and how it works really well even in extremely high dimensions.

As you might imagine, the more samples you take to get your average y value, the better your estimate is going to be. Unfortunately though, you have to quadruple the number of samples you have to cut the error in half, so it can take a while to get the correct answer (converge) if you need a high level of accuracy. (https://en.wikipedia.org/wiki/Monte_Carlo_method#Integration)

Here’s a C++ code snippet doing this process with 10,000 samples. Each time you run the program you’ll get a different estimate. If you take more samples, you’ll more reliably get a better answer.

double SimpleMonteCarlo()
{
    double rangeMin = 0;
    double rangeMax = 3.14159265359;

    size_t numSamples = 10000;

    std::random_device rd;
    std::mt19937 mt(rd());
    std::uniform_real_distribution<double> dist(rangeMin, rangeMax);

    double ySum = 0.0;
    for (size_t i = 1; i <= numSamples; ++i)
    {
        double x = dist(mt);
        double y = sin(x)*sin(x);
        ySum += y;
    }
    double yAverage = ySum / double(numSamples);

    double width = rangeMax - rangeMin;
    double height = yAverage;

    return width * height;
}

Below is the output of the code ran 5 times. Note that the real answer is \frac{\pi}{2} which is 1.57079632679.

  1. 1.548451
  2. 1.554312
  3. 1.576727
  4. 1.578759
  5. 1.598686

(I’m actually a bit disturbed that the 5 runs are actually sorted from low to high but whatever …)

A problem with this being based on regular old random numbers (white noise) is that sometimes the numbers will clump, giving too much weighting to one area of the function, and leave empty space where another part of the function wasn’t sampled at all.

There are many different ways to deal with this situation but two of my favorites are…

  1. Blue Noise: https://blog.demofox.org/2018/01/30/what-the-heck-is-blue-noise/
  2. Low discrepancy sequences: https://blog.demofox.org/2017/05/29/when-random-numbers-are-too-random-low-discrepancy-sequences/

Both of those things give more even coverage over the sampling space which means that you won’t have as large gaps of missing information from your samples.

Another way to help this is stratified sampling, where you break the sampling space up into some number of sections, and choose random numbers within each section, making sure to have samples in each of the sections. That keeps the randomness, but gives more even coverage over the sampling space.

You might be tempted to just say “If I’m taking 100 samples, i’ll just sample every 1/100th of the space evenly”. That uniform / regular sampling has some problems including aliasing, but also loses some of the positive mathematical properties that random numbers can give you (like, being able to sample from non rational numbered locations!).

A variation on stratified sampling is a technique invented by Pixar called “jittered grid” where you do even sampling, but add a small random value to each sample.

There are lots and lots of other techniques which could make up a long list of blog posts, so we’ll stop there! ๐Ÿ™‚

More General Monte Carlo Integration

The last section was actually a simplified version of a Monte Carlo integration which was able to be simplified because it was using uniform random numbers.

Monte Carlo integration works with random numbers that have arbitrary distributions as well, not just uniform random numbers.

The process works mostly the same but there are a couple differences.

In the previous section, we got an average height and then multiplied by the width to get an estimate of the area under the curve, pretending that it was a rectangle.

The first change is to move the multiplication by the width into the loop. Instead of calculating an average height, we are instead calculating average rectangle areas.

Mathematically you get the same answer, so there’s nothing crazy there.

The second change is that instead of multiplying by the width, you divide by the probability of the number being chosen, that you plugged into the equation.

In the case of our function that we are taking samples of between 0 and pi, the probability of any single number being chosen in that range is \frac{1}{\pi}. When we divide by that, it means we end up just multiplying by pi, so it’s mathematically equivalent to what were were doing before!

Here’s the steps for the more generalized monte carlo integration:

  1. Pick a random number between 0 and pi using any random number distribution you’d like to.
  2. Plug that value into the function y=\sin(x)^2 as x to get a y value.
  3. Divide that y value by the probability of having chosen that number (otherwise known as PDF(x)) to get an estimated area of the function.
  4. Do this multiple times and take the average to get your result.

Here is some code to do the more general Monte Carlo integration, still using uniformly distributed random numbers.

double GeneralMonteCarlo()
{
    size_t numSamples = 10000;

    std::random_device rd;
    std::mt19937 mt(rd());
    std::uniform_real_distribution<double> dist(0.0f, 1.0f);

    auto InverseCDF = [](double x) -> double
    {
        return x * c_pi;
    };

    auto PDF = [](double x) -> double
    {
        return 1.0f / c_pi;
    };

    double estimateSum = 0.0;
    for (size_t i = 1; i <= numSamples; ++i)
    {
        double rnd = dist(mt);
        double x = InverseCDF(rnd);
        double y = sin(x)*sin(x);
        double pdf = PDF(x);
        double estimate = y / pdf;

        estimateSum += estimate;
    }
    double estimateAverage = estimateSum / double(numSamples);

    return estimateAverage;
}

Interestingly, dividing by the PDF is the same mathematically as multiplying by width in the last section – it literally ends up being a multiplication by pi (the width). The only difference is that we pulled the multiply into the loop, instead of leaving it until the end.

As an optimization, you could definitely move the divide out again (and turn it into a multiply), but I wanted to present the code as close to the core concepts as possible.

Non Uniform Random Number Distributions

Let’s try sampling from a different random number distribution. Let’s generate random numbers which have a distribution of y=\sin(x). You can see it compared to the function we are integrating y=\sin(x)^2 below. They are fairly similarly shaped!

To use y=\sin(x) as a random number distribution for monte carlo integration, we’ll need to calculate the normalized PDF and we’ll also need to calculate the inverse CDF.

If you want to know more about PDFs and “whatever an inverse CDF may be”, give this a read: Generating Random Numbers From a Specific Distribution By Inverting the CDF

  • The function y=\sin(x) is normalized to this PDF: \mathit{PDF}(x) = \frac{\sin(x)}{2}
  • To generate numbers from that PDF, you take a random number x that is between 0 and 1 and plug it into this function, which is the inverse CDF: \mathit{CDF}^{-1}(x) = 2 \cdot \sin^{-1}(\sqrt{x})

Here is a code snippet doing monte carlo integration with this PDF and inverse CDF:

double ImportanceSampledMonteCarlo()
{
    size_t numSamples = 10000;

    std::random_device rd;
    std::mt19937 mt(rd());
    std::uniform_real_distribution<double> dist(0.0, 1.0);

    auto InverseCDF = [](double x) -> double
    {
        return 2.0 * asin(sqrt(x));
    };

    auto PDF = [](double x) -> double
    {
        return sin(x) / 2.0f;
    };

    double estimateSum = 0.0;
    for (size_t i = 1; i <= numSamples; ++i)
    {
        double rng = dist(mt);
        double x = InverseCDF(rng);
        double y = sin(x)*sin(x);
        double pdf = PDF(x);
        double estimate = y / pdf;

        estimateSum += estimate;
    }
    double estimateAverage = estimateSum / double(numSamples);

    return estimateAverage;
}

To compare this versus uniform random sampling, I'll show the progress it makes over 50,000,000 samples first using uniform random numbers, then using the y=\sin(x) shaped PDF.

Uniform aka 1/pi:

sin(x):

You may notice that every 4x samples, the standard deviation (which is the square root of variance) drops in half, like we talked about before. This is why path tracing takes so long. If you don’t know what path tracing is, this is why modern animated movies take so long to render.

In the results, you can see that the variance of the estimates is a lot lower using this PDF that is shaped more like the function we are trying to integrate. We got a better, more reliable answer with fewer samples. Is that pretty cool? You bet it is! When you use a PDF shaped like the function you are integrating, to get better results faster, that is called importance sampling.

Bad Random Number Distributions

If you use a PDF which is shaped very differently from the function you are trying to integrate, you will get more variance and it will take longer to converge, which is a total bummer.

Let’s try y=(\frac{x}{\pi})^5, which doesn’t look much like the function we are trying to integrate at all:

Here is the PDF and inverse CDF:

  • \mathit{PDF}(x)=(\frac{x}{\pi})^5 \cdot \frac{6}{\pi}
  • \mathit{CDF}^{-1}(x)= (x*\pi^6)^{\frac{1}{6}}

Here it is with 50,000,000 samples:

And here is the uniform sampling again as a comparison:

As you can see, it is approaching the right answer, but is taking about 10 times as long to get the same results (amount of variance) compared to uniform sampling. Ouch!

Perfect Random Number Distributions

Let’s say that we got really lucky and somehow got the PDF and inverse CDF for a function that perfectly matched the function we were trying to integrate. What would happen then?

Let’s check it out by integrating the function y=\sin(x) by using a random number distribution which has the form y=\sin(x).

We already calculated the PDF and inverse CDF of that function earlier:

  • \mathit{PDF}(x) = \frac{\sin(x)}{2}
  • \mathit{CDF}^{-1}(x) = 2 \cdot sin^{-1}(\sqrt{x})

Here we do that with 50,000,000 samples:

WOW! As you can see, it had the right answer from the first sample, with zero variance (randomness) and it kept steady at that answer for all 50,000,000 samples.

This is a pretty neat concept, and if you know about “cosine weighted hemisphere sampling”, that does this exact thing.

Cosine weighted hemisphere samples are weighted such that you can remove the \cos(\theta) from the lighting calculations, because the random number distribution handles it for you.

It basically removes that part of randomness from the equations.

Unfortunately there are more variables and randomness in path tracing than just that term, but it helps.

Beyond this, you’d start look at other variance reduction techniques if you were interested, including multiple importance sampling.

Closing

Going into this blog post I thought “hey no sweat, i’ll make a few simple functions, calculate their PDFs, inverse CDFs and be on my way”.

I can’t believe how almost all the simple functions I tried ended up being impossible to take through the process.

for instance, you can take x=\sin(y) and solve for y to get y=\sin^{-1}(x), but if you try to solve x=\sin(y)+y for y, you are going to have a bad day!

I think in the future if I need to do something like this, I’d like to try fitting a curve to the (x,y) data points reordered as (y,x) data points, but there are many other methods for doing this sort of thing as well.

BTW if wondering how I was calculating std dev (aka square root of variance) while integrating, variance is “The average of the squared differences from the mean”. That means that if you know the correct answer of what you are trying to integrate, you can calculate the std dev like this:

        // Variance is "The average of the squared differences from the mean"
        double difference = integration - actualAnswer;
        double differenceSquared = difference * difference;
        averageDifferenceSquared = Lerp(averageDifferenceSquared, differenceSquared, 1.0 / double(i));
        double stdDev = sqrt(averageDifferenceSquared);
  • integration is the current average estimate (if you have taken 100 samples, it’s the average of the 100 samples)
  • actualAnswer is the known right answer
  • averageDifferenceSquared is also the variance
  • i is the number of samples you have taken, including the current one (aka start at 1, not 0)
  • If you are confused about me doing a lerp to calculate an average, give this a read: Incremental Averaging

Hope you enjoyed this write up!

Anders Lindqvist (@anders_breakin) is writing up a blog post explaining monte carlo, importance sampling, and multiple importance sampling that you might be interested in if you enjoyed this. Give him a follow, and it’ll be coming out soon ๐Ÿ™‚

Also, here is a really nice twitter thread talking about why importance sampling actually works:
https://twitter.com/Atrix256/status/1003487338633105409

Taking a Stroll Between The Pixels

This post relates to a paper I wrote which talks about (ab)using linear texture interpolation to calculate points on Bezier curves. Extensions generalize it to Bezier surfaces and (multivariate) polynomials. All that can be found here: https://blog.demofox.org/2016/02/22/gpu-texture-sampler-bezier-curve-evaluation/

The original observation was that if you sample along the diagonal of a 2×2 texture, that as output you get points on a quadratic Bezier curve with the control points of the curve being the values of the pixels like in the image below. When I say you get a quadratic Bezier curve, I mean it literally, and exactly. One way of looking at what’s going on is that the texture interpolation is literally performing the De Casteljau algorithm. (Note: if the “B” values are not equal in the setup below, the 2nd control point will be the average of these two values, which an extension abuses to fit more curves into a smaller number of pixels.)

An item that’s been on my todo list for a while is to look and see what happens when you sample off of the 45 degree diagonal between the pixel values. I was curious about questions like:

  • What if we sampled across a different line?
  • What if we samples across a quadratic curve like by having y=x^2?
  • What if we sampled on a circle or a sine wave?
  • How does the changed sampling patterns work in higher dimensions – like trilinear or quadrilinear interpolation?

After accidentally coming across the answer to the first question, it was time to look into the other ones too!

PS – if wondering “what use can any of this possibly have?” the best answer I have there is data compression for data on the GPU. If you can fit your data with piecewise rational polynomials, the ideas of this technique could be useful for storing that data in a concise way (pixels in a texture) that are also quickly and easily decoded by the GPU. The ideas from this post allows for more curve types when fitting and storing your data, beyond piecewise rational polynomials. It’s also possible to store higher order curves and surfaces into smaller amounts of texture data.

Quick Setup: Bilinear Interpolation Formula

Bilinear interpolation is available on modern GPUs as a way of getting sub-pixel detail. In the olden days, when zooming into a texture, the square pixels just got larger because nearest neighbor filtering was used. In modern times, when looking at the space between pixel values, bilinear interpolation is used to fill in the details better than nearest neighbor does.

You can describe bilinear interpolation as interpolating two values across the x axis and interpolating between the results across the y axis (reversing the order of axes also works). Mathematically, that can look like this:

z = (A(1-x) + Bx)(1-y) + (C(1-x)+Dx)y

Where x and y are values between 0 and 1 describing where the point is between the pixels, and A,B,C,D are the values of the 4 nearest pixels, which form a box around the point we are calculating. A = (0,0), B = (1,0), C = (0,1) and D = (1,1).

With some algebra, you can get that equation into a power series form which is going to be easier to work with in our experiments:

z = (A-B-C+D)xy + (B-A)x + (C-A)y + A

For some deeper info on bilinear interpolation check out these links:
https://blog.demofox.org/2015/04/30/bilinear-filtering-bilinear-interpolation/
http://reedbeta.com/blog/quadrilateral-interpolation-part-1/
http://reedbeta.com/blog/quadrilateral-interpolation-part-2/
https://computergraphics.stackexchange.com/questions/7539/geometric-interpretation-of-this-bilinear-interpolation-equation/7541

Now that we have our formula, we can begin! ๐Ÿ™‚

Sampling Along Other Lines

So, if we sample along the diagonal from A to D, we know that we get a quadratic equation out. What happens if we sample along other lines though?

My guess before I knew the answer to this was that since the 45 degree angle line is quadratic (degree 2), and that horizontal and vertical lines were linear (degree 1), that sampling along other lines must be a fractional degree polynomial between 1 and 2. It turns out that isn’t the answer, but I wonder if there’s a way to interpret the “real answer” as a fractional polynomial?

Anyways, wikipedia clued me in: https://en.wikipedia.org/wiki/Bilinear_interpolation#Nonlinear

The interpolant is linear along lines parallel to either the x or the y direction, equivalently if x or y is set constant. Along any other straight line, the interpolant is quadratic

What that means is that if you walk along a horizontal or vertical line, it’s going to be linear. Any other line will be quadratic.

Let’s try it out.

Remembering that the equation for a linear function is y=mx+b let’s literally replace y with mx+b and see what we get out.

So, we start with the power series bilinear interpolation polynomial:

z = (A-B-C+D)xy + (B-A)x + (C-A)y + A

Which becomes this after substitution:

z = (A-B-C+D)x(mx+b) + (B-A)x + (C-A)(mx+b) + A

After some expansion and simplification we get this:

z = (Am-Bm-Cm+Dm)x^2+(Ab-Bb-Cb+Db+Cm-Am+B-A)x+Cb-Ab+A

This formula tells us the value we get if we have a bilinear interpolation of values A,B,C,D (aka a bilinear surface defined by those points), and we sample along the x,y line defined by y=mx+b.

It’s a very generalized function that’s hard to reason about much, but one thing is clear: it is a quadratic function! Whatever constant values you choose for A,B,C,D,m and b, you will get a quadratic polynomial (or lower degree, but never higher).

Here’s a shadertoy that shows curves generated by random sub pixel line segments on a random (white noise) RGB texture: https://www.shadertoy.com/view/XstBz7

(note that the rough edges of the curve are due to the fact that interpolation happens in X.8 fixed point format, so has pretty limited precision. Check the paper for more information and ways to address the issue.)

Let’s explore a bit by plugging in some values for m and b and see what happens for different types of lines.

m=0, b=0

Let’s see what happens when m is 0 and b is 0. In other words, lets see what happens when we sample along the line y=0.

Plugging those values in gives:

z = (B-A)x + A

interestingly, this is just a linear interpolation between A and B, which makes sense when looking at the graph of where we are sampling on the bilinear surface.

This goes along with what wikipedia told us: when one of the axes is constant (it’s a horizontal or vertical line) the result is linear.

m=1, b=0

Let’s try m = 1 and b = 0. That is the line: y=x. This graph shows where that is sampling from on the bilinear surface:

Plugging in the values gives us this equation:

z = (A-B-C+D)x^2+(C+B-2A)x+A

We get a quadratic out! This shouldn’t be too surprising. This is the original insight in the technique. This is also the formula for a quadratic Bezier curve with control points A, (B+C)/2, D.

m=2, b=1

Let’s try the line y=2x+1. Here’s the graph of where we are sampling on the bilinear surface:

Plugging in the values give us the equation:

z = (2A-2B-2C+2D)x^2+(C+D-2A)x+C

Once again we got a quadratic function when sampling along a line.

You might think it’s strange that the equation ends it “+C” instead of “+A”, but if you look at the graph it makes sense. The line literally starts at C when x is zero.

x=2u, y=3u

In the above examples we are only modifying the y variable, to be some function of x. What if we also want to modify the x variable?

One way to do this is to make a 3rd variable u that goes from 0 to 1. Then we can make x and y be based on that variable.

Let’s see what happens when we use these two equations:

y=2u

x=3u

That makes us sample this line on the bilinear surface.

Plugging the functions of u in for x and y we get:

z = (6A-6B-6C+6D)u^2+(2B+3C-5A)u+A

It’s still a quadratic!

What About a Quadratic Path?

So we now know that when moving along a straight line on a bilinear surface, that you will get a quadratic function as output, except in the case of the line being horizontal or vertical. Note: if the bilinear surface is a plane, all lines on that surface will be linear functions, so this is another way to get a linear result. It could also be degenerate and give you a point result. You will never get a cubic result (or higher) when going along a straight line though.

What would happen though if instead of sampling along straight lines, we sampled on other shapes, like quadratic curves?

y=x*x

Let’s start with the function y=x^2. The path that is sampled is:

Going back to the power series form of bilinear interpolation, let’s plug x^2 in for y and see what we get out.

The starting equation:

z = (A-B-C+D)xy + (B-A)x + (C-A)y + A

becomes:

z = (A-B-C+D)x(x^2) + (B-A)x + (C-A)(x^2) + A

Which becomes:

z = (A-B-C+D)x^3 + (C-A)x^2 + (B-A)x + A

It’s a cubic equation!

Here is a shadertoy which follows this sampling path on random pixels: https://www.shadertoy.com/view/4sdBz7

Something neat about this example specifically is that a cubic equation has 4 coefficients, which are basically 4 control points. This example makes use of the values of the 4 pixels involved to come up with the 4 coefficients, so “doesn’t leave anything on the table” so to speak.

This is unlike sampling along line segments where you have 3 control points stored in 4 pixel values. One is a bit redundant in that case.

You can make use of that fact (I have for instance!), but sampling along a quadratic path to get a cubic curve feels like a natural fit.

x=u*u, y=u*u

Let’s see what happens when we move along both x and y quadratically.

