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);


Next Generation Post Processing in Call Of Duty: Advanced Warfare
Dithering part three – 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 <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);

// calculate 2d dft (brute force, not using fast fourier transform) multithreadedly
std::atomic<size_t> nextRow(0);
{
[&] ()
{
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
}
}
);
}

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;
}

// write the data and close the 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;

{
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
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

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");
}

//======================================================================================
{
printf("\n%s\n", __FUNCTION__);

// make noise
SImageData noise;

// 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);
}
}

//======================================================================================
{
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;

// 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);
}
}

//======================================================================================
{
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;

// 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;
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);
}
}

//======================================================================================
{
printf("\n%s\n", __FUNCTION__);

// make noise
SImageData noiseSrc;

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;
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;
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;
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);
}
}

//======================================================================================
{
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;

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;
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;
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;
{
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);
{
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);
DitherBlueNoise(ditherImage, blueNoise[0]);

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

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

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

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

fclose(g_logFile);

return 0;
}


Transmuting White Noise To Blue, Red, Green, Purple

There are many algorithms for generating blue noise, and there are a lot of people working on new ways to do it.

It made me wonder: why don’t people just use the inverse discrete Fourier transform to make noise that has the desired frequency spectrum?

I knew there had to be a reason, since that is a pretty obvious thing to try, but I wasn’t sure if it was due to poor quality results, slower execution times, or some other reason.

After trying it myself and not getting great results I asked on twitter and Bart Wronski (@BartWronsk) clued me in.

It turns out that you can set up your frequency magnitudes such that there are only high frequencies, giving them random amplitudes, and random phases, but when you do the inverse DFT, the result isn’t guaranteed to use all possible color values (0-255) and even if it does, it may not use them evenly.

He pointed me at something that Timothy Lottes wrote up (@TimothyLottes), which talked about using some basic DSP operations to transform white noise into blue noise.

This post uses that technique to do some “Noise Alchemy” and turn white noise into a couple other types of noise. Simple single file standalone C++ source code included at bottom of the post!

Red Noise

We’ll start with red noise because it’s the simplest. Here’s how you do it:

2. Low pass filter the white noise
3. Re-normalize the histogram
4. Repeat from step 2 as many times as desired

That’s all there is to it.

If you are wondering how you low pass filter an image, that’s another way of saying “blur”. Blurring makes the high frequency details go away, leaving the low frequency smooth shapes.

There are multiple ways to do a blur, including: box blur (averaging pixels), Gaussian blur, sinc filtering. In this post I use a Gaussian blur and get decent results, but box blurring would be cheaper/faster, and sinc filtering would be the most correct results.

An important detail about doing the blur is that your blur needs to “wrap around”. If you are blurring a pixel on the edge of the image, it should smear over to the other side of the image.

You might be wondering how you would normalize the histogram. Normalizing the histogram just means that we want to make sure that the image uses the full range of greyscale values evenly. We don’t want the noise to only use bright colors or only use dark colors, or even just MOSTLY use dark colors, for instance. If we count each color used in the image (which is the histogram I’m referring to), the counts for each color should be roughly equal.

To fix the histogram, Timothy Lottes suggests making an array that contains each pixel location and the brightness of that pixel. You first shuffle the array and then sort by brightness (Timothy uses a 64 bit int to store the pixel information, so uses a radix sort which is more efficient for fixed size keys). Next set the brightness of each item in the array to be it’s index divided by the number of items in the list to put them in the 0 to 1 range. Lastly you write the brightness values back out to the image, using the pixel locations you stored off.

What this process does is makes sure that the full range of greyscale values are used, and that they are used evenly. It also preserves the order of the brightness of the pixels; if pixel A was darker than pixel B before this process, it still will be darker after this process.

You may wonder why the shuffle is done before the sort. That’s done so that if there are any ties between values that it will be random which one is darker after this process. This is important because if it wasn’t random, there may be obvious (ugly/incorrect) patterns in the results.