Just like in the linear case, we have our 3rd variable u that goes from 0 to 1 and we have x and y be based on that variable. We will use these equations:

x=u^2

y=u^2

The sampling path looks like this:

When we plug those in we get this quartic function:

z = (A-B-C+D)u^4 + (B+C-2A)u^2 + A

You might be surprised to see what looks like a linear path. It’s just because at all times, x is the same value as y, even though they travel down the line non linearly.

Shadertoy: https://www.shadertoy.com/view/Xdtfz7

Higher Order Curves: x=3u^2, y=2u^4

Let’s get a little more wild, using these equations:

x=3u^2

y=u^4

Which makes a sampling path of this:

Plugging in the equations, the bilinear interpolation equation:

z = (A-B-C+D)xy + (B-A)x + (C-A)y + A

becomes a hexic equation:

z = (3A-3B-3C+3D)u^6 + (C-A)u^4 + (3B-3A)u^2 + A

The shadertoy visualizes it on random pixels as per usual, but with u going from 0 to 1, it means that x goes from 0 to 3 (y is still 0-1), which makes some obvious discontinuities at the boundaries of pixels. In our pure math formulation, we wouldn’t have any of those, but since we are sampling a real texture, when we leave the safety of our (0,1) box, we enter a new box with different control points. https://www.shadertoy.com/view/4dtfz7

Trigonometric Function: y = sin(2*pi*x)

Let’s try y=sin(2\pi x), which takes this path on the bilinear surface:

The bilinear interpolation equation becomes a trigonometric polynomial:

z = (A-B-C+D)x*sin(2\pi x) + (B-A)x + (C-A)*sin(2\pi x) + A

That has disconuities in it when texture sampling again, due to leaving the original pixel region, so here’s a better looking shadertoy, which is for y=sin(2\pi x)*0.5+0.5. It scales and shifts the y values to be between 0 and 1. https://www.shadertoy.com/view/4stfz7

Circle

Lastly, here’s sampling on a circle.

x=sin(2 \pi u)*0.5+0.5

y=cos(2 \pi u)*0.5+0.5

It follows this path:

Plugging the functions into the power series bilinear equation gives:

z = (A-B-C+D)*(sin(2 \pi u)*0.5+0.5)*(cos(2 \pi u)*0.5+0.5) + (B-A)*(sin(2 \pi u)*0.5+0.5) + (C-A)*(cos(2 \pi u)*0.5+0.5) + A

Here’s the shadertoy: https://www.shadertoy.com/view/Xddfz7

Something neat about sampling in a circle is that it’s continuous – note how the left side of the curves line up with the right side seamlessly. That seems like a pretty useful property.

Moving On

We went off into the weeds a bit, but hopefully you can see how there are a ton of possibilities for encoding and decoding data in a very small number of pixels by carefully crafting the path you sample along.

Compared to the simple “sample along the diagonal” technique, there is some added complexity and shader instructions though. Namely, any work you do to modify x or y before passing them to the linear texture interpolator needs to happen in shader code. That means this technique takes more ALU, but can mean it takes even less texture memory than the other method.

The last question from the top of the post is “What does this all mean in higher dimensional interpolation, like trilinear or quadrilinear?”

Well, it works pretty much the same was as bilinear but there are more dimensions to work with.

We saw that in 2 dimensional bilinear interpolation that when we made x and y be functions (either of each other, or of a 3rd variable u), that the resulting polynomial had a degree that was the degree of x plus the degree of y.

In 3 dimensions with trilinear interpolation, the resulting polynomial would have a degree that is the degree of x, plus the degree of y, plus the degree of z.

In 4 dimensions with quadrilinear, add to that the degree of w.

Let’s consider the case when we don’t want a single curve though, but want a surface or (hyper) volume.

As we’ve seen in the extension dealing with surfaces and volumes, if you have a degree N polynomial, you can break it apart into a multivariate polynomial (aka a surface or hyper volume) so long as the sum of the degrees of each axis adds up to N.

It’s basically what we were just talking about but in reverse.

One thing I think would be interesting to explore further would be to see what the limitations are when you take this “too far”.

For instance, a 2×2 texture can give you a quadratic if you sample along any straight line in the uv coordinates. If you first put the u coordinate through a cubic function, and put the v coordinate through a different cubic function, I think you should be able to make a bicubic surface.

The surface will be constrained to a subset of what a general bicubic surface is able to be shaped like, but you will get a bicubic surface. (basically there will be implicit control points that you don’t have control over unless you add more pixels, and do more sampling, or higher dimensional linear interpolation)

I’d like to see what the constraints there are and see if there’s any chance of getting any real use out of something like that.

Anyhow, thanks for reading! Any ideas, corrections, usage cases you have, whatever, hit me up!

@Atrix256

Prefix Sums and Summed Area Tables

Prefix sums and summed area tables let you sum up regions of arrays or grids in constant time.

If that sounds like it might not have many uses, that is another way of saying that it does discrete integration in constant time, and can also be made to do some kinds of convolution.

These things come up quite a bit in game development and graphics so is pretty interesting for things like depth of field, glossy reflections, and maybe image based lighting. Check the links at the end of the post to see these things in action in some pretty interesting ways.

We’ll start with one dimension.

One Dimension – Prefix Sums

Say that you have 10 numbers:

\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \textbf{index} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} \\ \hline \textbf{value} & 8 & 3 & 7 & 4 & 12 & 6 & 4 & 10 & 1 & 2 \\ \hline \end{array}

To sum up numbers in a given range you have to manually add up the numbers in that range.

Summing the numbers at index 2 through 5 inclusively takes 3 adds and gives you the answer 29. (index 2 + index 3 + index 4 + index 5)

Summing the numbers at index 0 through index 9 inclusively (the whole table) takes 9 adds to get the answer 57.

Interestingly there is a way to preprocess this data such that summing any range takes only a single subtraction. The technique is called a prefix sum table and you make the table by having the number at each index be the sum from index 0 to that index inclusively.

Here is the prefix sum table for the numbers above:

\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \textbf{index} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} \\ \hline \textbf{value} & 8 & 11 & 18 & 22 & 34 & 40 & 44 & 54 & 55 & 57 \\ \hline \end{array}

Now, to find the sum of range a to b inclusively, you start with the value at index b, and subtract the value at index (a-1).

So, to sum the numbers at index 2 through 5 like we did before, we’d start with the value at index 5 which is 40, and we subtract the value at index (2-1) aka index 1, which is 11. That gives us a result of 29 like our manual summing did before.

To sum the numbers at index 0 through index 9, we’d start with the value at index 9, which is 57, and subtract the value at index -1. Since we don’t have anything before index 0, the sum for anything before index 0 is 0. That makes our result be 57-0 or 57, which we calculated before.

Let’s move on to 2D!

Two Dimensions – Making a Summed Area Table

In two dimensions, the same technique is called a summed area table, and things get only a little more complicated.

You start with a 2d grid of values like this:

i = \begin{array}{|c|c|c|c|} \hline 3 & 2 & 1 & 8 \\ \hline 9 & 11 & 15 & 0 \\ \hline 8 & 4 & 7 & 6 \\ \hline 12 & 7 & 8 & 3 \\ \hline \end{array}

Then you make a grid of the same size, where the value at a location is the sum of all the values in the rectangle going from (0,0) to (x,y) inclusive. Assuming that (0,0) is in the top left, that would give us this summed area table:

I =  \begin{array}{|c|c|c|c|} \hline 3 & 5 & 6 & 14 \\ \hline 12 & 25 & 41 & 49 \\ \hline 20 & 37 & 60 & 74 \\ \hline 32 & 56 & 87 & 104 \\ \hline \end{array}

You can literally sum up all the values for each index to make the table if you want to, but you can also use this formula which lets you iteratively create the table by starting at (0,0) and expand outwards from there. As before, when reading out of bounds values, just use zero.

I(x,y)=i(x,y)+I(x,y-1)+I(x-1,y)-I(x-1,y-1)

Two Dimensions – Using a Summed Area Table

So we know that I(x,y) is the sum of all the values in the rectangle from (0,0) to (x,y) inclusively, but what if we want to find the sum of a different rectangle? What if we have 4 points A,B,C,D and we want to know the sum of the numbers within that sub-rectangle?

With some cleverness we can calculate the sum inside this exact region.

First we get the value at point D, which gives us the sum of this rectangle:

Next, we subtract the value at point B, which gives us the sum of this rectangle:

The next step is to subtract the value at point C. The red area is a problem though as it has been subtracted out twice.

This is a problem that’s easily solved by adding the value at point A in, to give us our final result:

So, to summarize, using a summed area table to get the sum of all values in the rectangle defined by the points A,B,C,D is done by reading the values at points A,B,C,D and calculating: A+D-B-C

Storage Costs

When you want to store numbers added together, you are going to need storage larger than what you are storing the numbers in.

For instance, if you have the table below using 3 bits per value:
I =  \begin{array}{|c|c|} \hline 7 & 7 \\ \hline 7 & 7 \\ \hline \end{array}

Turning that into a summed area table, you are going to hit overflow problems:
I =  \begin{array}{|c|c|} \hline 7 & 6 \\ \hline 6 & 4 \\ \hline \end{array}

For summing up N items, you need log_2{(N)} more bits of storage which means we would need 2 more bits of storage in this case for the 2×2 grid (4 samples), making it be 5 bits total per value (3 bits of storage + 2 extra bits to hold the sum of 4 values). That would let us store the proper table:

I =  \begin{array}{|c|c|} \hline 7 & 14 \\ \hline 14 & 28 \\ \hline \end{array}

On twitter, Sam Littlewood (https://twitter.com/samlittlewood) shared some interesting info with me:

Using the previously shown 2×2 table of 3bit 7’s as an example, what this means is that if you are only ever going to want to ask about 1×1 ranges (which is pointless to use summed area tables for, but makes a nice simple example), you don’t need 2 extra bits, and in fact don’t need any extra bits in this case since a 1×1 range is just 1 sample, and log_2{(1)} is 0.

Looking back at the summed area table that had roll over problems:
I =  \begin{array}{|c|c|} \hline 7 & 6 \\ \hline 6 & 4 \\ \hline \end{array}

Let’s ask about the range (1,1) to (1,1). So we start with the value at index (1,1) which is 4. Next we add in the value at index (0,0) which is 7 and get 11. Keeping that in 3 bits (eg mod 8), that gives us a value of 3. Next we subtract the value at index (0,1) aka 6, which keeping it in 3 bits gives us 5. Subtracting index (1,0) from that (6 again) and keeping it in 3 bits gives us 7.

So, the sum of the numbers from (1,1) to (1,1) – aka the VALUE in the original table at (1,1) – is 7. Since we made the table, we know this is correct.

It works interestingly!

If we did a 2×2 lookup instead, it would fall apart. we’d need those 2 extra bits since we’d be summing 4 samples, and log_2{(4)} is 2.

So, just to re-iterate… summed area tables do need increased storage per data item to store the sums. However, while most descriptions base that increased storage on the size of the image being made into a summed area table, it is actually based on the largest range you want to sum from that table, which may be smaller than the total size.

I have an idea I’d like to try (next blog post?) where instead of storing the sum of the rectangle at each position, you store the sum divided by the area. In other words, you store the average value for the rectangle.

Calculating the sum for a specific rectangle then becomes getting the 4 values, multiplying by their area, and then doing the usual math.

Apparently this is similar to an idea of using floating point numbers in SAT, which also sounds interesting! Thread from Bart Wronski (https://twitter.com/BartWronsk):

While my idea is similar to using floating point, a handful of people (especially Tom Forsyth! https://twitter.com/tom_forsyth) have made sure I know that using floating point with large textures (~screen sized and above) is not a good idea.

Tom says:
“The entries in the bottom-right of the table start having very similar magnitudes, so the difference between them is very noisy. This is super obvious with float16s where you only have 10 bits of precision, which is less than most current screen widths.”

Other Stuff

Bilinear Interpolation
If you are wondering whether you should use bilinear interpolation when using this technique (sample between pixels) or not, the answer is that you should. Bilinear interpolation is compatible with this technique and gives you the correct values for sub pixel sample points.

Higher Dimensions
This technique extends to 3 dimensions and beyond. The table still contains the sum of the numbers for the (hyper)rectangle from the origin to that specific index. The way you calculate the sum of a specific range is different in each dimension, but it’s similar, and you should be able to figure it out using the logic described in the 2d case!

Integrating / Summing Over Other Shapes

I had a thought on this that might not be so bad.

My thought was that if you had some shape you wanted to sum values over (aka integrate values over), that you could sum over the bounding box of the shape, divide by the area of the bounding box to get an average sum per unit for that area, and then multiply by the area of the shape you want to sum over.

This makes the assumption that the bounding box is representative of the data inside of the shape, so that makes this an approximation, but it might be good enough depending on your needs.

You might even try having a couple different summed area tables made from rotated versions of the image. That would allow you to get a tighter fitting bounding box in some situations.

I’m definitely not the first to think about how to do this though, and this is not the only way to do it. There is a link in the next section that talks about a different way to do it “Fast and Exact Convolution With Polygonal Filters” that also references a few other ways to do it.

Uses in Graphics / Other Links

Here is the paper from Franklin Crow in 1984 that introduces summed area tables as a way to get box filtered mipmapping on the fly without having to generate mipmaps in advance:
http://www.florian-oeser.de/wordpress/wp-content/2012/10/crow-1984.pdf

Here is a neat paper that talks about how to generate summed area tables efficiently on the GPU, and some interesting ways to use them for things like depth of field, glossy reflections, and refraction through frosted glass:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.8836&rep=rep1&type=pdf

Here are some great reads from Fabien Giesen (https://twitter.com/rygorous) on doing fast blurs when the radius is very large. The second post also shows you how to do repeated box blurs to get tent filters, quadratic filters, cubic, etc and how they tend towards Gaussian. I’m sure there is some way to mix this concept with summed area tables to get higher order filters, but I haven’t found or worked out the details yet.
https://fgiesen.wordpress.com/2012/07/30/fast-blurs-1/
https://fgiesen.wordpress.com/2012/08/01/fast-blurs-2/

Here are some blog posts I made up explaining and demonstrating box blurs and Gaussian blurs:
https://blog.demofox.org/2015/08/18/box-blur/
https://blog.demofox.org/2015/08/19/gaussian-blur/

Bart also shared these really interesting links

“Fast and Exact Convolution With Polygonal Filters”
https://www.researchgate.net/publication/269699690_Fast_and_Exact_Convolution_with_Polygonal_Filters

“Fast Filter Spreading and its Applications”
https://www2.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-54.pdf

“Filtering by repeated integration”
https://www.researchgate.net/publication/220721661_Filtering_by_repeated_integration

“Cinematic Depth Of Field: How to make big filters cheap”
http://32ipi028l5q82yhj72224m8j.wpengine.netdna-cdn.com/wp-content/uploads/2017/03/GDC2017-Cinematic-Depth-Of-Field.pdf

What the Heck is Blue Noise?

This is a gentle explanation of blue noise and how it can be useful.

We’ll start with something simple that we can all get behind – not getting eaten by a cheetah!

Let’s talk about our eyes for a minute.

Our eyes have about 126 million photo receptors in them – about 6 million cones, 120 million rods (source). These photo receptors give your brain an image of the world around you. They are a bit like pixels because they are just small points of data that your brain combines into an image.

How those photo receptors are arranged in your eye can make a big difference. Imagine for a second that we only had 10 photo receptors. If they were laid out like these blue dots, we wouldn’t be able to see the cheetah and we’d become a tasty cat snack.

In the image above, white noise random numbers were used to place the points. White noise is what most people are talking about when they talk about random numbers. Using white noise to generate numbers, the numbers can clump up in some spots and leave empty holes in other spots. When using white noise to lay out photo receptors, that makes it so some photo receptors give redundant information when they are too close together, and leave big open spaces in your vision where you are not getting any information at all. Not good!

What if the dots were laid out like this instead?

The points are still randomly placed, but they are roughly evenly spaced. This makes it so we get the most bang for our buck from the photo receptors. We basically have the maximum amount of information we can get for the number of photo receptors we have to work with.

In this case, two of the photo receptors are on the cat, so we have some information about that predator, and we have a better chance at reacting before we become lunch!

Blue noise random numbers were used to place the points on this image, and this example shows exactly why blue noise can be better than white noise – you get maximal information with fewer samples.

Interestingly, our photo receptors (as well as other animals) are in fact laid out this way. Here is an image of a primate (macaque) retina (source)

You might also find this an interesting read about chicken eyes which also have blue noise properties:
https://www.princeton.edu/news/2014/02/24/eye-chicken-new-state-matter-comes-view

That’s blue noise in a nut shell, but continue on if you’d like to go just a tad bit deeper.

A Little More Technical

If maximizing information is the goal, you might wonder why blue noise is better than putting the sample points in a grid, or in a honeycomb structure or some other regular pattern. The short answer is that regular patterns have a problem called “aliasing”. Random numbers in general trade the problem of aliasing for the problem of noise, but blue noise random numbers in particular still get the benefits of “roughly even coverage”, so blue noise is the best of both worlds.

Blue noise is difficult / computationally intensive to generate though, compared to white noise or regular sampling. Generating better blue noise more efficient is in fact is an ongoing area of research!

For a deeper comparison of white noise, blue noise, and regular sampling, and also how to generate blue noise sample points, give this a read: https://blog.demofox.org/2017/10/20/generating-blue-noise-sample-points-with-mitchells-best-candidate-algorithm/

If you want at least some of the benefits of blue noise, but don’t want to spend the resources to compute it, a nice alternative might be low discrepancy sequences. You can read about them (and how to generate them) here: https://blog.demofox.org/2017/05/29/when-random-numbers-are-too-random-low-discrepancy-sequences/

You often hear about blue noise and low discrepancy sequences in graphics / in numerical integration. For low sample counts, the blue noise / LDS’s give you more even spaces for your samples in the sampling domain, but I’ve heard that white noise gives you better results for larger sample counts.

There is a whole rainbow of noises possible, each with their own unique usage cases. If you want to know a way to transmute white noise to other colors of noises, give this a read: https://blog.demofox.org/2017/10/25/transmuting-white-noise-to-blue-red-green-purple/

Lastly, the other day I found out that Tempurpedic beds are the best, because they have some secret formula/process they bought from NASA. This recipe allows them to make memory foam such that the bubbles are all roughly the same size. The foam is not arranged into any regular structure such as a grid or a honeycomb, so in essence, the memory foam is blue noise. More specifically, it’s basically the Voronoi diagram of blue noise distributed sample points in 3d.

So, Tempurpedic is the best because they have blue noise foam.

Weird, right?!

C++ Differentiable Programming: Searching For An Optimal Dither Pattern

The simple standalone C++ source code that implements this blog post and replicates the results shown is on github at: https://github.com/Atrix256/DitherFindGradientDescent

Neural networks are a hot topic right now. There is a lot of mystery and mystique surrounding them, but at their core, they are just simple programs where parameters are tuned using gradient descent.

(If curious about neural networks, you might find this interesting: How to Train Neural Networks With Backpropagation)

Gradient descent can be used in a lot of other situations though, and in fact, you can even generalize the core functionality of neural networks to work on other types of programs. That is exactly what we are doing in this post.

To be able to use gradient descent to optimize parameters of a program, your program has to be roughly of the form of:

  1. It has parameters that specify how it processes some other data
  2. There is some way for you to give a score to how well it did

Beyond those two points, much like as a shader program or a SIMD program, you want your program to be as branchless as possible. The reason for this is because ideally your entire program should be made up of differentiable operations. Branches (if statements) cause discontinuities and are not differentiable. There are ways to deal with branches, and some branches don’t actually impact the result, but it’s a good guideline to keep in mind. Because of this, you also want to stay away from non differentiable functions – such as a “step” function which you might be tempted to use instead of an if statement.