When normalizing the histogram, it affects the frequency composition of the image, but if doing this process multiple times, it seems to do an OK job of converging to decent results.

Red noise has low frequency content which means it doesn’t have sharp details / fast data changes. An interesting property of 2d red noise is that if you take a random walk on the 2d red noise texture, that the values you’d hit would be a random walk of 1d values. Also, if you draw a straight line randomly on the texture, the pixels it passes through will be a random walk. That is, you’ll get random numbers, but each number will be pretty close to the previous one.

The formal definition of red noise has a more strict definition about frequency content than what we are going for in this post. (Wikipedia: red noise)

Here’s red noise (top) and the frequency magnitudes (bottom) using 5 iterations of the described process, and a blur sigma (strength of blur) of 1.0:

Using different blur strengths controls what frequencies get attenuated. Weaker blurs leave higher frequency components.

Here is red noise generated the same way but using a blur sigma of 0.5:

And here is red noise generated using a blur sigma of 2.0

Here are some animated gifs showing the evolution of the noise as well as the frequencies over time:

Sigma 0.5:

Sigma 1.0:

Sigma 2.0:

Blue Noise

To make blue noise, you use the exact same method but instead of using a low pass filter you use a high pass filter.

An easy way to high pass filter an image is to do a low pass filter to get the low frequency content, and subtract that from the original image so that you are left with the high frequency content.

Blue noise has high frequency content which means it is only made up of sharp details / fast data changes. An interesting property of 2d blue noise is that if you take a random walk (or a straight line walk) on it in any direction, you’ll get a low discrepancy sequence. That is, you’ll get random numbers, but each number will be very different from the previous one.

The formal definition of blue noise has a more strict definition about frequency content than what we are going for in this post. (Wikipedia: blue noise)

Here is blue noise using 5 iterations and a blur sigma of 1.0:

Just like with red noise, changing the strength of the blur controls what frequencies get attenuated.

Here is a sigma of 0.5:

Here is a sigma of 2.0:

Animations of sigma 0.5:

Animations of sigma 1.0:

Animations of sigma 2.0:

Green Noise

Green noise is noise that doesn’t have either low or high frequency components, only mid frequency components.

To make green noise, use you a “band pass” filter, which is a filter that gets rid of both high and low frequency components leaving only the middle.

Here’s how to make a band pass filter:

1. Make a weak blur of the image – this is the image without the highest frequencies.
2. Make a strong blur of the image – this is the image with just the lowest frequencies.
3. Subtract the strong blur from the weak blur – this is the image with just the middle frequencies.

Here is 5 iterations using a sigma of 0.5 and 2.0:

Here is the animation of it evolving:

Nathan Reed (@ReedBeta) mentioned that the green noise looked a lot like Perlin noise, which made sense due to Perlin noise being designed to be band limited, which makes it easier to control the look of perlin noise by summing mulitple octaves. This makes sense to me because you basically can control what frequencies you put noise into by scaling the frequency ring.

Fabien Giesen (@rygorous) said this also helps with mipmapping. This makes sense to me because there can’t be (as much) aliasing with the higher frequencies missing from the noise.

Purple Noise

I’ve never heard of this noise so it may have another name, but what I’m calling purple noise is just noise which has high and low frequency content, but no middle frequency content. It’s basically red noise plus blue noise.

You could literally make red noise and add it to blue noise to make purple noise, but how I made it for this post is to use a “band stop” filter.

A band stop filter is a filter that gets rid of middle frequencies and leaves high and low frequencies alone.

To band stop filter an image, you do a band pass filter to get the middle frequencies (as described in the last section!), and then subtract that from the original image to get only the low and high frequencies.

Here is 5 iterations using a sigma of 0.5 and 2.0:

Here is the animation:

This technique might be useful if you ever need to generate specific types of noise quickly, but if you are just generating noise textures to use later in performance critical situations, there are better algorithms to use. When generating textures offline in advance, you have “all the time in the world”, so it is probably not worth the simplicity of this algorithm, when the trade off is less good noise results.