This post is going to go into detail about using differentiable programming in C++ for a specific goal. Results are shown, and the simple / no external dependency C++ code that generated them are at https://github.com/Atrix256/DitherFindGradientDescent.

First, let’s have a short introduction to gradient descent.

One Dimensional Gradient Descent

If you have a function of the form f(x), it takes one input so is one dimensional.

You can think of a function like this as having a value for every point on the number line.

You can visualize those values as a height, which gives you a function of the form y=f(x) which we are still going to call one dimensional, despite it now having two dimensions.

Let’s look at a function y=3x+1

You might remember that the equation of a line is y=mx+b where m is the slope of the line (\frac{\text{rise}}{\text{run}} or \frac{y}{x}) and b is where the line crosses the y axis.

In calculus, you learn that the slope m is also the derivative of the function: \frac{dy}{dx}

The slope / derivative tells you how much is added to y for every 1 you add to x.

Let’s say that you were on this graph at the point x=1 (which puts you at y=4), and let’s say that you want to go downhill from where you were at. You could do that by looking at the slope / derivative at that point, which is 3 (it’s 3 for every point on the line). Since the derivative is positive, that means going to the right will make the y value larger (you’ll go up hill) and going to the left will make the y value smaller (you’ll go down hill).

So, if you want to go downhill to a smaller y value, you know that you need to subtract values from x.

A simpler way to think of this is that you need to subtract the derivative from your x value to make your y value smaller.

That is a core fact that will help guide you through things as they get more difficult: subtract the derivative (later, subtract the gradient) to make your value smaller. The value subtracted is often multiplied by some scalar value to make it move faster or slower.

What happens if you have a more complex function, such as y=(x-2)^2?

Let’s say that you are on this graph at the point x=1, which puts you at y=1. Now, which way do you move to go downhill?

The derivative of this function is y=2x-4, which you can plug your x value into to get the slope / derivative at that point: -2.

Remembering that we subtract that derivative to go down hill, that means we need to subtract a negative value from our x; aka we need to ADD a value to our x.

As you can see, adding a value to x and making it move to the right does in fact make us go down hill.

The rule works, hooray!

Two Dimensional Gradient Descent

Things do get a little more complex when there’s more than one dimension, but not really that much more complex, so hang in there!

Let’s look at the function z=xy

Let’s say that we are at the (x,y) point (1,1) – in the upper right corner – which puts us at z=1, and let’s say that we want to go down hill. Instead of just having one variable to take the derivative of (x), we now have two variables (x and y). How are we going to deal with this?

The answer is PARTIAL derivatives.

First up, we are going to pretend that y is a constant value, and not actually a variable. This will give us the partial derivative for x: \frac{\partial z}{\partial x}. That tells us how much we would add to z if we added one to x. It’s a slope that is specifically down the x axis.

In this case, the partial derivative of z with respect to x is just: y.

Doing the same thing for the other variable, the partial derivative of z with respect to y is just: x.

Now that we have partial derivatives for each variable, we put them into a vector. This vector is called the gradient, and has some intimidating notation that looks like this:

\nabla z = \nabla f(x,y) = (\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y})

For this function, the gradient is:

\nabla z = \nabla f(x,y) = (y,x)

That makes the gradient at our specific point:

\nabla z = \nabla f(1,1) = (1,1)

In the last section we saw that the derivative / slope pointed to where the function got larger. The same thing is true of gradients, they point in the direction where the function gets larger too!

So, if we want to go downhill, we need to subtract values from our x and our y to go there. In fact, we know that the steepest way down from our current point is when we subtract the same value from both x and y. This is because the gradient doesn’t just point to where it gets larger, it points to where it gets larger the FASTEST. So, the reverse of the gradient also points to where it gets smaller the fastest.

Pretty cool huh?

You can confirm this visually by looking at the graph of the function.

One last things about slopes, derivatives and gradients before moving on. While they do point in the direction of greatest increase, they are only valid for an infinitely small point on the graph for functions that are non linear. This will be important later when we move in the opposite direction of the gradients, but do so with very small steps to help make sure we find the lowest points on the graph.

Why Gradient Descent?

Why do we want to use gradient descent? Imagine that we have a function:

w=f(x,y,z)

Sure, we can pick some random starting values for x,y and z, and then use gradient descent to find the smallest w, but who cares?

Let’s give some other names to these variables and see if the value becomes a little more apparent:

DamageTakenMultiplier = CalculateDamageTakenMultiplier(Armor, Dodge, Resist)

Now, by only changing the names of the variables, we can see that we could use gradient descent to find what amount of Armor, Dodge and Resist would make it so our character takes the least amount of damage. This can now tell you how to distribute stat points to a character to get the best results ๐Ÿ˜›

Note that if you are ever trying to find the highest number possible, instead of the lowest, you can just multiply your function by -1 and do everything else the same way. You could also do gradient ASCENT, but it’s equivalent to multiplying by -1 and doing gradient descent.

Problems

Here are a few common problems you can encounter when doing gradient descent.

  • Local minima – when you get to the bottom of a bowl, but it isn’t the deepest bowl.
  • Flat derivatives – these make it hard to escape a local area because the derivatives are very small, which will make each movement also very small.
  • Discontinuities – The problem space (graph) changes abruptly without warning, making gradient descent do the wrong thing

Here’s an example of a local minima versus a global minima. You can see that depending on where you start on this graph, you might end up in the deeper bowl, or the shallower bowl if your only rule is “move downhill”.

(Image from wikipedia By KSmrq – http://commons.wikimedia.org/wiki/File:Extrema_example.svg, GFDL 1.2, https://commons.wikimedia.org/w/index.php?curid=6870865)

Here’s an example of a flat derivative. You can imagine that if you were at x=1, that you could see that the derivative would tell you to go to the left to decrease the y value, but it’s a very, very small number. This is a problem because it’s common to multiply the derivative or gradient by a multiplier before subtracting it, so you’d only take a very small step towards the goal.

It’s also possible to hit a perfectly flat derivative, which will be exactly 0. In this case, no matter how big or small of a number you multiply the derivative by, you won’t move AT ALL!

Below is a discontinuous function where if x is less than 0.5, the value is 1, otherwise the value is x. This essentially shows you what happens when you use if statements in differentiable programming. If you start on the right side, it’s going to correctly tell you that you should move left to improve your score. However, it’ll keep telling you to move left, until you get to x being less than 0.5, at which point your score will suddenly get a lot worse and your derivative will become 0. You will now be stuck!

There are ways to deal with these problems, but they are deep topics. If nothing else, you should know these problems exist, so you can know when they are affecting you, and/or why you should avoid them if you have a choice.

What If I Want to Avoid Calculus?

Let’s say that you don’t get a kick out of calculating all these partial derivatives. Or, more pragmatically, you don’t want to sit down and manually calculate the gradient function of some generic C++ code!

I have some great news for you.

While we do need partial derivatives for our gradients, we aren’t going to have to do all this calculus to get them!

Here are a few other ways to get partial derivatives:

  • Finite Differences – Conceptually super simple, but slow to calculate and not always very precise. More info: Finite Differences
  • Backpropagation – What neural networks use. Also called backwards mode automatic differentiation. Fast but a bit complex mentally. I linked this already but for more info: How to Train Neural Networks With Backpropagation
  • Dual Numbers – Also called forward mode automatic differentiation. Not as fast as backwards mode, but in the same neighborhood for speed. Super, super convinient and awesome for programmers. I love these. More info: Dual Numbers & Automatic Differentiation

Care to guess which one we are going to use? Yep, Dual Numbers!

In a nutshell, if you have code that uses floats, you can change it to use a templated type instead. Then, you put dual numbers through your code instead of floats. The output you get will be the specific value output from your code, but also the GRADIENT of your code at that value. Even better, this isn’t a numerical method (it’s not an approximation), it’s analytical (it’s exact).

That is seriously all there is to it. Dual numbers are amazing!

Since you made the code templated, you can still use it for floats when you don’t want or need the gradient.

Differentiable Programming / Gradient Descent Skeleton

Here’s the general skeleton we are going to be following for using gradient descent with our differentiable program.

  1. Initialize the parameters to random (but valid) values, storing them in dual numbers.
  2. Run the code that does our work, taking dual numbers as input for the parameters of how it does the work.
  3. Put the result (which is dual numbers) into a scoring function to give us a score. Usually the score is such that smaller numbers are better. If not, just multiply the score by -1 so it is.
  4. Since we did the work and calculated the score using dual numbers, we now have a gradient which describes how we need to adjust the parameters to make our score better.
  5. Adjust our parameters using the gradient and go back to step 2. Repeating until whatever exit condition we want is hit: maybe when a certain number of iterations happen, or maybe when our score gets below a certain value.

That’s our game plan. Let’s dive into the specific problem we are going to be attacking.

Searching For an Ideal Dithering Pattern

Here is the problem we want to tackle:

We want to find a 3×3 dithering pattern such that when we use it to dither an image (by repeating the 3×3 pattern over and over across the image), and then blur the result by a specific amount, that it’s as close as possible to the original image being blurred by that same amount.

That sounds a bit challenging right? It’s not actually that bad, don’t worry (:

The steps the code has to do (differentiably) are:

  1. Dither the source image
  2. Blur the results
  3. Blur the source image
  4. Calculate a score for how similar they are
  5. Use all this with Gradient Descent to optimize the dither pattern

Once again, we need to do this stuff differentiably, using dual numbers, so that we get a gradient for how to modify the dither pattern to better our score.

Step 1 – Dither Source Image

Dithering an image is a pretty simple process.

We are going to be dithering it such that we take a greyscale image as input and convert it to a black and white image using the dither pattern.

(If you are starting with a color image, this shows how to convert it to greyscale: Converting RGB to Grayscale)

For every pixel (x,y) in the source image, you look at pixel (x%3, y%3) in the dither pattern, and if the dither pattern pixel is less than the source, you write a black pixel out, else you write a white pixel out.

if (sourcePixel(x,y) < ditherPixel(x%3, y%3))
    pixelOut(x,y) = 0.0;
else
    pixelOut(x,y) = 1.0;

There’s a problem though… this is a branch, which makes a discontinuity, which will make it so we can’t have good derivatives to help us get to the goal.

Another way to write the dithering operation above is to write it like this:

difference = ditherPixel(x%3, y%3) - sourcePixel(x,y);
pixelOut(x,y) = step(difference);

Where “step” is the “heaviside step function”, which is 1 if x >= 0, otherwise is 0.

(Image from Wikipedia By Omegatron (Own work) [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)%5D, via Wikimedia Commons)

That got rid of the branch (if statement), but we still have a discontinuous function.

Luckily we can approximate a step function with other functions. I decided to use the formula 0.5+atan(100*x)/pi which looks like this:

Unfortunately, I found that my results weren’t that good, so i switched it to 0.5+atan(10000*x)/pi which ended up working better for me:

This function does have the problem of having flat derivatives, but I found that it worked pretty well anyways. The flat derivatives don’t seem to be a big problem in this case luckily.

To put it all together, the differentiable version of dithering a pixel that I use looks like this:

difference = ditherPixel(x%3, y%3) - sourcePixel(x,y);
pixelOut(x,y) = 0.5+atan(10000.0f * difference) / pi;

As input to this dithering process, we take:

  • The source image
  • a 3×3 dither pattern, where each pixel is a dual number

As output this dithering process gives us:

  • A dithered image that is converted to black and white (either a 1.0 or 0.0 value per pixel)
  • It’s the same size as the source image
  • Each pixel is a dual number with 9 derivatives. There is one derivative per dither pixel.

Step 2 – Blur the Results

Blurring the results of the dither wasn’t that difficult. I used a Gaussian blur, but other blurs could be used easily.

I had some Gaussian blur code laying around (from this blog post: Gaussian Blur) and I converted it to using a templated type instead of floats/pixels where appropriate, also making sure there were no branches or anything discontinuous.

It turned out there wasn’t a whole lot to fix up here luckily so wasn’t too difficult.

This allowed me to take the dithered results which are a dual number per pixel, and do a Gaussian blur on them, preserving and correctly modifying the gradient (derivatives) as it did the Blur.

Step 3 – Blur the Source Image

Blurring the source image was easy since the last step made a generic gaussian blur function. I used the generic Gaussian blur function to blur the image. This doesn’t need to be done as dual numbers, so it was regular pixels in and regular pixels out.

You might wonder why this part doesn’t need to be done as dual numbers.

The simple answer is because these values are in no way dependant on the dither pattern (which are what we are tracking with the derivatives).

The more mathematical explanation is that you could in fact consider these dual numbers, which just have a gradient of zero because they are essentially constants that have nothing to do (yet) with the parameters of the function. The gradient would just implicitly be zero, like any other constant value you might introduce to the function.

Step 4 – Calculating a Similarity Score

Next up I needed to calculate a similarity score between the dithered then blurred results (which is made up of dual numbers), and the source image which was blurred (and is made up of regular pixels).

The similarity score I went with is just MSE or “Mean Squared Error”.

To calculate MSE, for every pixel you do this:

error = ditheredBlurredImage(x,y) - blurredImage(x,y);
errorSquared = error * error;

After you have the squared error for every pixel, you just take the average of them to get the MSE.

An interesting thing about MSE is that because errors are squared, it will favor smaller errors much more than larger errors, which is a nice property.

A not so nice property about MSE is that it might decide something is a small difference mathematically even though a human would decide that it was a huge difference perceptually. The reverse is also true. Despite this, I chose it because it is simple and I ended up getting decent results with it.

If you want to go down the rabbit hole of looking at “perceptual similarity scores of images” check out these links:

After this step, we have an MSE value which says how similar the images are. A lower value means lower average squared error, so lower numbers are indeed better.

What else is nice is that the MSE value is a dual number with a gradient that has the 9 partial derivatives that describe how much the MSE changes as you adjust each parameter.

That gradient tells us how to adjust the parameters (the 3×3 dither pixels!) to lower the MSE!

Step 5 – Putting it All Together

Now it’s time to put all of this together and use gradient descent to make our dither pattern better.

Here’s how the program operates:

  1. Initialize the 3×3 dither pattern to random values, setting the derivatives to 1.0 in the gradient, for the variable that they represent.
  2. do 1000 iterations of this loop:
    1. Dither and blur the source image
    2. Calculate MSE of this result compared to the source image blurred
    3. Using the gradient from the MSE value, subtract the respective partial derivative from each of the pixels in the dither pattern, but scaling the partial derivative by a “learning rate”.
  3. Output the best result found

The learning rate starts at 3.0 at loop iteration 0, but decays with each iteration, down to 0.1 at iteration 999. It starts above 1 to help escape local minima, and uses a very small rate at the end to try and get deeper into whatever minimum it has found.

After adjusting the dither pattern pixels, I clamp them to be between 0 and 1.

Something else I ought to mention is that while I’m doing the gradient descent, I keep track of the best scoring dither pattern seen.

This way, after the 1000 iterations are up, if we ever saw anything better than where we are at currently, we just use that instead of the final result.

Presumably, if you tune your parameters (learning rate, iterations, etc!) correctly, this won’t come up often, but it’s always a possibility that your final state is not the best state encountered, so this is a nice way to get better results more often.

Results

Did you notice that I called this post “searching for an ideal dither pattern” instead of “finding an ideal dither pattern”? (:

The results are decent, but I know they could be better. Even so, I think the techniques talked about here are a good start going down the path of differentiable programming, and similar topics.

Here are some results I was able to get with the code. Click to see the full size images. The shrunken down images have aliasing issues.

The images left to right are: The original, the dither pattern used (repeated), the dithered image, the blurred dither image, and lastly the blurred original image. The program aims to make the last two images look as close as possible as it can, using MSE as the metric for how close they are.

Here is the starting state of using a Gaussian blur with a sigma of 10:

Here it is after the 1000 iterations of gradient descent. Notice the black blob at the top is gone compared to where it started.

Here’s the starting state when using a Gaussian blur sigma of 1:

And here it is after 1000 iterations, which is pretty decent results:

Lastly, here it is with no blurring whatsoever:

And after 1000 iterations, I think it actually looks worse!

Using no blur at all makes for some really awful results. The blur gives the algorithm more freedom on how it can succeed, whereas with no blur, there is a lot less wiggle room for finding a solution.

Another benefit of using the blur before MSE calculation is that a blur is a low pass filter. That means that higher frequencies are gone before the MSE calculation. The result of this is that the algorithm will favor results which are closer to blue noise dithering. Pretty neat right?!

Closing

I hope you enjoyed this journey through differentiable programming and gradient descent, and I hope you were able to follow along.

Here are some potentially interesting things to do beyond what we talked about here:

  • Have it learn from a set of images, instead of only this single image. This should help prevent “over fitting” and let it find a dither pattern which works well for all images instead of just this one specific image.
  • Use a separate set of images to gauge the accuracy of the result that weren’t used as part of the training, to help prove that it really hasn’t overfit the training data.
  • Try applying “small corruption” in the learning to help prevent overfitting or getting stuck in local minima – one idea would be to have some percentage chance per derivative that you don’t apply the change to the dither pattern pixel. This would add some randomness to the gradient descent instead of it only being down the steepest direction all of the time.
  • Instead of optimizing the dithering patterns, you could make a formula that generated the dithering patterns, and instead optimize the coefficients / terms of that formula. If you get good results, you’ll end up with a formula you can use for dithering instead of a pattern, which might be nice for the case of avoiding a texture read in a pixel shader to do the dithering.

I’m not a data scientist or machine learning expert by any means, so there are plenty of improvements to be made. There is a lot of overlap with what is being done here and other algorithms – both in the machine learning realm and outside of the machine learning realm.

For one, you can use Newton’s method for gradient descent. It can find minima faster by using the second derivative in the calculations as well.

However, this algorithm is almost purely “exploitative” in that wherever you start with your parameters, it will try to go from there to the deepest point in whatever valley it’s already in. Some other types of algorithms differ from this in that they are more “explorative” and try to find other valleys, but aren’t always as good at finding the deepest part of the valleys that they do find. More explorative algorithms include simulated annealing, differential evolution, and genetic algorithms.

If you enjoyed this post, check out this book for deeper details on algorithms relating to gradient descent (simulated annealing, genetic algorithms, etc!). It’s a very good book and very easy to read!
Essentials of Metaheuristics

Any corrections to what i’ve said, the code, or suggestions for improvements, please let me know by leaving a comment here, or hitting me up on twitter: https://twitter.com/Atrix256

Demystifying Floating Point Precision

Floating point numbers have limited precision. If you are a game programmer, you have likely encountered bugs where things start breaking after too much time has elapsed, or after something has moved too far from the origin.

This post aims to show you how to answer the questions:

  1. What precision do I have at a number?
  2. When will I hit precision issues?

First, a very quick look at the floating point format.

Floating Point Format

Floating point numbers (Wikipedia: IEEE 754) have three components:

  1. Sign bit – whether the number is positive or negative
  2. Exponent bits – the magnitude of the number
  3. Mantissa bits – the fractional bits

32 bit floats use 1 bit for sign, 8 bits for exponent and 23 bits for mantissa. Whatever number is encoded in the exponent bits, you subtract 127 to get the actual exponent, meaning the exponent can be from -126 to +127.

64 bit doubles use 1 bit for sign, 11 bits for exponent and 52 bits for mantissa. Whatever number is encoded in the exponent bits, you subtract 1023 to get the actual exponent, meaning the exponent can be from -1022 to +1023.