Dithering part two – golden ratio sequence, blue noise and highpass-and-remap (Bart Wronski)

VDR Follow Up – Fine Art of Film Grain (Timothy Lottes)

Gaussian Blur (Me)

Image DFT / IDFT (me)

Blue-noise Dithered Sampling (Solid Angle) – a better way to generate colored noises

Apparently there is a relation between blue noise, turing patterns / reaction diffusion and these filtering techniques. (Thanks @R4_Unit!)
Turing Patterns in Photoshop

Point Sampling with General Noise Spectrum

Here is an interesting shadertoy which uses the mip map of a noise texture to get the low frequency content to do a high pass filter: (found by @paniq, who unfortunately got nerd sniped by this noise generation stuff hehe)
pseudo blue noise 2

Source Code

The source code to generate the images is below, but is also on githib at Atrix256 – NoiseShaping

#define _CRT_SECURE_NO_WARNINGS

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

typedef uint8_t uint8;
typedef int64_t int64;

const float c_pi = 3.14159265359f;

// settings
const size_t    c_imageSize = 256;
const bool      c_doDFT = true;
const float     c_blurThresholdPercent = 0.005f; // lower numbers give higher quality results, but take longer. This is 0.5%
const float     c_numBlurs = 5;

//======================================================================================
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 SImageDataFloat
{
SImageDataFloat()
: m_width(0)
, m_height(0)
{ }

size_t m_width;
size_t m_height;
std::vector<float> m_pixels;
};

//======================================================================================
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);

// calculate 2d dft (brute force, not using fast fourier transform) multithreadedly
std::atomic<size_t> nextRow(0);
{
[&] ()
{
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
}
}
);
}

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;
}

// write the data and close the 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;

{
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
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

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);
}

//======================================================================================
void ImageFloatInit (SImageDataFloat& image, size_t width, size_t height)
{
image.m_width = width;
image.m_height = height;
image.m_pixels.resize(image.m_width * image.m_height);
std::fill(image.m_pixels.begin(), image.m_pixels.end(), 0.0f);
}

//======================================================================================
int PixelsNeededForSigma (float sigma)
{
// returns the number of pixels needed to represent a gaussian kernal that has values
// down to the threshold amount.  A gaussian function technically has values everywhere
// on the image, but the threshold lets us cut it off where the pixels contribute to
// only small amounts that aren't as noticeable.
return int(floor(1.0f + 2.0f * sqrtf(-2.0f * sigma * sigma * log(c_blurThresholdPercent)))) + 1;
}

//======================================================================================
float Gaussian (float sigma, float x)
{
return expf(-(x*x) / (2.0f * sigma*sigma));
}

//======================================================================================
float GaussianSimpsonIntegration (float sigma, float a, float b)
{
return
((b - a) / 6.0f) *
(Gaussian(sigma, a) + 4.0f * Gaussian(sigma, (a + b) / 2.0f) + Gaussian(sigma, b));
}

//======================================================================================
std::vector<float> GaussianKernelIntegrals (float sigma, int taps)
{
std::vector<float> ret;
float total = 0.0f;
for (int i = 0; i < taps; ++i)
{
float x = float(i) - float(taps / 2);
float value = GaussianSimpsonIntegration(sigma, x - 0.5f, x + 0.5f);
ret.push_back(value);
total += value;
}
// normalize it
for (unsigned int i = 0; i < ret.size(); ++i)
{
ret[i] /= total;
}
return ret;
}

//======================================================================================
const float* GetPixelWrapAround (const SImageDataFloat& image, int x, int y)
{
if (x >= (int)image.m_width)
{
x = x % (int)image.m_width;
}
else
{
while (x < 0)
x += (int)image.m_width;
}

if (y >= (int)image.m_height)
{
y = y % (int)image.m_height;
}
else
{
while (y < 0)
y += (int)image.m_height;
}

return &image.m_pixels[(y * image.m_width) + x];
}

//======================================================================================
void ImageGaussianBlur (const SImageDataFloat& srcImage, SImageDataFloat &destImage, float xblursigma, float yblursigma, unsigned int xblursize, unsigned int yblursize)
{
// allocate space for copying the image for destImage and tmpImage
ImageFloatInit(destImage, srcImage.m_width, srcImage.m_height);

SImageDataFloat tmpImage;
ImageFloatInit(tmpImage, srcImage.m_width, srcImage.m_height);

// horizontal blur from srcImage into tmpImage
{
auto row = GaussianKernelIntegrals(xblursigma, xblursize);

int startOffset = -1 * int(row.size() / 2);

for (int y = 0; y < tmpImage.m_height; ++y)
{
for (int x = 0; x < tmpImage.m_width; ++x)
{
float blurredPixel = 0.0f;
for (unsigned int i = 0; i < row.size(); ++i)
{
const float *pixel = GetPixelWrapAround(srcImage, x + startOffset + i, y);
blurredPixel += pixel[0] * row[i];
}

float *destPixel = &tmpImage.m_pixels[y * tmpImage.m_width + x];
destPixel[0] = blurredPixel;
}
}
}

// vertical blur from tmpImage into destImage
{
auto row = GaussianKernelIntegrals(yblursigma, yblursize);

int startOffset = -1 * int(row.size() / 2);

for (int y = 0; y < destImage.m_height; ++y)
{
for (int x = 0; x < destImage.m_width; ++x)
{
float blurredPixel = 0.0f;
for (unsigned int i = 0; i < row.size(); ++i)
{
const float *pixel = GetPixelWrapAround(tmpImage, x, y + startOffset + i);
blurredPixel += pixel[0] * row[i];
}

float *destPixel = &destImage.m_pixels[y * destImage.m_width + x];
destPixel[0] = blurredPixel;
}
}
}
}

//======================================================================================
void SaveImageFloatAsBMP (const SImageDataFloat& imageFloat, const char* fileName)
{
printf("\n%s\n", fileName);

// init the image
SImageData image;
ImageInit(image, imageFloat.m_width, imageFloat.m_height);

// write the data to the image
const float* srcData = &imageFloat.m_pixels[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)
{
uint8 value = uint8(255.0f * srcData[0]);

pixel->Set(value, value, value);

++pixel;
++srcData;
}
}

// save the image
ImageSave(image, fileName);

// also save a DFT of the image
if (c_doDFT)
{
SImageDataComplex dftData;
ImageDFT(image, dftData);

SImageData DFTMagImage;
GetMagnitudeData(dftData, DFTMagImage);

char buffer[256];
sprintf(buffer, "%s.mag.bmp", fileName);

ImageSave(DFTMagImage, buffer);
}
}

//======================================================================================
void NormalizeHistogram (SImageDataFloat& image)
{
struct SHistogramHelper
{
float value;
size_t pixelIndex;
};
static std::vector<SHistogramHelper> pixels;
pixels.resize(image.m_width * image.m_height);

// put all the pixels into the array
for (size_t i = 0, c = image.m_width * image.m_height; i < c; ++i)
{
pixels[i].value = image.m_pixels[i];
pixels[i].pixelIndex = i;
}

// shuffle the pixels to randomly order ties. not as big a deal with floating point pixel values though
static std::random_device rd;
static std::mt19937 rng(rd());
std::shuffle(pixels.begin(), pixels.end(), rng);

// sort the pixels by value
std::sort(
pixels.begin(),
pixels.end(),
[] (const SHistogramHelper& a, const SHistogramHelper& b)
{
return a.value < b.value;
}
);