16 bit half floats use 1 bit for sign, 5 bits for exponent and 10 bits for mantissa. Whatever number is encoded in the exponent bits, you subtract 15 to get the actual exponent, meaning the exponent can be from -14 to +15.

For all of the above, an exponent of all zeros has the special meaning “exponent 0” (and this is where the denormals / subnormals come into play) and all ones has the special meaning “infinity”

The exponent bits tell you which power of two numbers you are between – [2^{exponent}, 2^{exponent+1}) – and the mantissa tells you where you are in that range.

What precision do I have at a number?

Let’s look at the number 3.5.

To figure out the precision we have at that number, we figure out what power of two range it’s between and then subdivide that range using the mantissa bits.

3.5 is between 2 and 4. That means we are diving the range of numbers 2 to 4 using the mantissa bits. A float has 23 bits of mantissa, so the precision we have at 3.5 is:

\frac{4-2}{2^{23}} = \frac{2}{8388608} \approx 0.000000238418579

3.5 itself is actually exactly representable by a float, double or half, but the amount of precision numbers have at that scale is that value. The smallest number you can add or subtract to a value between 2 and 4 is that value. That is the resolution of the values you are working with when working between 2 and 4 using a float.

Using a double instead of a float gives us 52 bits of mantissa, making the precision:

\frac{4-2}{2^{52}} = \frac{2}{4503599627370496} \approx 0.00000000000000044408921

Using a half float with 10 bits of mantissa it becomes:

\frac{4-2}{2^{10}} = \frac{2}{1024} = 0.001953125

Here’s a table showing the amount of precision you get with each data type at various exponent values. N/A is used when an exponent is out of range for the specific data type.

\begin{array}{c|c|c|c|c} exponent & range & half & float & double \\ \hline 0 & [1,2) & 0.0009765625 & 0.00000011920929 & 0.0000000000000002220446 \\ 1 & [2,4) & 0.001953125 & 0.000000238418579 & 0.00000000000000044408921 \\ 2 & [4,8) & 0.00390625 & 0.000000476837158 & 0.00000000000000088817842 \\ 9 & [512, 1024) & 0.5 & 0.00006103515 & 0.00000000000011368684 \\ 10 & [1024,2048) & 1 & 0.00012207031 & 0.00000000000022737368 \\ 11 & [2048,4096) & 2 & 0.00024414062 & 0.00000000000045474735 \\ 12 & [4096,8192) & 4 & 0.00048828125 & 0.0000000000009094947 \\ 15 & [32768, 65536) & 32 & 0.00390625 & 0.0000000000072759576 \\ 16 & [65536, 131072) & N/A & 0.0078125 & 0.0000000000014551915 \\ 17 & [131072, 262144) & N/A & 0.015625 & 0.00000000002910383 \\ 18 & [262144, 524288) & N/A & 0.03125 & 0.000000000058207661 \\ 19 & [524288, 1048576) & N/A & 0.0625 & 0.00000000011641532 \\ 23 & [8388608,16777216) & N/A & 1 & 0.00000000186264515 \\ 52 & [4503599627370496, 9007199254740992) & N/A & 536870912 & 1 \\ \end{array}

A quick note on the maximum number you can store in floating point numbers, by looking at the half float specifically:

A half float has a maximum exponent of 15, which you can see above puts the number range between 32768 and 65536. The precision is 32 which is the smallest step that can be made in a half float at that scale. That range includes the smaller number but not the larger number. That means that the largest number a half float can store is one step away (32) from the high side of that range. So, the largest number that can be stored is 65536 – 32 = 65504.

How Many Digits Can I Rely On?

Another helpful way of looking at floating point precision is how many digits of precision you can rely on.

A float has 23 bits of mantissa, and 2^23 is 8,388,608. 23 bits let you store all 6 digit numbers or lower, and most of the 7 digit numbers. This means that floating point numbers have between 6 and 7 digits of precision, regardless of exponent.

That means that from 0 to 1, you have quite a few decimal places to work with. If you go into the hundreds or thousands, you’ve lost a few. When you get up into the tens of millions, you’ve run out of digits for anything beyond the decimal place.

You can actually see that this is true in the table in the last section. With floating point numbers, it’s at exponent 23 (8,388,608 to 16,777,216) that the precision is at 1. The smallest value that you can add to a floating point value in that range is in fact 1. It’s at this point that you have lost all precision to the right of the decimal place. Interestingly, you still have perfect precision of the integers though.

Half floats have 10 mantissa bits and 2^10 = 1024, so they just barely have 3 digits of precision.

Doubles have 52 mantissa bits and 2^52 = 4,503,599,627,370,496. That means doubles have between 15 and 16 digits of precision.

This can help you understand how precision will break down for you when using a specific data type for a specific magnitude of numbers.

When will I hit precision issues?

Besides the loose rules above about how many digits of precision you can count on, you can also solve to see when precision will break down for you.

Let’s say that you are tracking how long your game has been running (in seconds), and you do so by adding your frame delta (in seconds) to a variable every frame.

If you have a 30fps game, your frame delta is going to be 0.0333.

Adding that each frame to a float will eventually cause the float to reach a value where that number is smaller than the smallest difference representable (smaller than the precision), at which point things will start breaking. At first your accuracy will drop and your time will be wrong, but eventually adding your frame delta to the current time won’t even change the value of the current time. Time will effectively stop!

When will this happen though?

We’ll start with the formula we saw earlier and do one step of simple algebra to get us an equation which can give us this answer.

\frac{range}{mantissa} = precision \\ \\ range = mantissa * precision

How we use this formula is we put the precision we want into “precision” and we put the size of the mantissa (2^{MantissaBits}) into “mantissa”. The result tells us the range that we’ll get the precision at.

Let’s plug in our numbers:

range = 8388608 * 0.0333 = 279340.6464

This tells us the range of the floating point numbers where we’ll have our problems, but this isn’t the value that we’ll have our problems at, so we have another step to do. We need to find what exponent has this range.

Looking at the table earlier in the post you might notice that the range at an exponent also happens to be just 2^{exponent}.

That’s helpful because that just means we take log2 of the answer we got:

log2(279340.6464) = 18.0916659875

Looking at the table again, we can see that floating point numbers have a precision of 0.03125 at exponent value 18. So, exponent 18 is close, but it’s precision is smaller than what we want – aka the precision is still ok.

That means we need to ceil() the number we got from the log2.

Doing that, we see that things break down at exponent 19, which has precision of 0.0625. This actual value it has this problem at is 528,288 (which is 2^{19}).

So, our final formula for “where does precision become this value?” becomes:

value = pow(2, ceil(log2(mantissa * precision)))

Note that at exponent 18, there is still imprecision happening. When adding 1/30 to 264144, It goes from 264144 to 264144.031 to 264144.063, instead of 264144, 264144.033, 264144.066. There is error, but it’s fairly small.

At exponent 19 though, things fall apart a lot more noticeably. When adding 1/30 to 528288, it goes from 528288 to 528288.063 to 528288.125. Time is actually moving almost twice as fast in this case!

At exponent 20, we start at 1056576.00 and adding 1/30 doesn’t even change the value. Time is now stopped.

It does take 6.1 days (528,288 seconds) to reach exponent 19 though, so that’s quite a long time.

If we use half floats, it falls apart at value 64. That’s right, it only takes 64 seconds for this to fall apart when using 16 bit half floats, compared to 6.1 days when using 32 bit floats!

With doubles, it falls apart at value 281,474,976,710,656. That is 8,925,512 years!

Let’s check out that equation again:

value = pow(2, ceil(log2(mantissa * precision)))

A possibly more programmer friendly way to do the above would be to calculate mantissa * precision and then round up to the next power of 2. That’s exactly what the formula is doing above, but in math terms, not programming terms.

Storing Integers

I recently learned that floating point numbers can store integers surprisingly well. It blows my mind that I never knew this. Maybe you are in the same boat ๐Ÿ˜›

Here’s the setup:

  1. For any exponent, the range of numbers it represents is a power of 2.
  2. The mantissa will always divide that range into a power of 2 different values.

It might take some time and/or brain power to soak that up (it did for me!) but what that ends up ultimately meaning is that floating point numbers can exactly represent a large number of integers.

In fact, a floating point number can EXACTLY store all integers from -2^{MantissaBits+1} to +2^{MantissaBits+1}.

For half floats that means you can store all integers between (and including) -2048 to +2048. (\pm 2^{11})

For floats, it’s -16,777,216 to +16,777,216. (\pm 2^{24})

For doubles it’s -9,007,199,254,740,992 to +9,007,199,254,740,992. (\pm 2^{53})

Doubles can in fact exactly represent any 32 bit unsigned integer, since 2^32 = 4,294,967,296.

Links

Here are some links you might find interesting!

Floating point visually explained:
http://fabiensanglard.net/floating_point_visually_explained/

What Every Computer Scientist Should Know About Floating-Point Arithmetic:
https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

A matter of precision:
http://tomforsyth1000.github.io/blog.wiki.html#[[A%20matter%20of%20precision]]

Denormal numbers – aka very small numbers that make computations slow when you use them:
https://en.m.wikipedia.org/wiki/Denormal_number

Catastrophic Cancellation – a problem you can run into when doing floating point math:
https://en.wikipedia.org/wiki/Loss_of_significance

A handy web page that lets you play with the binary representation of a float and what number it comes out as:
https://www.h-schmidt.net/FloatConverter/IEEE754.html

Half precision floating point format:
https://en.wikipedia.org/wiki/Half-precision_floating-point_format

What is the first integer that a float is incapable of representing?
https://stackoverflow.com/questions/3793838/which-is-the-first-integer-that-an-ieee-754-float-is-incapable-of-representing-e

Ready to go deeper? Bruce Dawson has some amazing write ups on deeper floating point issues:
https://randomascii.wordpress.com/category/floating-point/

This talks about how to use floating point precision limits as an activation function in a neural network (?!)
https://blog.openai.com/nonlinear-computation-in-linear-networks/

Animating Noise For Integration Over Time 2: Uniform Over Time

After I put out the last post, Mikkel Gjoel (@pixelmager), made an interesting observation that you can see summarized in his image below. (tweet / thread here)

BTW Mikkel has an amazing presentation about rendering the beautiful game “Inside” that you should check out. Lots of interesting techniques used, including some enlightening uses of noise.
YouTube –
Low Complexity, High Fidelity: The Rendering of INSIDE

The images left to right are:

  • One frame of white noise
  • N frames of white noise averaged.
  • N frames averaged where the first frame is white noise, and a per frame random number is added to all pixels every frame.
  • N frames averaged where the first frame is white noise, and 1/N is added to all pixels every frame.
  • N frames averaged where the first frame is white noise, and the golden ratio is added to all pixels every frame.

In the above, the smoother and closer to middle grey that an image is, the better it is – that means it converged to the true result of the integral better.

Surprisingly it looks like adding 1/N outperforms the golden ratio, which means that regular spaced samples are outperforming a low discrepancy sequence!

To compare apples to apples, we’ll do the “golden ratio” tests we did last post, but instead do them with adding this uniform value instead.

To be explicit, there are 8 frames and they are:

  • Frame 0: The noise
  • Frame 1: The noise + 1/8
  • Frame 2: The noise + 2/8
  • Frame 7: the noise + 7/8

Modulus is used to keep the values between 0 and 1.

Below is how white noise looks animated with golden ratio (top) vs uniform values (bottom). There are 8 frames and it’s played at 8fps so it loops every second.

Interleaved Gradient Noise. Top is golden ratio, bottom is uniform.

Blue Noise. Top is golden ratio, bottom is uniform.

The uniform ones look pretty similar. Maybe a little smoother, but it’s hard to tell by just looking at it. Interestingly, the frequency content of the blue noise seems more stable using these uniform values instead of golden ratio.

The histogram data of the noise was the same for all frames of animation, just like in last post, which is a good thing. The important bit is that adding a uniform value doesn’t modify the histogram shape, other than changing which counts go to which specific buckets. Ideally the histogram would start out perfectly even like the blue noise does, but since this post is about the “adding uniform values” process, and not about the starting noise, this shows that the process does the right thing with the histogram.

  • White Noise – min 213, max 306, average 256, std dev 16.51
  • Interleaved Gradient Noise – min 245, max 266, average 256, std dev 2.87
  • Blue Noise – min, max, average are 256, std dev 0.

Let’s look at the integrated animations.

White noise. Top is golden ratio, bottom is uniform.

Interleaved gradient noise. Top is golden ratio, bottom is uniform.

Blue noise. Top is golden ratio, bottom is uniform.

The differences between these animations are subtle unless you know what you are looking for specifically so let’s check out the final frames and the error graphs.

Each noise comparison below has three images. The first image is the “naive” way to animate the noise. The second uses golden ratio instead. The third one uses 1/N. The first two images (and techniques) are from (and explained in) the last post, and the third image is the technique from this post.

White noise. Naive (top), golden ratio (mid), uniform (bottom).


Interleaved gradient noise. Naive (top), golden ratio (mid), uniform (bottom).


Blue noise. Naive (top), golden ratio (mid), uniform (bottom).


So, what’s interesting is that the uniform sampling over time has lower error and standard deviation (variance) than golden ratio, which has less than the naive method. However, it’s only at the end that the uniform sampling over time has the best results, and it’s actually quite terrible until then.

The reason for this is that uniform has good coverage over the sample space, but it takes until the last frame to get that good coverage because each frame takes a small step over the remaining sample space.

What might work out better would be if our first frame was the normal noise, but then the second frame was the normal noise plus a half, so we get the most information we possibly can from that sample by splitting the sample space in half. We would then want to cut the two halves of the space space in half, and so the next two frames would be the noise plus 1/4 and the noise plus 3/4. We would then continue with 1/8, 5/8, 3/8 and 7/8 (note we didn’t do these 1/8 steps in order. We did it in the order that gives us the most information the most quickly!). At the end of all this, we would have our 8 uniformly spaced samples over time, but we would have taken the samples in an order that makes our intermediate frames look better hopefully.

Now, interestingly, that number sequence I just described has a name. It’s the base 2 Van Der Corput sequence, which is a type of low discrepancy sequence. It’s also the 1D version of the Halton sequence, and is related to other sequences as well. More info here: When Random Numbers Are Too Random: Low Discrepancy Sequences

Mikkel mentioned he thought this would be helpful, and I was thinking the same thing too. Let’s see how it does!

White noise. Uniform (top), Van Der Corput (bottom).

Interleaved gradient noise. Uniform (top), Van Der Corput (bottom).

Blue noise. Uniform (top), Van Der Corput (bottom).

The final frames look the same as before (and the same as each other), so I won’t show those again but here are the updated graphs.



Interestingly, using the Van Der Corput sequence has put intermediate frames more in line with golden ratio, while of course still being superior at the final frame.

I’ve been trying to understand why uniform sampling over time out performs the golden ratio which acts more like blue noise over time. I still don’t grasp why it works as well as it does, but the proof is in the pudding.

Theoretically, this uniform sampling over time should lead to the possibility of aliasing on the time axis, since blue noise / white noise (and other randomness) get rid of the aliasing in exchange for noise.

Noise over the time dimension would mean missing details that were smaller than the sample spacing size. in our case, we are using the time sampled values (noise + uniform value) to threshold a source image to make a sample. It may be that since we are thresholding, that aliasing isn’t possible since our sample represents everything below or equal to the value?

I’m not really sure, but will be thinking about it for a while. If you have any insights please let me know!

It would be interesting to try an actual 1d blue noise sequence and see how it compares. If it does better, it sounds like it would be worth while to try jittering the uniform sampled values on the time axis to try and approximate blue noise a bit. Mikkel tried the jittering and said it gave significantly worse results, so that seems like a no go.

Lastly, some other logical experiments from here seem to be…

  • See how other forms of noise and ordered dithers do, including perhaps a Bayer Matrix. IG noise seems to naturally do better on the time axis for some reason I don’t fully understand yet. There may be some interesting properties of other noise waiting to be found.
  • Do we get any benefits in this context by arranging the interleaved gradient noise in a spiral like Jorge mentions in his presentation? (Next Generation Post Processing In Call Of Duty: Advanced Warfare
  • It would be interesting to see how this works in a more open ended case – such as if you had temporal AA which was averaging a variable number of pixels each frame. Would doing a van Der Corput sequence give good results there? Would you keep track of sample counts per pixel and keep marching the Van Der Corput forward for each pixel individually? Or would you just pick something like an 8 Van Der Corput sequence, adding the current sequence to all pixels and looping that sequence every 8 frames? It really would be interesting to see what is best in that sort of a setup.

I’m sure there are all sorts of other things to try to. This is a deep, interesting and important topic for graphics and beyond (:

Code

Source code below, but it’s also available on github, along with the source images used: Github:
Atrix256/RandomCode/AnimatedNoise

#define _CRT_SECURE_NO_WARNINGS

#include <windows.h>  // for bitmap headers.  Sorry non windows people!
#include <stdint.h>
#include <vector>
#include <random>
#include <atomic>
#include <thread>
#include <complex>
#include <array>

typedef uint8_t uint8;

const float c_pi = 3.14159265359f;

// settings
const bool c_doDFT = true;

// globals 
FILE* g_logFile = nullptr;

//======================================================================================
inline float Lerp (float A, float B, float t)
{
    return A * (1.0f - t) + B * t;
}

//======================================================================================
struct SImageData
{
    SImageData ()
        : m_width(0)
        , m_height(0)
    { }
   
    size_t m_width;
    size_t m_height;
    size_t m_pitch;
    std::vector<uint8> m_pixels;
};
 
//======================================================================================
struct SColor
{
    SColor (uint8 _R = 0, uint8 _G = 0, uint8 _B = 0)
        : R(_R), G(_G), B(_B)
    { }

    inline void Set (uint8 _R, uint8 _G, uint8 _B)
    {
        R = _R;
        G = _G;
        B = _B;
    }
 
    uint8 B, G, R;
};

//======================================================================================
struct SImageDataComplex
{
    SImageDataComplex ()
        : m_width(0)
        , m_height(0)
    { }
  
    size_t m_width;
    size_t m_height;
    std::vector<std::complex<float>> m_pixels;
};
 
//======================================================================================
std::complex<float> DFTPixel (const SImageData &srcImage, size_t K, size_t L)
{
    std::complex<float> ret(0.0f, 0.0f);
  
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            // Get the pixel value (assuming greyscale) and convert it to [0,1] space
            const uint8 *src = &srcImage.m_pixels[(y * srcImage.m_pitch) + x * 3];
            float grey = float(src[0]) / 255.0f;
  
            // Add to the sum of the return value
            float v = float(K * x) / float(srcImage.m_width);
            v += float(L * y) / float(srcImage.m_height);
            ret += std::complex<float>(grey, 0.0f) * std::polar<float>(1.0f, -2.0f * c_pi * v);
        }
    }
  
    return ret;
}
  
//======================================================================================
void ImageDFT (const SImageData &srcImage, SImageDataComplex &destImage)
{
    // NOTE: this function assumes srcImage is greyscale, so works on only the red component of srcImage.
    // ImageToGrey() will convert an image to greyscale.
 