// use the pixel's place in the array as the new value, and write it back to the image
for (size_t i = 0, c = image.m_width * image.m_height; i < c; ++i)
{
float value = float(i) / float(c - 1);
image.m_pixels[pixels[i].pixelIndex] = value;
}
}

//======================================================================================
void BlueNoiseTest (float blurSigma)
{
// calculate the blur size from our sigma
int blurSize = PixelsNeededForSigma(blurSigma) | 1;

// setup the randon number generator
std::random_device rd;
std::mt19937 rng(rd());
std::uniform_real_distribution<float> dist(0.0f, 1.0f);

// generate some white noise
SImageDataFloat noise;
ImageFloatInit(noise, c_imageSize, c_imageSize);
for (float& v : noise.m_pixels)
{
v = dist(rng);
}

// save off the starting white noise
const char* baseFileName = "bluenoise_%i_%zu.bmp";
char fileName[256];

sprintf(fileName, baseFileName, int(blurSigma * 100.0f), 0);
SaveImageFloatAsBMP(noise, fileName);

// iteratively high pass filter and rescale histogram to the 0 to 1 range
SImageDataFloat blurredImage;
for (size_t blurIndex = 0; blurIndex < c_numBlurs; ++blurIndex)
{
// get a low passed version of the current image
ImageGaussianBlur(noise, blurredImage, blurSigma, blurSigma, blurSize, blurSize);

// subtract the low passed version to get the high passed version
for (size_t pixelIndex = 0; pixelIndex < c_imageSize * c_imageSize; ++pixelIndex)
noise.m_pixels[pixelIndex] -= blurredImage.m_pixels[pixelIndex];

// put all pixels between 0.0 and 1.0 again
NormalizeHistogram(noise);

// save this image
sprintf(fileName, baseFileName, int(blurSigma * 100.0f), blurIndex + 1);
SaveImageFloatAsBMP(noise, fileName);
}
}

//======================================================================================
void RedNoiseTest (float blurSigma)
{
// calculate the blur size from our sigma
int blurSize = PixelsNeededForSigma(blurSigma) | 1;

// setup the randon number generator
std::random_device rd;
std::mt19937 rng(rd());
std::uniform_real_distribution<float> dist(0.0f, 1.0f);

// generate some white noise
SImageDataFloat noise;
ImageFloatInit(noise, c_imageSize, c_imageSize);
for (float& v : noise.m_pixels)
{
v = dist(rng);
}

// save off the starting white noise
const char* baseFileName = "rednoise_%i_%zu.bmp";
char fileName[256];

sprintf(fileName, baseFileName, int(blurSigma * 100.0f), 0);
SaveImageFloatAsBMP(noise, fileName);

// iteratively high pass filter and rescale histogram to the 0 to 1 range
SImageDataFloat blurredImage;
for (size_t blurIndex = 0; blurIndex < c_numBlurs; ++blurIndex)
{
// get a low passed version of the current image
ImageGaussianBlur(noise, blurredImage, blurSigma, blurSigma, blurSize, blurSize);

// set noise image to the low passed version
noise.m_pixels = blurredImage.m_pixels;

// put all pixels between 0.0 and 1.0 again
NormalizeHistogram(noise);

// save this image
sprintf(fileName, baseFileName, int(blurSigma * 100.0f), blurIndex + 1);
SaveImageFloatAsBMP(noise, fileName);
}
}

//======================================================================================
void BandPassTest (float blurSigma1, float blurSigma2)
{
// calculate the blur size from our sigma
int blurSize1 = PixelsNeededForSigma(blurSigma1) | 1;
int blurSize2 = PixelsNeededForSigma(blurSigma2) | 1;

// setup the randon number generator
std::random_device rd;
std::mt19937 rng(rd());
std::uniform_real_distribution<float> dist(0.0f, 1.0f);

// generate some white noise
SImageDataFloat noise;
ImageFloatInit(noise, c_imageSize, c_imageSize);
for (float& v : noise.m_pixels)
{
v = dist(rng);
}

// save off the starting white noise
const char* baseFileName = "bandpass_%i_%i_%zu.bmp";
char fileName[256];

sprintf(fileName, baseFileName, int(blurSigma1 * 100.0f), int(blurSigma2 * 100.0f), 0);
SaveImageFloatAsBMP(noise, fileName);

// iteratively high pass filter and rescale histogram to the 0 to 1 range
SImageDataFloat blurredImage1;
SImageDataFloat blurredImage2;
for (size_t blurIndex = 0; blurIndex < c_numBlurs; ++blurIndex)
{
// get two low passed versions of the current image
ImageGaussianBlur(noise, blurredImage1, blurSigma1, blurSigma1, blurSize1, blurSize1);
ImageGaussianBlur(noise, blurredImage2, blurSigma2, blurSigma2, blurSize2, blurSize2);

// subtract one low passed version from the other
for (size_t pixelIndex = 0; pixelIndex < c_imageSize * c_imageSize; ++pixelIndex)
noise.m_pixels[pixelIndex] = blurredImage1.m_pixels[pixelIndex] - blurredImage2.m_pixels[pixelIndex];

// put all pixels between 0.0 and 1.0 again
NormalizeHistogram(noise);

// save this image
sprintf(fileName, baseFileName, int(blurSigma1 * 100.0f), int(blurSigma2 * 100.0f), blurIndex + 1);
SaveImageFloatAsBMP(noise, fileName);
}
}

//======================================================================================
void BandStopTest (float blurSigma1, float blurSigma2)
{
// calculate the blur size from our sigma
int blurSize1 = PixelsNeededForSigma(blurSigma1) | 1;
int blurSize2 = PixelsNeededForSigma(blurSigma2) | 1;

// setup the randon number generator
std::random_device rd;
std::mt19937 rng(rd());
std::uniform_real_distribution<float> dist(0.0f, 1.0f);

// generate some white noise
SImageDataFloat noise;
ImageFloatInit(noise, c_imageSize, c_imageSize);
for (float& v : noise.m_pixels)
{
v = dist(rng);
}

// save off the starting white noise
const char* baseFileName = "bandstop_%i_%i_%zu.bmp";
char fileName[256];

sprintf(fileName, baseFileName, int(blurSigma1 * 100.0f), int(blurSigma2 * 100.0f), 0);
SaveImageFloatAsBMP(noise, fileName);

// iteratively high pass filter and rescale histogram to the 0 to 1 range
SImageDataFloat blurredImage1;
SImageDataFloat blurredImage2;
for (size_t blurIndex = 0; blurIndex < c_numBlurs; ++blurIndex)
{
// get two low passed versions of the current image
ImageGaussianBlur(noise, blurredImage1, blurSigma1, blurSigma1, blurSize1, blurSize1);
ImageGaussianBlur(noise, blurredImage2, blurSigma2, blurSigma2, blurSize2, blurSize2);

// subtract one low passed version from the other to get the pandpass noise, and subtract that from the original noise to get the band stop noise
for (size_t pixelIndex = 0; pixelIndex < c_imageSize * c_imageSize; ++pixelIndex)
noise.m_pixels[pixelIndex] -= (blurredImage1.m_pixels[pixelIndex] - blurredImage2.m_pixels[pixelIndex]);

// put all pixels between 0.0 and 1.0 again
NormalizeHistogram(noise);

// save this image
sprintf(fileName, baseFileName, int(blurSigma1 * 100.0f), int(blurSigma2 * 100.0f), blurIndex + 1);
SaveImageFloatAsBMP(noise, fileName);
}
}

//======================================================================================
int main (int argc, char ** argv)
{
BlueNoiseTest(0.5f);
BlueNoiseTest(1.0f);
BlueNoiseTest(2.0f);

RedNoiseTest(0.5f);
RedNoiseTest(1.0f);
RedNoiseTest(2.0f);

BandPassTest(0.5f, 2.0f);

BandStopTest(0.5f, 2.0f);

return 0;
}


Generating Blue Noise Sample Points With Mitchell’s Best Candidate Algorithm

Lately I’ve been eyeball deep in noise, ordered dithering and related topics, and have been learning some really interesting things.