    // size the output dft data
    destImage.m_width = srcImage.m_width;
    destImage.m_height = srcImage.m_height;
    destImage.m_pixels.resize(destImage.m_width*destImage.m_height);
 
    size_t numThreads = std::thread::hardware_concurrency();
    //if (numThreads > 0)
        //numThreads = numThreads - 1;
 
    std::vector<std::thread> threads;
    threads.resize(numThreads);
 
    printf("Doing DFT with %zu threads...\n", numThreads);
 
    // calculate 2d dft (brute force, not using fast fourier transform) multithreadedly
    std::atomic<size_t> nextRow(0);
    for (std::thread& t : threads)
    {
        t = std::thread(
            [&] ()
            {
                size_t row = nextRow.fetch_add(1);
                bool reportProgress = (row == 0);
                int lastPercent = -1;
 
                while (row < srcImage.m_height)
                {
                    // calculate the DFT for every pixel / frequency in this row
                    for (size_t x = 0; x < srcImage.m_width; ++x)
                    {
                        destImage.m_pixels[row * destImage.m_width + x] = DFTPixel(srcImage, x, row);
                    }
 
                    // report progress if we should
                    if (reportProgress)
                    {
                        int percent = int(100.0f * float(row) / float(srcImage.m_height));
                        if (lastPercent != percent)
                        {
                            lastPercent = percent;
                            printf("            \rDFT: %i%%", lastPercent);
                        }
                    }
 
                    // go to the next row
                    row = nextRow.fetch_add(1);
                }
            }
        );
    }
 
    for (std::thread& t : threads)
        t.join();
 
    printf("\n");
}
 
//======================================================================================
void GetMagnitudeData (const SImageDataComplex& srcImage, SImageData& destImage)
{
    // size the output image
    destImage.m_width = srcImage.m_width;
    destImage.m_height = srcImage.m_height;
    destImage.m_pitch = 4 * ((srcImage.m_width * 24 + 31) / 32);
    destImage.m_pixels.resize(destImage.m_pitch*destImage.m_height);
  
    // get floating point magnitude data
    std::vector<float> magArray;
    magArray.resize(srcImage.m_width*srcImage.m_height);
    float maxmag = 0.0f;
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            // Offset the information by half width & height in the positive direction.
            // This makes frequency 0 (DC) be at the image origin, like most diagrams show it.
            int k = (x + (int)srcImage.m_width / 2) % (int)srcImage.m_width;
            int l = (y + (int)srcImage.m_height / 2) % (int)srcImage.m_height;
            const std::complex<float> &src = srcImage.m_pixels[l*srcImage.m_width + k];
  
            float mag = std::abs(src);
            if (mag > maxmag)
                maxmag = mag;
  
            magArray[y*srcImage.m_width + x] = mag;
        }
    }
    if (maxmag == 0.0f)
        maxmag = 1.0f;
  
    const float c = 255.0f / log(1.0f+maxmag);
  
    // normalize the magnitude data and send it back in [0, 255]
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            float src = c * log(1.0f + magArray[y*srcImage.m_width + x]);
  
            uint8 magu8 = uint8(src);
  
            uint8* dest = &destImage.m_pixels[y*destImage.m_pitch + x * 3];
            dest[0] = magu8;
            dest[1] = magu8;
            dest[2] = magu8;
        }
    }
}

//======================================================================================
bool ImageSave (const SImageData &image, const char *fileName)
{
    // open the file if we can
    FILE *file;
    file = fopen(fileName, "wb");
    if (!file) {
        printf("Could not save %s\n", fileName);
        return false;
    }
   
    // make the header info
    BITMAPFILEHEADER header;
    BITMAPINFOHEADER infoHeader;
   
    header.bfType = 0x4D42;
    header.bfReserved1 = 0;
    header.bfReserved2 = 0;
    header.bfOffBits = 54;
   
    infoHeader.biSize = 40;
    infoHeader.biWidth = (LONG)image.m_width;
    infoHeader.biHeight = (LONG)image.m_height;
    infoHeader.biPlanes = 1;
    infoHeader.biBitCount = 24;
    infoHeader.biCompression = 0;
    infoHeader.biSizeImage = (DWORD) image.m_pixels.size();
    infoHeader.biXPelsPerMeter = 0;
    infoHeader.biYPelsPerMeter = 0;
    infoHeader.biClrUsed = 0;
    infoHeader.biClrImportant = 0;
   
    header.bfSize = infoHeader.biSizeImage + header.bfOffBits;
   
    // write the data and close the file
    fwrite(&header, sizeof(header), 1, file);
    fwrite(&infoHeader, sizeof(infoHeader), 1, file);
    fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
    fclose(file);
  
    return true;
}

//======================================================================================
bool ImageLoad (const char *fileName, SImageData& imageData)
{
    // open the file if we can
    FILE *file;
    file = fopen(fileName, "rb");
    if (!file)
        return false;
 
    // read the headers if we can
    BITMAPFILEHEADER header;
    BITMAPINFOHEADER infoHeader;
    if (fread(&header, sizeof(header), 1, file) != 1 ||
        fread(&infoHeader, sizeof(infoHeader), 1, file) != 1 ||
        header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
    {
        fclose(file);
        return false;
    }
 
    // read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
    imageData.m_pixels.resize(infoHeader.biSizeImage);
    fseek(file, header.bfOffBits, SEEK_SET);
    if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
    {
        fclose(file);
        return false;
    }
 
    imageData.m_width = infoHeader.biWidth;
    imageData.m_height = infoHeader.biHeight;
    imageData.m_pitch = 4 * ((imageData.m_width * 24 + 31) / 32);
 
    fclose(file);
    return true;
}

//======================================================================================
void ImageInit (SImageData& image, size_t width, size_t height)
{
    image.m_width = width;
    image.m_height = height;
    image.m_pitch = 4 * ((width * 24 + 31) / 32);
    image.m_pixels.resize(image.m_pitch * image.m_height);
    std::fill(image.m_pixels.begin(), image.m_pixels.end(), 0);
}

//======================================================================================
template <typename LAMBDA>
void ImageForEachPixel (SImageData& image, const LAMBDA& lambda)
{
    size_t pixelIndex = 0;
    for (size_t y = 0; y < image.m_height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < image.m_width; ++x)
        {
            lambda(*pixel, pixelIndex);
            ++pixel;
            ++pixelIndex;
        }
    }
}

//======================================================================================
template <typename LAMBDA>
void ImageForEachPixel (const SImageData& image, const LAMBDA& lambda)
{
    size_t pixelIndex = 0;
    for (size_t y = 0; y < image.m_height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < image.m_width; ++x)
        {
            lambda(*pixel, pixelIndex);
            ++pixel;
            ++pixelIndex;
        }
    }
}

//======================================================================================
void ImageConvertToLuma (SImageData& image)
{
    ImageForEachPixel(
        image,
        [] (SColor& pixel, size_t pixelIndex)
        {
            float luma = float(pixel.R) * 0.3f + float(pixel.G) * 0.59f + float(pixel.B) * 0.11f;
            uint8 lumau8 = uint8(luma + 0.5f);
            pixel.R = lumau8;
            pixel.G = lumau8;
            pixel.B = lumau8;
        }
    );
}

//======================================================================================
void ImageCombine2 (const SImageData& imageA, const SImageData& imageB, SImageData& result)
{
    // put the images side by side. A on left, B on right
    ImageInit(result, imageA.m_width + imageB.m_width, max(imageA.m_height, imageB.m_height));
    std::fill(result.m_pixels.begin(), result.m_pixels.end(), 0);

    // image A on left
    for (size_t y = 0; y < imageA.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch];
        SColor* srcPixel = (SColor*)&imageA.m_pixels[y * imageA.m_pitch];
        for (size_t x = 0; x < imageA.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image B on right
    for (size_t y = 0; y < imageB.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3];
        SColor* srcPixel = (SColor*)&imageB.m_pixels[y * imageB.m_pitch];
        for (size_t x = 0; x < imageB.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }
}

//======================================================================================
void ImageCombine3 (const SImageData& imageA, const SImageData& imageB, const SImageData& imageC, SImageData& result)
{
    // put the images side by side. A on left, B in middle, C on right
    ImageInit(result, imageA.m_width + imageB.m_width + imageC.m_width, max(max(imageA.m_height, imageB.m_height), imageC.m_height));
    std::fill(result.m_pixels.begin(), result.m_pixels.end(), 0);

    // image A on left
    for (size_t y = 0; y < imageA.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch];
        SColor* srcPixel = (SColor*)&imageA.m_pixels[y * imageA.m_pitch];
        for (size_t x = 0; x < imageA.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image B in middle
    for (size_t y = 0; y < imageB.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3];
        SColor* srcPixel = (SColor*)&imageB.m_pixels[y * imageB.m_pitch];
        for (size_t x = 0; x < imageB.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image C on right
    for (size_t y = 0; y < imageC.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3 + imageC.m_width * 3];
        SColor* srcPixel = (SColor*)&imageC.m_pixels[y * imageC.m_pitch];
        for (size_t x = 0; x < imageC.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }
}

//======================================================================================
float GoldenRatioMultiple (size_t multiple)
{
    return float(multiple) * (1.0f + std::sqrtf(5.0f)) / 2.0f;
}

//======================================================================================
void IntegrationTest (const SImageData& dither, const SImageData& groundTruth, size_t frameIndex, const char* label)
{
    // calculate min, max, total and average error
    size_t minError = 0;
    size_t maxError = 0;
    size_t totalError = 0;
    size_t pixelCount = 0;
    for (size_t y = 0; y < dither.m_height; ++y)
    {
        SColor* ditherPixel = (SColor*)&dither.m_pixels[y * dither.m_pitch];
        SColor* truthPixel = (SColor*)&groundTruth.m_pixels[y * groundTruth.m_pitch];
        for (size_t x = 0; x < dither.m_width; ++x)
        {
            size_t error = 0;
            if (ditherPixel->R > truthPixel->R)
                error = ditherPixel->R - truthPixel->R;
            else
                error = truthPixel->R - ditherPixel->R;

            totalError += error;

            if ((x == 0 && y == 0) || error < minError)
                minError = error;

            if ((x == 0 && y == 0) || error > maxError)
                maxError = error;

            ++ditherPixel;
            ++truthPixel;
            ++pixelCount;
        }
    }
    float averageError = float(totalError) / float(pixelCount);

    // calculate standard deviation
    float sumSquaredDiff = 0.0f;
    for (size_t y = 0; y < dither.m_height; ++y)
    {
        SColor* ditherPixel = (SColor*)&dither.m_pixels[y * dither.m_pitch];
        SColor* truthPixel = (SColor*)&groundTruth.m_pixels[y * groundTruth.m_pitch];
        for (size_t x = 0; x < dither.m_width; ++x)
        {
            size_t error = 0;
            if (ditherPixel->R > truthPixel->R)
                error = ditherPixel->R - truthPixel->R;
            else
                error = truthPixel->R - ditherPixel->R;

            float diff = float(error) - averageError;

            sumSquaredDiff += diff*diff;
        }
    }
    float stdDev = std::sqrtf(sumSquaredDiff / float(pixelCount - 1));

    // report results
    fprintf(g_logFile, "%s %zu error\n", label, frameIndex);
    fprintf(g_logFile, "  min error: %zu\n", minError);
    fprintf(g_logFile, "  max error: %zu\n", maxError);
    fprintf(g_logFile, "  avg error: %0.2f\n", averageError);
    fprintf(g_logFile, "  stddev: %0.2f\n", stdDev);
    fprintf(g_logFile, "\n");
}

//======================================================================================
void HistogramTest (const SImageData& noise, size_t frameIndex, const char* label)
{
    std::array<size_t, 256> counts;
    std::fill(counts.begin(), counts.end(), 0);

    ImageForEachPixel(
        noise,
        [&] (const SColor& pixel, size_t pixelIndex)
        {
            counts[pixel.R]++;
        }
    );

    // calculate min, max, total and average
    size_t minCount = 0;
    size_t maxCount = 0;
    size_t totalCount = 0;
    for (size_t i = 0; i < 256; ++i)
    {
        if (i == 0 || counts[i] < minCount)
            minCount = counts[i];

        if (i == 0 || counts[i] > maxCount)
            maxCount = counts[i];

        totalCount += counts[i];
    }
    float averageCount = float(totalCount) / float(256.0f);

    // calculate standard deviation
    float sumSquaredDiff = 0.0f;
    for (size_t i = 0; i < 256; ++i)
    {
        float diff = float(counts[i]) - averageCount;
        sumSquaredDiff += diff*diff;
    }
    float stdDev = std::sqrtf(sumSquaredDiff / 255.0f);

    // report results
    fprintf(g_logFile, "%s %zu histogram\n", label, frameIndex);
    fprintf(g_logFile, "  min count: %zu\n", minCount);
    fprintf(g_logFile, "  max count: %zu\n", maxCount);
    fprintf(g_logFile, "  avg count: %0.2f\n", averageCount);
    fprintf(g_logFile, "  stddev: %0.2f\n", stdDev);
    fprintf(g_logFile, "  counts: ");
    for (size_t i = 0; i < 256; ++i)
    {
        if (i > 0)
            fprintf(g_logFile, ", ");
        fprintf(g_logFile, "%zu", counts[i]);
    }

    fprintf(g_logFile, "\n\n");
}

//======================================================================================
void GenerateWhiteNoise (SImageData& image, size_t width, size_t height)
{
    ImageInit(image, width, height);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_int_distribution<unsigned int> dist(0, 255);

    ImageForEachPixel(
        image,
        [&] (SColor& pixel, size_t pixelIndex)
        {
            uint8 value = dist(rng);
            pixel.R = value;
            pixel.G = value;
            pixel.B = value;
        }
    );
}

//======================================================================================
void GenerateInterleavedGradientNoise (SImageData& image, size_t width, size_t height, float offsetX, float offsetY)
{
    ImageInit(image, width, height);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_int_distribution<unsigned int> dist(0, 255);

    for (size_t y = 0; y < height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < width; ++x)
        {
            float valueFloat = std::fmodf(52.9829189f * std::fmod(0.06711056f*float(x + offsetX) + 0.00583715f*float(y + offsetY), 1.0f), 1.0f);
            size_t valueBig = size_t(valueFloat * 256.0f);
            uint8 value = uint8(valueBig % 256);
            pixel->R = value;
            pixel->G = value;
            pixel->B = value;
            ++pixel;
        }
    }
}

//======================================================================================
template <size_t NUM_SAMPLES>
void GenerateVanDerCoruptSequence (std::array<float, NUM_SAMPLES>& samples, size_t base)
{
    for (size_t i = 0; i < NUM_SAMPLES; ++i)
    {
        samples[i] = 0.0f;
        float denominator = float(base);
        size_t n = i;
        while (n > 0)
        {
            size_t multiplier = n % base;
            samples[i] += float(multiplier) / denominator;
            n = n / base;
            denominator *= base;
        }
    }
}

//======================================================================================
void DitherWithTexture (const SImageData& ditherImage, const SImageData& noiseImage, SImageData& result)
{
    // init the result image
    ImageInit(result, ditherImage.m_width, ditherImage.m_height);

    // make the result image
    for (size_t y = 0; y < ditherImage.m_height; ++y)
    {
        SColor* srcDitherPixel = (SColor*)&ditherImage.m_pixels[y * ditherImage.m_pitch];
        SColor* destDitherPixel = (SColor*)&result.m_pixels[y * result.m_pitch];

        for (size_t x = 0; x < ditherImage.m_width; ++x)
        {
            // tile the noise in case it isn't the same size as the image we are dithering
            size_t noiseX = x % noiseImage.m_width;
            size_t noiseY = y % noiseImage.m_height;
            SColor* noisePixel = (SColor*)&noiseImage.m_pixels[noiseY * noiseImage.m_pitch + noiseX * 3];

            uint8 value = 0;
            if (noisePixel->R < srcDitherPixel->R)
                value = 255;

            destDitherPixel->R = value;
            destDitherPixel->G = value;
            destDitherPixel->B = value;

            ++srcDitherPixel;
            ++destDitherPixel;
        }
    }
}

//======================================================================================
void DitherWhiteNoise (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noise;
    GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, noise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, noise, dither, combined);
    ImageSave(combined, "out/still_whitenoise.bmp");
}

//======================================================================================
void DitherInterleavedGradientNoise (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noise;
    GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, noise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, noise, dither, combined);
    ImageSave(combined, "out/still_ignoise.bmp");
}

//======================================================================================
void DitherBlueNoise (const SImageData& ditherImage, const SImageData& blueNoise)
{
    printf("\n%s\n", __FUNCTION__);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, blueNoise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, blueNoise, dither, combined);
    ImageSave(combined, "out/still_bluenoise.bmp");
}

//======================================================================================
void DitherWhiteNoiseAnimated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_whitenoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1000.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_ignoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, dist(rng), dist(rng));

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimated (const SImageData& ditherImage, const SImageData blueNoise[8])
{
    printf("\n%s\n", __FUNCTION__);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_bluenoise%zu.bmp", i);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, blueNoise[i], dither);

        // save the results
        SImageData combined;
        ImageCombine2(blueNoise[i], dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_whitenoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1000.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_ignoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, dist(rng), dist(rng));

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&](SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i + 1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedIntegrated (const SImageData& ditherImage, const SImageData blueNoise[8])
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_bluenoise%zu.bmp", i);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, blueNoise[i], dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(blueNoise[i], dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedGoldenRatio (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_whitenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedGoldenRatio (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_ignoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedGoldenRatio (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_bluenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedUniform (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuni_whitenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedUniform (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuni_ignoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedUniform (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuni_bluenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_whitenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_ignoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_bluenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedUniformIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuniint_whitenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedUniformIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuniint_ignoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedUniformIntegrated (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animuniint_bluenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = float(i) / 8.0f;
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedVDCIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // Make Van Der Corput sequence
    std::array<float, 8> VDC;
    GenerateVanDerCoruptSequence(VDC, 2);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animvdcint_whitenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = VDC[i];
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedVDCIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // Make Van Der Corput sequence
    std::array<float, 8> VDC;
    GenerateVanDerCoruptSequence(VDC, 2);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animvdcint_ignoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = VDC[i];
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedVDCIntegrated (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // Make Van Der Corput sequence
    std::array<float, 8> VDC;
    GenerateVanDerCoruptSequence(VDC, 2);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animvdcint_bluenoise%zu.bmp", i);

        // add uniform value to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = VDC[i];
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
int main (int argc, char** argv)
{
    // load the dither image and convert it to greyscale (luma)
    SImageData ditherImage;
    if (!ImageLoad("src/ditherimage.bmp", ditherImage))
    {
        printf("Could not load src/ditherimage.bmp");
        return 0;
    }
    ImageConvertToLuma(ditherImage);

    // load the blue noise images.
    SImageData blueNoise[8];
    for (size_t i = 0; i < 8; ++i)
    {
        char buffer[256];
        sprintf(buffer, "src/BN%zu.bmp", i);
        if (!ImageLoad(buffer, blueNoise[i]))
        {
            printf("Could not load %s", buffer);
            return 0;
        }

        // They have different values in R, G, B so make R be the value for all channels
        ImageForEachPixel(
            blueNoise[i],
            [] (SColor& pixel, size_t pixelIndex)
            {
                pixel.G = pixel.R;
                pixel.B = pixel.R;
            }
        );
    }

    g_logFile = fopen("log.txt", "w+t");
    
    // still image dither tests
    DitherWhiteNoise(ditherImage);
    DitherInterleavedGradientNoise(ditherImage);
    DitherBlueNoise(ditherImage, blueNoise[0]);

    // Animated dither tests
    DitherWhiteNoiseAnimated(ditherImage);
    DitherInterleavedGradientNoiseAnimated(ditherImage);
    DitherBlueNoiseAnimated(ditherImage, blueNoise);

    // Golden ratio animated dither tests
    DitherWhiteNoiseAnimatedGoldenRatio(ditherImage);
    DitherInterleavedGradientNoiseAnimatedGoldenRatio(ditherImage);
    DitherBlueNoiseAnimatedGoldenRatio(ditherImage, blueNoise[0]);

    // Uniform animated dither tests
    DitherWhiteNoiseAnimatedUniform(ditherImage);
    DitherInterleavedGradientNoiseAnimatedUniform(ditherImage);
    DitherBlueNoiseAnimatedUniform(ditherImage, blueNoise[0]);

    // Animated dither integration tests
    DitherWhiteNoiseAnimatedIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedIntegrated(ditherImage);
    DitherBlueNoiseAnimatedIntegrated(ditherImage, blueNoise);

    // Golden ratio animated dither integration tests
    DitherWhiteNoiseAnimatedGoldenRatioIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedGoldenRatioIntegrated(ditherImage);
    DitherBlueNoiseAnimatedGoldenRatioIntegrated(ditherImage, blueNoise[0]);

    // Uniform animated dither integration tests
    DitherWhiteNoiseAnimatedUniformIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedUniformIntegrated(ditherImage);
    DitherBlueNoiseAnimatedUniformIntegrated(ditherImage, blueNoise[0]);

    // Van der corput animated dither integration tests
    DitherWhiteNoiseAnimatedVDCIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedVDCIntegrated(ditherImage);
    DitherBlueNoiseAnimatedVDCIntegrated(ditherImage, blueNoise[0]);

    fclose(g_logFile);

    return 0;
}

Animating Noise For Integration Over Time

You can use noise textures (like the ones from the last post) to do dithering.

For instance, you can do the process below to make a 1 bit (black and white) dithered image using a gray scale source image and a gray scale noise texture. This would be useful if you had a 1 bit display that you were trying to display an image on.

  1. For each pixel in the source image…
  2. If the source image pixel is brighter than the noise texture, put a white pixel.
  3. Else put a black pixel.

(info on converting images to grey scale here: Converting RGB to Grayscale)

The quality of the result depends on the type of noise you use.

If you use pure random numbers (white noise) it looks like this:

You could also use something called “Interleaved Gradient Noise” which would look like this:

Or you could use blue noise which would look like this:

As you can see, white noise was the worst looking, interleaved gradient noise is is the middle, and blue noise looked the best.

White noise is very cheap to generate and can be done in real time on either the CPU or GPU – you just use random numbers.

Blue noise is more expensive to generate and usually must be done in advance, but gives high quality results.

Interleaved gradient noise, which gives middle results, is actually very similar in generation costs as white noise believe it or not, and so can also be done in real time on either the CPU or GPU.

If you have X and Y pixel coordinates (not uv coordinates), you can generate the noise value for the pixel by using this formula:

float noise = std::fmodf(52.9829189f * std::fmodf(0.06711056f*float(x) + 0.00583715f*float(y), 1.0f), 1.0f);

Interleaved gradient noise was made by Jorge Jimenez (@iryoku1) and you can read more about it at these links:
Next Generation Post Processing in Call Of Duty: Advanced Warfare
Dithering part three โ€“ real world 2D quantization dithering (Bart Wronksi)

Dithering still images is fun, but in the context of video games, we are more interested in animated images, so let’s look at things in motion.

Animated Noise

Let’s start by just animating those three noise types over 8 frames.

For white noise, we’ll generate a new white noise texture every frame.

For interleaved gradient noise, we’ll add a random offset (0 to 1000) to the pixel each frame, so we get 8 different interleaved gradient noise textures.

For blue noise, we’ll just have 8 different blue noise textures that we generate in advance.

These are playing at 8 fps, so loop every second.

White Noise:

IG Noise:

Blue Noise:

Once again we can see that white noise is worst, blue noise is best, and interleaved gradient noise is in the middle.

When you think about it though, despite these animations all using different types of noise over SPACE, they all use white noise over time. What i mean by that is if you isolate any individual pixel in any of the images and look at it over the 8 frames, that single pixel will look like white noise.

Let’s see if we can improve that.

Golden Ratio Animated Noise

In a conversation on twitter, @R4_Unit told me that in the past he had good success by adding the golden ratio to blue noise textures to make the noise more blue over time.

The background here is that repeatedly adding the golden ratio to any number will make a low discrepancy sequence (details: When Random Numbers Are Too Random: Low Discrepancy Sequences)

The golden ratio is \frac{1+\sqrt{5}}{2} or approximately 1.61803398875, and interestingly is THE MOST irrational number that there is. Weird right?

For each of the noise types, we’ll generate a single texture for frame 0, and each subsequent frame we will add the golden ratio to each pixel. The pixel values are in the 0 to 1 space when adding the golden ratio (not 0 to 255) and we use modulus to wrap it around.

The DFT magnitude is shown on the left to show how adding the golden ratio affects frequency components.

White Noise:

IG Noise:

Blue Noise:

When I look at these side by side with the previous animations, it’s hard for me to see much of a difference. That is interesting for the case of blue noise, where it’s difficult to generate multiple blue noise textures. It means that you can get a fairly decent “blue noise” texture by adding multiples of the golden ratio to an existing blue noise texture (aka recycling!).

It’s interesting that the white noise and interleaved gradient noise don’t change their frequency spectrum much over time. On the other hand, it’s a bit sad to see that the blue noise texture gains some low frequency content so the blue noise becomes lower quality. You aren’t just getting more blue noise textures for free by adding the golden ratio, even though they are blue-ish.

Another important aspect to look at is the histogram of colors used of these images when adding golden ratio. The ideal situation is that the starting images have roughly the same number of every color in the image, and that when adding the golden ratio for each frame, that we still use roughly the same number of every color. That turns out to be the case luckily.

The white noise histogram has a minimum count of 213, a maximum count of 303, an average count of 256 (the image is 256×256), and a standard deviation of 15.64. Those values are the same for each frame of the animation.

For interleaved gradient noise, it has a minimum count of 245, a maximum count of 266, an average count of 256 and a standard deviation of 2.87. Those values are the same for the entire animation.

Lastly, for blue noise, it has a minimum, maximum, and average count of 256, and a standard deviation of 0. This also remains true for the entire animation.

Integration Over Time

A big reason we might want animated noise in graphics is because we are taking multiple samples and want to numerically integrate them.

Lets analyze how these two types of animations (regular and golden ratio) compare for integration.

These animations are the same as before, but on frame 1, we show the average of frame 0 and 1. On frame 2 we show the average of frame 0, 1 and 2. And so on to frame 7 which is the average of all 8 frames. This is an integration of our black and white sample points we are taking, where the correct value of the integration is the greyscale image we started with.

Here is white noise, IG noise and blue noise animated (new noise each frame), integrated over those 8 frames, playing at 8 frames a second:



Here is the same using the golden ratio to animate the noise instead:



Since it can be a little difficult to compare these things while they are in motion, here is the final frames of each method and some graphs to show the average error and standard deviation of the error, compared to the ground truth source image.

White Noise vs White Noise Golden Ratio:


IG Noise vs IG Noise Golden Ratio:


Blue Noise vs Blue Noise Golden Ratio:


Interestingly, the golden ratio average error and standard deviation (from the ground truth) are pretty even for all types of noise by frame 7, even though the blue noise is perceptually superior. This also happens for the non golden ratio integrations of blue noise and white noise. That’s part of the value of blue noise, that even if it has the same amount of error as say, white noise, it still looks better.

Another interesting observation is that interleaved gradient noise performs better at integration (at least numerically) than white or blue noise, when not using the golden ratio. The only way I can explain this is that when picking random pixel offsets to generate each frame of interleaved gradient noise, it’s somehow more blue over time than the other two methods. It’s a strange but pretty useful property.

Despite IG having success when looking at the numbers, it has very visible directional patterns which are not so nice. The fact that it is as cheap as white noise to generate, but has results much closer to blue noise perceptually is pretty awesome though.

Something else important to note is that white noise beats blue noise in the long run (higher sample counts). It’s only at these lower sample counts that blue noise is the clear winner.

Lastly, it seems like the ideal setup for integrating some values over time with a lower sample count would be to have N blue noise textures to use over N frames, but *somehow* have a constraint on those textures generated such that each individual pixel over time has blue noise distributed values.

I’m not sure how to generate that, or if it’s even possible to do so, but doing that seems like it would be pretty near the ideal for doing integration like the above.

Taking a guess at how the DFT’s would look, each individual slice seems like it should look like a totally normal blue noise texture where it’s black in the middle (low frequencies) and noisy elsewhere (high frequencies). If you had N slices of these it would look like a black cylinder surrounded by noise when taking the 3D DFT. I’m not sure though how having the constraint on individual pixels would modify the DFT, or if it even would.

This “ideal” I’m describing is different than vanilla 3d blue noise. The 3d DFT of 3d blue noise is a black sphere surrounded by noise. What I’m describing is a cylinder instead of a sphere.

3d blue noise turns out not to be great for these needs. You can read about that here:

The problem with 3D blue noise

That author also has some an interesting post on blue noise, and a zip file full of blue noise textures that you can take and use.

Free Blue Noise Textures

I have some thoughts on generating this blue noise cylinder that if they work out may very well be the next blog post.

Code

Here is the code used to generate the images in this post. It’s also on github, which also contains the source images used.

Atrix256: RandomCode/AnimatedNoise

#define _CRT_SECURE_NO_WARNINGS

#include <windows.h>  // for bitmap headers.  Sorry non windows people!
#include <stdint.h>
#include <vector>
#include <random>
#include <atomic>
#include <thread>
#include <complex>
#include <array>

typedef uint8_t uint8;

const float c_pi = 3.14159265359f;

// settings
const bool c_doDFT = true;

// globals 
FILE* g_logFile = nullptr;

//======================================================================================
inline float Lerp (float A, float B, float t)
{
    return A * (1.0f - t) + B * t;
}

//======================================================================================
struct SImageData
{
    SImageData ()
        : m_width(0)
        , m_height(0)
    { }
   
    size_t m_width;
    size_t m_height;
    size_t m_pitch;
    std::vector<uint8> m_pixels;
};
 
//======================================================================================
struct SColor
{
    SColor (uint8 _R = 0, uint8 _G = 0, uint8 _B = 0)
        : R(_R), G(_G), B(_B)
    { }

    inline void Set (uint8 _R, uint8 _G, uint8 _B)
    {
        R = _R;
        G = _G;
        B = _B;
    }
 
    uint8 B, G, R;
};

//======================================================================================
struct SImageDataComplex
{
    SImageDataComplex ()
        : m_width(0)
        , m_height(0)
    { }
  
    size_t m_width;
    size_t m_height;
    std::vector<std::complex<float>> m_pixels;
};
 
//======================================================================================
std::complex<float> DFTPixel (const SImageData &srcImage, size_t K, size_t L)
{
    std::complex<float> ret(0.0f, 0.0f);
  
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            // Get the pixel value (assuming greyscale) and convert it to [0,1] space
            const uint8 *src = &srcImage.m_pixels[(y * srcImage.m_pitch) + x * 3];
            float grey = float(src[0]) / 255.0f;
  
            // Add to the sum of the return value
            float v = float(K * x) / float(srcImage.m_width);
            v += float(L * y) / float(srcImage.m_height);
            ret += std::complex<float>(grey, 0.0f) * std::polar<float>(1.0f, -2.0f * c_pi * v);
        }
    }
  
    return ret;
}
  
//======================================================================================
void ImageDFT (const SImageData &srcImage, SImageDataComplex &destImage)
{
    // NOTE: this function assumes srcImage is greyscale, so works on only the red component of srcImage.
    // ImageToGrey() will convert an image to greyscale.
 
    // size the output dft data
    destImage.m_width = srcImage.m_width;
    destImage.m_height = srcImage.m_height;
    destImage.m_pixels.resize(destImage.m_width*destImage.m_height);
 
    size_t numThreads = std::thread::hardware_concurrency();
    //if (numThreads > 0)
        //numThreads = numThreads - 1;
 
    std::vector<std::thread> threads;
    threads.resize(numThreads);
 
    printf("Doing DFT with %zu threads...\n", numThreads);
 
    // calculate 2d dft (brute force, not using fast fourier transform) multithreadedly
    std::atomic<size_t> nextRow(0);
    for (std::thread& t : threads)
    {
        t = std::thread(
            [&] ()
            {
                size_t row = nextRow.fetch_add(1);
                bool reportProgress = (row == 0);
                int lastPercent = -1;
 
                while (row < srcImage.m_height)
                {
                    // calculate the DFT for every pixel / frequency in this row
                    for (size_t x = 0; x < srcImage.m_width; ++x)
                    {
                        destImage.m_pixels[row * destImage.m_width + x] = DFTPixel(srcImage, x, row);
                    }
 
                    // report progress if we should
                    if (reportProgress)
                    {
                        int percent = int(100.0f * float(row) / float(srcImage.m_height));
                        if (lastPercent != percent)
                        {
                            lastPercent = percent;
                            printf("            \rDFT: %i%%", lastPercent);
                        }
                    }
 
                    // go to the next row
                    row = nextRow.fetch_add(1);
                }
            }
        );
    }
 
    for (std::thread& t : threads)
        t.join();
 
    printf("\n");
}
 
//======================================================================================
void GetMagnitudeData (const SImageDataComplex& srcImage, SImageData& destImage)
{
    // size the output image
    destImage.m_width = srcImage.m_width;
    destImage.m_height = srcImage.m_height;
    destImage.m_pitch = 4 * ((srcImage.m_width * 24 + 31) / 32);
    destImage.m_pixels.resize(destImage.m_pitch*destImage.m_height);
  
    // get floating point magnitude data
    std::vector<float> magArray;
    magArray.resize(srcImage.m_width*srcImage.m_height);
    float maxmag = 0.0f;
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            // Offset the information by half width & height in the positive direction.
            // This makes frequency 0 (DC) be at the image origin, like most diagrams show it.
            int k = (x + (int)srcImage.m_width / 2) % (int)srcImage.m_width;
            int l = (y + (int)srcImage.m_height / 2) % (int)srcImage.m_height;
            const std::complex<float> &src = srcImage.m_pixels[l*srcImage.m_width + k];
  
            float mag = std::abs(src);
            if (mag > maxmag)
                maxmag = mag;
  
            magArray[y*srcImage.m_width + x] = mag;
        }
    }
    if (maxmag == 0.0f)
        maxmag = 1.0f;
  
    const float c = 255.0f / log(1.0f+maxmag);
  
    // normalize the magnitude data and send it back in [0, 255]
    for (size_t x = 0; x < srcImage.m_width; ++x)
    {
        for (size_t y = 0; y < srcImage.m_height; ++y)
        {
            float src = c * log(1.0f + magArray[y*srcImage.m_width + x]);
  
            uint8 magu8 = uint8(src);
  
            uint8* dest = &destImage.m_pixels[y*destImage.m_pitch + x * 3];
            dest[0] = magu8;
            dest[1] = magu8;
            dest[2] = magu8;
        }
    }
}

//======================================================================================
bool ImageSave (const SImageData &image, const char *fileName)
{
    // open the file if we can
    FILE *file;
    file = fopen(fileName, "wb");
    if (!file) {
        printf("Could not save %s\n", fileName);
        return false;
    }
   
    // make the header info
    BITMAPFILEHEADER header;
    BITMAPINFOHEADER infoHeader;
   
    header.bfType = 0x4D42;
    header.bfReserved1 = 0;
    header.bfReserved2 = 0;
    header.bfOffBits = 54;
   
    infoHeader.biSize = 40;
    infoHeader.biWidth = (LONG)image.m_width;
    infoHeader.biHeight = (LONG)image.m_height;
    infoHeader.biPlanes = 1;
    infoHeader.biBitCount = 24;
    infoHeader.biCompression = 0;
    infoHeader.biSizeImage = (DWORD) image.m_pixels.size();
    infoHeader.biXPelsPerMeter = 0;
    infoHeader.biYPelsPerMeter = 0;
    infoHeader.biClrUsed = 0;
    infoHeader.biClrImportant = 0;
   
    header.bfSize = infoHeader.biSizeImage + header.bfOffBits;
   
    // write the data and close the file
    fwrite(&header, sizeof(header), 1, file);
    fwrite(&infoHeader, sizeof(infoHeader), 1, file);
    fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
    fclose(file);
  
    return true;
}

//======================================================================================
bool ImageLoad (const char *fileName, SImageData& imageData)
{
    // open the file if we can
    FILE *file;
    file = fopen(fileName, "rb");
    if (!file)
        return false;
 
    // read the headers if we can
    BITMAPFILEHEADER header;
    BITMAPINFOHEADER infoHeader;
    if (fread(&header, sizeof(header), 1, file) != 1 ||
        fread(&infoHeader, sizeof(infoHeader), 1, file) != 1 ||
        header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
    {
        fclose(file);
        return false;
    }
 
    // read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
    imageData.m_pixels.resize(infoHeader.biSizeImage);
    fseek(file, header.bfOffBits, SEEK_SET);
    if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
    {
        fclose(file);
        return false;
    }
 
    imageData.m_width = infoHeader.biWidth;
    imageData.m_height = infoHeader.biHeight;
    imageData.m_pitch = 4 * ((imageData.m_width * 24 + 31) / 32);
 
    fclose(file);
    return true;
}

//======================================================================================
void ImageInit (SImageData& image, size_t width, size_t height)
{
    image.m_width = width;
    image.m_height = height;
    image.m_pitch = 4 * ((width * 24 + 31) / 32);
    image.m_pixels.resize(image.m_pitch * image.m_height);
    std::fill(image.m_pixels.begin(), image.m_pixels.end(), 0);
}

//======================================================================================
template <typename LAMBDA>
void ImageForEachPixel (SImageData& image, const LAMBDA& lambda)
{
    size_t pixelIndex = 0;
    for (size_t y = 0; y < image.m_height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < image.m_width; ++x)
        {
            lambda(*pixel, pixelIndex);
            ++pixel;
            ++pixelIndex;
        }
    }
}

//======================================================================================
template <typename LAMBDA>
void ImageForEachPixel (const SImageData& image, const LAMBDA& lambda)
{
    size_t pixelIndex = 0;
    for (size_t y = 0; y < image.m_height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < image.m_width; ++x)
        {
            lambda(*pixel, pixelIndex);
            ++pixel;
            ++pixelIndex;
        }
    }
}

//======================================================================================
void ImageConvertToLuma (SImageData& image)
{
    ImageForEachPixel(
        image,
        [] (SColor& pixel, size_t pixelIndex)
        {
            float luma = float(pixel.R) * 0.3f + float(pixel.G) * 0.59f + float(pixel.B) * 0.11f;
            uint8 lumau8 = uint8(luma + 0.5f);
            pixel.R = lumau8;
            pixel.G = lumau8;
            pixel.B = lumau8;
        }
    );
}

//======================================================================================
void ImageCombine2 (const SImageData& imageA, const SImageData& imageB, SImageData& result)
{
    // put the images side by side. A on left, B on right
    ImageInit(result, imageA.m_width + imageB.m_width, max(imageA.m_height, imageB.m_height));
    std::fill(result.m_pixels.begin(), result.m_pixels.end(), 0);

    // image A on left
    for (size_t y = 0; y < imageA.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch];
        SColor* srcPixel = (SColor*)&imageA.m_pixels[y * imageA.m_pitch];
        for (size_t x = 0; x < imageA.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image B on right
    for (size_t y = 0; y < imageB.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3];
        SColor* srcPixel = (SColor*)&imageB.m_pixels[y * imageB.m_pitch];
        for (size_t x = 0; x < imageB.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }
}

//======================================================================================
void ImageCombine3 (const SImageData& imageA, const SImageData& imageB, const SImageData& imageC, SImageData& result)
{
    // put the images side by side. A on left, B in middle, C on right
    ImageInit(result, imageA.m_width + imageB.m_width + imageC.m_width, max(max(imageA.m_height, imageB.m_height), imageC.m_height));
    std::fill(result.m_pixels.begin(), result.m_pixels.end(), 0);

    // image A on left
    for (size_t y = 0; y < imageA.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch];
        SColor* srcPixel = (SColor*)&imageA.m_pixels[y * imageA.m_pitch];
        for (size_t x = 0; x < imageA.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image B in middle
    for (size_t y = 0; y < imageB.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3];
        SColor* srcPixel = (SColor*)&imageB.m_pixels[y * imageB.m_pitch];
        for (size_t x = 0; x < imageB.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }

    // image C on right
    for (size_t y = 0; y < imageC.m_height; ++y)
    {
        SColor* destPixel = (SColor*)&result.m_pixels[y * result.m_pitch + imageA.m_width * 3 + imageC.m_width * 3];
        SColor* srcPixel = (SColor*)&imageC.m_pixels[y * imageC.m_pitch];
        for (size_t x = 0; x < imageC.m_width; ++x)
        {
            destPixel[0] = srcPixel[0];
            ++destPixel;
            ++srcPixel;
        }
    }
}

//======================================================================================
float GoldenRatioMultiple (size_t multiple)
{
    return float(multiple) * (1.0f + std::sqrtf(5.0f)) / 2.0f;
}

//======================================================================================
void IntegrationTest (const SImageData& dither, const SImageData& groundTruth, size_t frameIndex, const char* label)
{
    // calculate min, max, total and average error
    size_t minError = 0;
    size_t maxError = 0;
    size_t totalError = 0;
    size_t pixelCount = 0;
    for (size_t y = 0; y < dither.m_height; ++y)
    {
        SColor* ditherPixel = (SColor*)&dither.m_pixels[y * dither.m_pitch];
        SColor* truthPixel = (SColor*)&groundTruth.m_pixels[y * groundTruth.m_pitch];
        for (size_t x = 0; x < dither.m_width; ++x)
        {
            size_t error = 0;
            if (ditherPixel->R > truthPixel->R)
                error = ditherPixel->R - truthPixel->R;
            else
                error = truthPixel->R - ditherPixel->R;

            totalError += error;

            if ((x == 0 && y == 0) || error < minError)
                minError = error;

            if ((x == 0 && y == 0) || error > maxError)
                maxError = error;

            ++ditherPixel;
            ++truthPixel;
            ++pixelCount;
        }
    }
    float averageError = float(totalError) / float(pixelCount);

    // calculate standard deviation
    float sumSquaredDiff = 0.0f;
    for (size_t y = 0; y < dither.m_height; ++y)
    {
        SColor* ditherPixel = (SColor*)&dither.m_pixels[y * dither.m_pitch];
        SColor* truthPixel = (SColor*)&groundTruth.m_pixels[y * groundTruth.m_pitch];
        for (size_t x = 0; x < dither.m_width; ++x)
        {
            size_t error = 0;
            if (ditherPixel->R > truthPixel->R)
                error = ditherPixel->R - truthPixel->R;
            else
                error = truthPixel->R - ditherPixel->R;

            float diff = float(error) - averageError;

            sumSquaredDiff += diff*diff;
        }
    }
    float stdDev = std::sqrtf(sumSquaredDiff / float(pixelCount - 1));

    // report results
    fprintf(g_logFile, "%s %zu error\n", label, frameIndex);
    fprintf(g_logFile, "  min error: %zu\n", minError);
    fprintf(g_logFile, "  max error: %zu\n", maxError);
    fprintf(g_logFile, "  avg error: %0.2f\n", averageError);
    fprintf(g_logFile, "  stddev: %0.2f\n", stdDev);
    fprintf(g_logFile, "\n");
}

//======================================================================================
void HistogramTest (const SImageData& noise, size_t frameIndex, const char* label)
{
    std::array<size_t, 256> counts;
    std::fill(counts.begin(), counts.end(), 0);

    ImageForEachPixel(
        noise,
        [&] (const SColor& pixel, size_t pixelIndex)
        {
            counts[pixel.R]++;
        }
    );

    // calculate min, max, total and average
    size_t minCount = 0;
    size_t maxCount = 0;
    size_t totalCount = 0;
    for (size_t i = 0; i < 256; ++i)
    {
        if (i == 0 || counts[i] < minCount)
            minCount = counts[i];

        if (i == 0 || counts[i] > maxCount)
            maxCount = counts[i];

        totalCount += counts[i];
    }
    float averageCount = float(totalCount) / float(256.0f);

    // calculate standard deviation
    float sumSquaredDiff = 0.0f;
    for (size_t i = 0; i < 256; ++i)
    {
        float diff = float(counts[i]) - averageCount;
        sumSquaredDiff += diff*diff;
    }
    float stdDev = std::sqrtf(sumSquaredDiff / 255.0f);

    // report results
    fprintf(g_logFile, "%s %zu histogram\n", label, frameIndex);
    fprintf(g_logFile, "  min count: %zu\n", minCount);
    fprintf(g_logFile, "  max count: %zu\n", maxCount);
    fprintf(g_logFile, "  avg count: %0.2f\n", averageCount);
    fprintf(g_logFile, "  stddev: %0.2f\n", stdDev);
    fprintf(g_logFile, "  counts: ");
    for (size_t i = 0; i < 256; ++i)
    {
        if (i > 0)
            fprintf(g_logFile, ", ");
        fprintf(g_logFile, "%zu", counts[i]);
    }

    fprintf(g_logFile, "\n\n");
}

//======================================================================================
void GenerateWhiteNoise (SImageData& image, size_t width, size_t height)
{
    ImageInit(image, width, height);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_int_distribution<unsigned int> dist(0, 255);

    ImageForEachPixel(
        image,
        [&] (SColor& pixel, size_t pixelIndex)
        {
            uint8 value = dist(rng);
            pixel.R = value;
            pixel.G = value;
            pixel.B = value;
        }
    );
}

//======================================================================================
void GenerateInterleavedGradientNoise (SImageData& image, size_t width, size_t height, float offsetX, float offsetY)
{
    ImageInit(image, width, height);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_int_distribution<unsigned int> dist(0, 255);

    for (size_t y = 0; y < height; ++y)
    {
        SColor* pixel = (SColor*)&image.m_pixels[y * image.m_pitch];
        for (size_t x = 0; x < width; ++x)
        {
            float valueFloat = std::fmodf(52.9829189f * std::fmod(0.06711056f*float(x + offsetX) + 0.00583715f*float(y + offsetY), 1.0f), 1.0f);
            size_t valueBig = size_t(valueFloat * 256.0f);
            uint8 value = uint8(valueBig % 256);
            pixel->R = value;
            pixel->G = value;
            pixel->B = value;
            ++pixel;
        }
    }
}

//======================================================================================
void DitherWithTexture (const SImageData& ditherImage, const SImageData& noiseImage, SImageData& result)
{
    // init the result image
    ImageInit(result, ditherImage.m_width, ditherImage.m_height);

    // make the result image
    for (size_t y = 0; y < ditherImage.m_height; ++y)
    {
        SColor* srcDitherPixel = (SColor*)&ditherImage.m_pixels[y * ditherImage.m_pitch];
        SColor* destDitherPixel = (SColor*)&result.m_pixels[y * result.m_pitch];

        for (size_t x = 0; x < ditherImage.m_width; ++x)
        {
            // tile the noise in case it isn't the same size as the image we are dithering
            size_t noiseX = x % noiseImage.m_width;
            size_t noiseY = y % noiseImage.m_height;
            SColor* noisePixel = (SColor*)&noiseImage.m_pixels[noiseY * noiseImage.m_pitch + noiseX * 3];

            uint8 value = 0;
            if (noisePixel->R < srcDitherPixel->R)
                value = 255;

            destDitherPixel->R = value;
            destDitherPixel->G = value;
            destDitherPixel->B = value;

            ++srcDitherPixel;
            ++destDitherPixel;
        }
    }
}

//======================================================================================
void DitherWhiteNoise (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noise;
    GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, noise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, noise, dither, combined);
    ImageSave(combined, "out/still_whitenoise.bmp");
}

//======================================================================================
void DitherInterleavedGradientNoise (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noise;
    GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, noise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, noise, dither, combined);
    ImageSave(combined, "out/still_ignoise.bmp");
}

//======================================================================================
void DitherBlueNoise (const SImageData& ditherImage, const SImageData& blueNoise)
{
    printf("\n%s\n", __FUNCTION__);

    // dither the image
    SImageData dither;
    DitherWithTexture(ditherImage, blueNoise, dither);

    // save the results
    SImageData combined;
    ImageCombine3(ditherImage, blueNoise, dither, combined);
    ImageSave(combined, "out/still_bluenoise.bmp");
}

//======================================================================================
void DitherWhiteNoiseAnimated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_whitenoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1000.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_ignoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, dist(rng), dist(rng));

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimated (const SImageData& ditherImage, const SImageData blueNoise[8])
{
    printf("\n%s\n", __FUNCTION__);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/anim_bluenoise%zu.bmp", i);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, blueNoise[i], dither);

        // save the results
        SImageData combined;
        ImageCombine2(blueNoise[i], dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_whitenoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateWhiteNoise(noise, ditherImage.m_width, ditherImage.m_height);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1000.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_ignoise%zu.bmp", i);

        // make noise
        SImageData noise;
        GenerateInterleavedGradientNoise(noise, ditherImage.m_width, ditherImage.m_height, dist(rng), dist(rng));

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&](SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i + 1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedIntegrated (const SImageData& ditherImage, const SImageData blueNoise[8])
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animint_bluenoise%zu.bmp", i);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, blueNoise[i], dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(blueNoise[i], dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedGoldenRatio (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_whitenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedGoldenRatio (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_ignoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedGoldenRatio (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    SImageDataComplex noiseDFT;
    SImageData noiseDFTMag;

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgr_bluenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // DFT the noise
        if (c_doDFT)
        {
            ImageDFT(noise, noiseDFT);
            GetMagnitudeData(noiseDFT, noiseDFTMag);
        }
        else
        {
            ImageInit(noiseDFTMag, noise.m_width, noise.m_height);
            std::fill(noiseDFTMag.m_pixels.begin(), noiseDFTMag.m_pixels.end(), 0);
        }

        // Histogram test the noise
        HistogramTest(noise, i, __FUNCTION__);

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // save the results
        SImageData combined;
        ImageCombine3(noiseDFTMag, noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherWhiteNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateWhiteNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_whitenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherInterleavedGradientNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    // make noise
    SImageData noiseSrc;
    GenerateInterleavedGradientNoise(noiseSrc, ditherImage.m_width, ditherImage.m_height, 0.0f, 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_ignoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
void DitherBlueNoiseAnimatedGoldenRatioIntegrated (const SImageData& ditherImage, const SImageData& noiseSrc)
{
    printf("\n%s\n", __FUNCTION__);

    std::vector<float> integration;
    integration.resize(ditherImage.m_width * ditherImage.m_height);
    std::fill(integration.begin(), integration.end(), 0.0f);

    SImageData noise;
    ImageInit(noise, noiseSrc.m_width, noiseSrc.m_height);

    // animate 8 frames
    for (size_t i = 0; i < 8; ++i)
    {
        char fileName[256];
        sprintf(fileName, "out/animgrint_bluenoise%zu.bmp", i);

        // add golden ratio to the noise after each frame
        noise.m_pixels = noiseSrc.m_pixels;
        float add = GoldenRatioMultiple(i);
        ImageForEachPixel(
            noise,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float valueFloat = (float(pixel.R) / 255.0f) + add;
                size_t valueBig = size_t(valueFloat * 255.0f);
                uint8 value = uint8(valueBig % 256);
                pixel.R = value;
                pixel.G = value;
                pixel.B = value;
            }
        );

        // dither the image
        SImageData dither;
        DitherWithTexture(ditherImage, noise, dither);

        // integrate and put the current integration results into the dither image
        ImageForEachPixel(
            dither,
            [&] (SColor& pixel, size_t pixelIndex)
            {
                float pixelValueFloat = float(pixel.R) / 255.0f;
                integration[pixelIndex] = Lerp(integration[pixelIndex], pixelValueFloat, 1.0f / float(i+1));

                uint8 integratedPixelValue = uint8(integration[pixelIndex] * 255.0f);
                pixel.R = integratedPixelValue;
                pixel.G = integratedPixelValue;
                pixel.B = integratedPixelValue;
            }
        );

        // do an integration test
        IntegrationTest(dither, ditherImage, i, __FUNCTION__);

        // save the results
        SImageData combined;
        ImageCombine2(noise, dither, combined);
        ImageSave(combined, fileName);
    }
}

//======================================================================================
int main (int argc, char** argv)
{
    // load the dither image and convert it to greyscale (luma)
    SImageData ditherImage;
    if (!ImageLoad("src/ditherimage.bmp", ditherImage))
    {
        printf("Could not load src/ditherimage.bmp");
        return 0;
    }
    ImageConvertToLuma(ditherImage);

    // load the blue noise images.
    SImageData blueNoise[8];
    for (size_t i = 0; i < 8; ++i)
    {
        char buffer[256];
        sprintf(buffer, "src/BN%zu.bmp", i);
        if (!ImageLoad(buffer, blueNoise[i]))
        {
            printf("Could not load %s", buffer);
            return 0;
        }

        // They have different values in R, G, B so make R be the value for all channels
        ImageForEachPixel(
            blueNoise[i],
            [] (SColor& pixel, size_t pixelIndex)
            {
                pixel.G = pixel.R;
                pixel.B = pixel.R;
            }
        );
    }

    g_logFile = fopen("log.txt", "w+t");
    
    // still image dither tests
    DitherWhiteNoise(ditherImage);
    DitherInterleavedGradientNoise(ditherImage);
    DitherBlueNoise(ditherImage, blueNoise[0]);

    // Animated dither tests
    DitherWhiteNoiseAnimated(ditherImage);
    DitherInterleavedGradientNoiseAnimated(ditherImage);
    DitherBlueNoiseAnimated(ditherImage, blueNoise);

    // Golden ratio animated dither tests
    DitherWhiteNoiseAnimatedGoldenRatio(ditherImage);
    DitherInterleavedGradientNoiseAnimatedGoldenRatio(ditherImage);
    DitherBlueNoiseAnimatedGoldenRatio(ditherImage, blueNoise[0]);

    // Animated dither integration tests
    DitherWhiteNoiseAnimatedIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedIntegrated(ditherImage);
    DitherBlueNoiseAnimatedIntegrated(ditherImage, blueNoise);

    // Golden ratio animated dither integration tests
    DitherWhiteNoiseAnimatedGoldenRatioIntegrated(ditherImage);
    DitherInterleavedGradientNoiseAnimatedGoldenRatioIntegrated(ditherImage);
    DitherBlueNoiseAnimatedGoldenRatioIntegrated(ditherImage, blueNoise[0]);

    fclose(g_logFile);

    return 0;
}

Calculating the Distance Between Points in “Wrap Around” (Toroidal) Space

Let’s say you are trying to find the distance between two points in 2D, but that these points are in a universe that “wraps around” like old video games – leaving the screen on the right, left, top or bottom side makes you re-appear on the opposite edge.

This universe is actually shaped like a toroid, also known as a doughnut. It’s actually an impossible object, a “flat torus”, so not exactly a doughnut, but whatever.

If you imagine yourself on the surface of a doughnut, it would behave exactly this way. If you go “down” you end up where you previously considered “up”. If you go far enough “left” you end up where you previously considered “right”.

How would you calculate the distance between two points in a universe like this?

Let’s imagine the situation below where we are trying to find the distance between the red point and the green point:

One way to do this would be to pick one of the points (I’m picking red in this case) and clone it 8 times to surround the cell like the below. You’d calculate the distance from the green point to each of the 9 red points, and whatever distance was smallest would be the answer.

Something not so desirable about this is that it takes 9 distance calculations to find the minimum distance. You can work with squared distances instead of regular distances to avoid a square root on each of these distance calculations, but that’s still a bit of calculation to do.

Going up in dimensions makes the problem even worse. In 3D, it requires 27 distance calculations to find the shortest point, and 81 distance calculations in 4D!

Luckily there’s a better way to approach this.

Let’s say that our universe (image) is 1 unit by 1 unit big (aka we are working in texture UVs). If you look at the image with 9 copies of the red dot, you can see that they are just the 9 possible combinations of having -1, +0, +1 on each axis added to the red dot’s coordinates. All possible combinations of the x and y axis having -1, +0 or +1 added to them are valid locations of the red dot.

Looking at the distance formula we can see that if we minimize each axis individually, that we will also end up with the minimal distance overall.

d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

So, the better way is to minimize each axis individually.

On the x axis you’d find if the x axis distance between the red and green point is minimal when you subtract 1 from the red dot’s x axis position, leave it alone, or add 1.

Whichever x axis value of the red dot gives you the minimal x axis 1D distance is the x axis location to use.

You’d repeat for the y axis to get the y axis location to use (and would repeat for any further axes for higher dimensions).

This gives you the closest point which you can then plug into the distance formula to get the distance between the points in this wrap around space.

You can actually do better though.

Still working on each axis individually, you can calculate the absoluate value of the 1D distance between the two points on that axis. If that distance is greater than 0.5, the real distance for that axis is 1-distance.

The intuition here is that if you are in a 1d repeating space, if going from A to B is more than half the distance, it means that you went the wrong way, and that going the other way is shorter. The distance of that other way is one minus whatever distance you just calculated since the distance from one point to itself is 1!

Do that for each axis and use those 1d distances in the distance formula to get the actual distance.

This lets you minimize the distance without having to explicitly figure out which combination makes the point closest.

More importantly, it lets you efficiently calculate the distance between the two points in toroidal space (doughnut space!)

The computational complexity is a lot better. It’s now linear in the number of dimensions: O(N), instead of O(3^N).

Here is some C++ to show you how it would work in 2D.

float ToroidalDistance (float x1, float y1, float x2, float y2)
{
    float dx = std::abs(x2 - x1);
    float dy = std::abs(y2 - y1);

    if (dx > 0.5f)
        dx = 1.0f - dx;

    if (dy > 0.5f)
        dy = 1.0f - dy;

    return std::sqrt(dx*dx + dy*dy);
}

I hit this problem trying to make a tileable texture. I needed to place a few circles on a texture such that the circles weren’t too close to each other, even when the texture was tiled.

The calculations above gave me the basic tool needed to be able to calculate distances between points. Subtracting circle radii from the distance between points let me get toroidal distance between circles and make sure I didn’t place them too closely to each other.

That let me make an image that kept the distance constraints even when it was tiled.

Here’s an example image by itself:

Here is the image tiled:

Generating Random Numbers From a Specific Distribution With Rejection Sampling

The last post showed how to transform uniformly generated random numbers into any random number distribution you desired.

It did so by turning the PDF (probability density function) into a CDF (cumulative density function) and then inverting it – either analytically (making a function) or numerically (making a look up table).

This post will show you how to generate numbers from a PDF as well, but will do so using rejection sampling.

Dice

Let’s say you wanted to simulate a fair five sided die but that you only had a six sided die.

You can use rejection sampling for this by rolling a six sided die and ignoring the roll any time a six came up. Doing that, you do in fact get a fair five sided die roll!

This shows doing that to get 10,000 five sided die rolls:

One disadvantage to this method is that you are throwing away die rolls which can be a source of inefficiency. In this setup it takes 1.2 six sided die rolls on average to get a valid five sided die roll since a roll will be thrown away 1/6 of the time.

Another disadvantage is that each time you need a new value, there are an unknown number of die rolls needed to get it. On average it’s true that you only need 1.2 die rolls, but in reality, it’s possible you may roll 10 sixes in a row. Heck it’s even technically possible (but very unlikely) that you could be rolling dice until the end of time and keep getting sixes. (Using PRNG’s in computers, this won’t happen, but it does take a variable number of rolls).

This is just to say: there is uneven and unpredictable execution time of this algorithm, and it needs an unknown (but somewhat predictable) amount of random numbers to work. This is true of the other forms of sampling methods I talk about lower down as well.

Instead of using a six sided die you could use a die of any size that is greater than (or equal to…) five. Here shows a twenty sided die simulating a five sided die:

It looks basically the same as using a six sided die, which makes sense (that shows that it works), but in this case, it actually took 4 rolls on average to make a valid five sided die roll, since the roll fails 15/20 times (3 out of 4 rolls will fail).

Quick Asides:

  • If straying from rejection sampling ideas for a minute, in the case of the twenty sided die, you could use modulus to get a fair five sided die roll each time: ((roll - 1) \% 5) + 1. This works because there is no remainder for 20 % 5. If there was a remainder it would bias the rolls towards the numbers <= the remainder, making them more likely to come up than the other numbers.
  • You could also get a four sided die roll at the same time if you didn’t want to waste any of this precious random information: ((roll - 1) / 5) + 1
  • Another algorithm to check out for discrete (integer) weighted random numbers is Vose’s method: Vose’s Method.

Box Around PDF

Moving back into the world of continuous valued random numbers and PDF’s, a simple version of how rejection sampling can be used is like this:

  1. Graph your PDF
  2. Draw a box around the PDF
  3. Generate a (uniform) random point in that box
  4. If the point is under the curve of the PDF, use the x axis value as your random number, else throw it out and go to 1

That’s all there is to it!

This works because the x axis value of your 2d point is the random number you might be choosing. The y axis value of your 2d point is a probability of choosing that point. Since the PDF graph is higher in places that are more probable, those places are more likely to accept your 2d point than places that have lower PDF values.

Furthermore, the average number of rejected samples vs accepted samples is based on the area under the PDF compared to the area of the box.

The number of samples on average will be the area of the box divided by the area of the PDF.

Since PDF’s by definition have to integrate to 1, that means that you are dividing by 1. So, to simplify: The number of samples on average will be the same as the area of the box!

If it’s hard to come up with the exact size of the box for the PDF, the box doesn’t have to fit exactly, but of course the tighter you can fit the box around the PDF, the fewer rejected samples you’ll have.

You don’t actually need to graph the PDF and draw a box to do this though. Just generate a 2d random number (a random x and a random y) and reject the point if PDF(x) < y.

Here I'm using this technique with the PDF y=2x where x is in [0,1) and I'm using a box that goes from (0,0) to (1,2) to get 100,000 samples.

As expected, it took on average 2 points to get a single valid point since the area of the box is 2. Here are how many failed tests each histogram bucket had. Unsurprisingly, lower values of the PDF have more failed tests!

Moving to a more complex PDF, let’s look at y=\frac{x^3-10x^2+5x+11}{10.417}

Here are 10 million samples (lots of samples to minimize the noise), using a box height of 1.2, which unsurprisingly takes 1.2 samples on average to get a valid sample:

Here is the graph of the failure counts:

Here the box has a height of 2.8. It still works, but uses 2.8 samples on average which is less efficient:

Here’s the graph of failure counts:

Something interesting about this technique is that technically, the distribution you are sampling from doesn’t even have to be a PDF! If you have negative parts of the graph, they will be treated as zero, assuming your box has a minimum y of 0. Also, the fact that your function may not integrate to (have an area of) 1 doesn’t matter at all.

Here we take the PDF from the last examples, and take off the division by a constant, so that it doesn’t integrate to 1: y=x^3-10x^2+5x+11

The interesting thing is that we get as output a normalized PDF (the red line), even though the distribution we were using to sample was not normalized (the blue line, which is mostly hidden behind the yellow line).

Here are the rejection counts:

Generating One PDF from Another PDF

In the last section we showed how to enclose a PDF in a box, make uniformly random 2d points, and use them to generate points from the PDF.

By enclosing it in a box, all we were really doing is putting it under a uniform distribition that was scaled up to be larger than the PDF at all points.

Now here’s the interesting thing: We aren’t limited to using the uniform distribution!

To generalize this technique, if you are trying to sample from a PDF f(x), you can use any PDF g(x) to do so, so long as you multiply g(x) by a scalar value M so that M*g(x)>= f(x) for all values of x. In other words: scale up g so that it’s always bigger than f.

Using this more generalized technique has one or two more steps than the other way, but allows for a tighter fit of a generating function, resulting in fewer samples thrown away.

Here’s how to do it:

  1. Generate a random number from the distribution g, and call it x.
  2. Calculate the percentage chance of x being chosen by getting a ratio of how likely that number is to be chosen in each PDF: \frac{f(x)}{M*g(x)}
  3. Generate a uniform random number from 0 to 1. If it’s less than the value you just calculated, accept x as the random number, else reject it and go back to 1.

Let’s see this in action!

We’ll generate numbers in a Gaussian distribution with a mean of 15 and a standard deviation of 5. We’ll truncate it to +/- 3 standard deviations so we want to generate random numbers from [0,30).

To generate these numbers, we’ll draw random numbers from the PDF y=x*0.002222. We’ll use an M value of 3 to scale up this PDF to always be greater than the Gaussian one.

Here is how it looks doing this with 20,000 samples:

We generate random numbers along the red line, multiply them by 3 to make them be the yellow line. Then, at whatever point we are at on the x axis, we divide the blue line value by the yellow line value and use that as an acceptance probability. Doing this and counting numbers in a histogram gives us our result – the green line. Since the end goal is the blue line, you can see it is indeed working! With a larger number of samples, the green line would more closely match the blue line.

Here’s the graph of the failed tests:

We have to take on average 3 samples before we get a valid random number. That shouldn’t be too surprising because both PDF’s start with area of 1, but we are multiplying one of them by 3 to make it always be larger than the other.

Something else interesting you might notice is that we have a lot fewer failed tests where the two PDF functions are more similar.

That is the power of this technique: If you can cheaply and easily generate samples that are “pretty close” to a harder distribution to sample from, you can use this technique to more cheaply sample from it.

Something to note is that just like in the last section, the target PDF doesn’t necessarily need to be a real PDF with only positive values and integrating to 1. It would work just the same with a non PDF function, just so long as the PDF generating the random numbers you start with is always above the function.

Some Other Notes

There is family of techniques called “adaptive rejection sampling” that will change the PDF they are drawing from whenever there is a failed test.

Basically, if you imagine the PDF you are drawing from as being a bunch of line segments connected together, you could imagine that whenever you failed a test, you moved a line segment down to be closer to the curve, so that when you sampled from that area again, the chances would be lower that you’d fail the test again.

Taking this to the limit, your sampling PDF will eventually become the PDF you are trying to sample from, and then using this PDF will be a no-op.

These techniques are a continued area of research.

Something else to note is that rejection sampling can be used to find random points within shapes.

For instance, a random point on a triangle, ellipse or circle could be done by putting a (tight) bounding box around the shape, generating points randomly in that box, and only accepting ones within the inner shape.

This can be extended to 3d shapes as well.

Some shapes have better ways to generate points within them that don’t involve iteration and rejected samples, but if all else fails, rejection sampling does indeed work!

At some point in the future I’d like to look into “Markov Chain Monte Carlo” more deeply. It seems like a very interesting technique to approach this same problem, but I have no idea if it’s used often in graphics, especially real time graphics.

Code

Here is the code that generated all the data from this post. The data was visualized with open office.

#define _CRT_SECURE_NO_WARNINGS
 
#include <stdio.h>
#include <random>
#include <array>
#include <unordered_map>

template <size_t NUM_TEST_SAMPLES, size_t SIMULATED_DICE_SIDES, size_t ACTUAL_DICE_SIDES>
void TestDice (const char* fileName)
{
    // seed the random number generator
    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_int_distribution<size_t> dist(0, ACTUAL_DICE_SIDES-1);

    // generate the histogram
    std::array<size_t, SIMULATED_DICE_SIDES> histogram = { 0 };
    size_t rejectedSamples = 0;
    for (size_t i = 0; i < NUM_TEST_SAMPLES; ++i)
    {
        size_t roll = dist(rng);
        while (roll >= SIMULATED_DICE_SIDES)
        {
            ++rejectedSamples;
            roll = dist(rng);
        }
        histogram[roll]++;
    }

    // write the histogram and rejected sample count to a csv
    // an extra 0 data point forces the graph to include 0 in the scale. hack to make the data not look noisier than it really is.
    FILE *file = fopen(fileName, "w+t");
    fprintf(file, "Actual Count, Expected Count, , %0.2f samples needed per roll on average.\n", (float(NUM_TEST_SAMPLES) + float(rejectedSamples)) / float(NUM_TEST_SAMPLES));
    for (size_t value : histogram)
        fprintf(file, "%zu,%zu,0\n", value, (size_t)(float(NUM_TEST_SAMPLES) / float(SIMULATED_DICE_SIDES)));
    fclose(file);
}
 
template <size_t NUM_TEST_SAMPLES, size_t NUM_HISTOGRAM_BUCKETS, typename PDF_LAMBDA>
void Test (const char* fileName, float maxPDFValue, const PDF_LAMBDA& PDF)
{
    // seed the random number generator
    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1.0f);
 
    // generate the histogram
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> histogram = { 0 };
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> failedTestCounts = { 0 };
    size_t rejectedSamples = 0;
    for (size_t i = 0; i < NUM_TEST_SAMPLES; ++i)
    {
        // Generate a sample from the PDF by generating a random 2d point.
        // If the y axis of the value is <= the value returned by PDF(x), accept it, else reject it.
        // NOTE: this takes an unknown number of iterations, and technically may NEVER finish.
        float pointX = 0.0f;
        float pointY = 0.0f;
        bool validPoint = false;
        while (!validPoint)
        {
            pointX = dist(rng);
            pointY = dist(rng) * maxPDFValue;
            float pdfValue = PDF(pointX);
            validPoint = (pointY <= pdfValue);

            // track number of failed tests per histogram bucket
            if (!validPoint)
            {
                size_t bin = (size_t)std::floor(pointX * float(NUM_HISTOGRAM_BUCKETS));
                failedTestCounts[std::min(bin, NUM_HISTOGRAM_BUCKETS - 1)]++;
                ++rejectedSamples;
            }
        }
 
        // increment the correct bin in the histogram
        size_t bin = (size_t)std::floor(pointX * float(NUM_HISTOGRAM_BUCKETS));
        histogram[std::min(bin, NUM_HISTOGRAM_BUCKETS -1)]++;
    }
 
    // write the histogram and pdf sample to a csv
    FILE *file = fopen(fileName, "w+t");
    fprintf(file, "PDF, Simulated PDF, Generating Function, Failed Tests, %0.2f samples needed per value on average.\n", (float(NUM_TEST_SAMPLES) + float(rejectedSamples)) / float(NUM_TEST_SAMPLES));
    for (size_t i = 0; i < NUM_HISTOGRAM_BUCKETS; ++i)
    {
        float x = (float(i) + 0.5f) / float(NUM_HISTOGRAM_BUCKETS);
        float pdfSample = PDF(x);
        fprintf(file, "%f,%f,%f,%f\n",
            pdfSample,
            NUM_HISTOGRAM_BUCKETS * float(histogram[i]) / float(NUM_TEST_SAMPLES),
            maxPDFValue,
            float(failedTestCounts[i])
        );
    }
    fclose(file);
}

template <size_t NUM_TEST_SAMPLES, size_t NUM_HISTOGRAM_BUCKETS, typename PDF_LAMBDA>
void TestNotPDF (const char* fileName, float maxPDFValue, float normalizationConstant, const PDF_LAMBDA& PDF)
{
    // seed the random number generator
    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1.0f);
 
    // generate the histogram
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> histogram = { 0 };
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> failedTestCounts = { 0 };
    size_t rejectedSamples = 0;
    for (size_t i = 0; i < NUM_TEST_SAMPLES; ++i)
    {
        // Generate a sample from the PDF by generating a random 2d point.
        // If the y axis of the value is <= the value returned by PDF(x), accept it, else reject it.
        // NOTE: this takes an unknown number of iterations, and technically may NEVER finish.
        float pointX = 0.0f;
        float pointY = 0.0f;
        bool validPoint = false;
        while (!validPoint)
        {
            pointX = dist(rng);
            pointY = dist(rng) * maxPDFValue;
            float pdfValue = PDF(pointX);
            validPoint = (pointY <= pdfValue);

            // track number of failed tests per histogram bucket
            if (!validPoint)
            {
                size_t bin = (size_t)std::floor(pointX * float(NUM_HISTOGRAM_BUCKETS));
                failedTestCounts[std::min(bin, NUM_HISTOGRAM_BUCKETS - 1)]++;
                ++rejectedSamples;
            }
        }
 
        // increment the correct bin in the histogram
        size_t bin = (size_t)std::floor(pointX * float(NUM_HISTOGRAM_BUCKETS));
        histogram[std::min(bin, NUM_HISTOGRAM_BUCKETS -1)]++;
    }
 
    // write the histogram and pdf sample to a csv
    FILE *file = fopen(fileName, "w+t");
    fprintf(file, "Function, Simulated PDF, Scaled Simulated PDF, Generating Function, Failed Tests, %0.2f samples needed per value on average.\n", (float(NUM_TEST_SAMPLES) + float(rejectedSamples)) / float(NUM_TEST_SAMPLES));
    for (size_t i = 0; i < NUM_HISTOGRAM_BUCKETS; ++i)
    {
        float x = (float(i) + 0.5f) / float(NUM_HISTOGRAM_BUCKETS);
        float pdfSample = PDF(x);
        fprintf(file, "%f,%f,%f,%f,%f\n",
            pdfSample,
            NUM_HISTOGRAM_BUCKETS * float(histogram[i]) / float(NUM_TEST_SAMPLES),
            NUM_HISTOGRAM_BUCKETS * float(histogram[i]) / float(NUM_TEST_SAMPLES) * normalizationConstant,
            maxPDFValue,
            float(failedTestCounts[i])
        );
    }
    fclose(file);
}

template <size_t NUM_TEST_SAMPLES, size_t NUM_HISTOGRAM_BUCKETS, typename PDF_F_LAMBDA, typename PDF_G_LAMBDA, typename INVERSE_CDF_G_LAMBDA>
void TestPDFToPDF (const char* fileName, const PDF_F_LAMBDA& PDF_F, const PDF_G_LAMBDA& PDF_G, float M, const INVERSE_CDF_G_LAMBDA& Inverse_CDF_G, float rngRange)
{
    // We generate a sample from PDF F by generating a sample from PDF G, and accepting it with probability PDF_F(x)/(M*PDF_G(x))

    // seed the random number generator
    std::random_device rd;
    std::mt19937 rng(rd());
    std::uniform_real_distribution<float> dist(0.0f, 1.0f);
 
    // generate the histogram
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> histogram = { 0 };
    std::array<size_t, NUM_HISTOGRAM_BUCKETS> failedTestCounts = { 0 };
    size_t rejectedSamples = 0;
    for (size_t i = 0; i < NUM_TEST_SAMPLES; ++i)
    {
        // generate random points until we have one that's accepted
        // NOTE: this takes an unknown number of iterations, and technically may NEVER finish.
        float sampleG = 0.0f;
        bool validPoint = false;
        while (!validPoint)
        {
            // Generate a sample from the soure PDF G
            sampleG = Inverse_CDF_G(dist(rng));

            // calculate the ratio of how likely we are to accept this sample
            float acceptChance = PDF_F(sampleG) / (M * PDF_G(sampleG));

            // see if we should accept it
            validPoint = dist(rng) <= acceptChance;

            // track number of failed tests per histogram bucket
            if (!validPoint)
            {
                size_t bin = (size_t)std::floor(sampleG * float(NUM_HISTOGRAM_BUCKETS) / rngRange);
                failedTestCounts[std::min(bin, NUM_HISTOGRAM_BUCKETS - 1)]++;
                ++rejectedSamples;
            }
        }

        // increment the correct bin in the histogram
        size_t bin = (size_t)std::floor(sampleG * float(NUM_HISTOGRAM_BUCKETS) / rngRange);
        histogram[std::min(bin, NUM_HISTOGRAM_BUCKETS - 1)]++;
    }
 
    // write the histogram and pdf sample to a csv
    FILE *file = fopen(fileName, "w+t");
    fprintf(file, "PDF F,PDF G,Scaled PDF G,Simulated PDF,Failed Tests,%0.2f samples needed per value on average.\n", (float(NUM_TEST_SAMPLES) + float(rejectedSamples)) / float(NUM_TEST_SAMPLES));
    for (size_t i = 0; i < NUM_HISTOGRAM_BUCKETS; ++i)
    {
        float x = (float(i) + 0.5f) * rngRange / float(NUM_HISTOGRAM_BUCKETS);
        
        fprintf(file, "%f,%f,%f,%f,%f\n",
            PDF_F(x),
            PDF_G(x),
            PDF_G(x)*M,
            NUM_HISTOGRAM_BUCKETS * float(histogram[i]) / (float(NUM_TEST_SAMPLES)*rngRange),
            float(failedTestCounts[i])
        );
    }
    fclose(file);
}
 
int main(int argc, char **argv)
{
    // Dice
    {
        // Simulate a 5 sided dice with a 6 sided dice
        TestDice<10000, 5, 6>("test1_5_6.csv");

        // Simulate a 5 sided dice with a 20 sided dice
        TestDice<10000, 5, 20>("test1_5_20.csv");
    }

    // PDF y=2x, simulated with a uniform distribution
    {
        auto PDF = [](float x) { return 2.0f * x; };

        Test<1000, 100>("test2_1k.csv", 2.0f, PDF);
        Test<100000, 100>("test2_100k.csv", 2.0f, PDF);
        Test<1000000, 100>("test2_1m.csv", 2.0f, PDF);
    }

    // PDF y=(x^3-10x^2+5x+11)/10.417, simulated with a uniform distribution
    {
        auto PDF = [](float x) {return (x*x*x - 10.0f*x*x + 5.0f*x + 11.0f) / (10.417f); };
        Test<10000000, 100>("test3_10m_1_15.csv", 1.15f, PDF);
        Test<10000000, 100>("test3_10m_1_5.csv", 1.5f, PDF);
        Test<10000000, 100>("test3_10m_2_8.csv", 2.8f, PDF);
    }

    // function (not PDF, Doesn't integrate to 1!) y=(x^3-10x^2+5x+11), simulated with a scaled up uniform distribution
    {
        auto PDF = [](float x) {return (x*x*x - 10.0f*x*x + 5.0f*x + 11.0f); };
        TestNotPDF<10000000, 100>("test4_10m_12_5.csv", 12.5f, 10.417f, PDF);
    }

    // Generate samples from PDF F using samples from PDF G.  random numbers are from 0 to 30.
    // F PDF = gaussian distribution, mean 15, std dev of 5.  Truncated to +/- 3 stddeviations.
    // G PDF = x*0.002222
    // G CDF = 0.001111 * x^2
    // G inverted CDF = (1000 * sqrt(x)) / sqrt(1111)
    // M = 3
    {
        // gaussian PDF F
        const float mean = 15.0f;
        const float stddev = 5.0f;
        auto PDF_F = [=] (float x) -> float
        {
            return (1.0f / (stddev * sqrt(2.0f * (float)std::_Pi))) * std::exp(-0.5f * pow((x - mean) / stddev, 2.0f));
        };

        // PDF G
        auto PDF_G = [](float x) -> float
        {
            return x * 0.002222f;
        };

        // Inverse CDF of G
        auto Inverse_CDF_G = [] (float x) -> float
        {
            return 1000.0f * std::sqrtf(x) / std::sqrtf(1111.0f);
        };

        TestPDFToPDF<20000, 100>("test5.csv", PDF_F, PDF_G, 3.0f, Inverse_CDF_G, 30.0f);
    }

    return 0;
}