As the information coalesces it’ll become apparent whether there is going to be a huge mega post coming, or if there will be several smaller ones.

In the meantime, I wanted to share this bite sized chunk of information.

Three sampling patterns that are most often used when sampling (say, when numerically integrating a lighting function for graphics/rendering purposes) are: regular samples, white noise samples, and blue noise samples.

Regular Sampling

Regular sampling just means evenly spacing the samples. This sampling strategy can suffer from aliasing, but gives good coverage of the sample space.

Here are 256, 1024 and 4096 samples:

Here are those samples taken from a source image:

Here is the DFT (frequency amplitude) of those samples:

White Noise Sampling

White noise sampling just chooses random numbers for where to place the samples.
The random numbers are uniformly distributed, meaning they are just plain vanilla random numbers where each number is equally likely to come up.

White noise sampling can make for noisy results, and suffers from the fact that white noise sample points can clump together and leave empty space. Clumped sample points give redundant information while empty space is information that you are lacking in the sample space. In general, noise is often desired over aliasing though, so white noise samples are generally preferred over regular sampling. Monte Carlo integration also requires random samples.

White noise is called white noise because it contains all frequencies approximately evenly, like how white light is made up of all frequencies of light.

Here are 256, 1024 and 4096 samples:

Here are those samples taken from a source image:

Here is the DFT (frequency amplitude) of those samples:

Blue Noise Sampling

Lastly is blue noise sampling which is somewhere between regular sampling and white noise sampling. Blue noise sampling has randomly placed points like white noise does, but the randomly placed points are approximately evenly spaced, which is more like regular sampling.

Things like low discrepancy sequences, stratified sampling, and jittered regular sampling mimic blue noise, and are often a cheaper alternative when an approximation is acceptable. More info on low discrepancy sequences is available on my post here: When Random Numbers Are Too Random: Low Discrepancy Sequences

Blue noise is called blue noise because it contains higher amounts of higher frequencies and lower amounts of lower frequencies. This is the same of blue light, which contains higher frequency (bluer) light.

Here are 256, 1024 and 4096 samples:

Here are those samples taken from a source image:

Here is the DFT (frequency amplitude) of those samples:

Comparison

Imagine you were a robot with 4096 light/color sensors. Which of the arrangements below do you think would give you the best information about the world around you with that number of sensors?

To me, the regular grid and the blue noise are a bit of a coin toss, while the white noise version is awful.

The regular grid does seem to show me straight lined things better (the road, sidewalk, etc), but it makes non straight lined things – like the trees – look blocky. The blue noise grid does the reverse and makes straight things look wavy, while making it easier to see the true shape of non straight things.

Mathematically, blue noise is superior sampling, so maybe this example isn’t the best way to show the value of blue noise.

Here is the real image:

Apparently the photo-receptors in our eyes are arranged in a blue noise pattern. Some people say this is why blue noise is more agreeable with our perception, but since it also helps numerical integration converge faster for lower sample counts (compared to white noise – which wins out with larger sample counts BTW!), it seems like there is a more fundamental reason which would cause an evolutionary advantage for them to be arranged that way in the first place.

Generating Blue Noise Sample Points

The obvious question is: I know how to make uniform and random sample points. How do I make blue noise sample points?

There are multiple ways to do it, but a method that I find very easy to understand and to implement is “Mitchell’s Best Candidate Algorithm”.

The algorithm is as follows:

1. Place a random sample point as the first sample point.
2. Generate some number of random dots as candidates to be the next sample point.
3. Whichever of these dots is farthest away from the closest existing sample point is the winner. Place that dot as the new sample point.
4. GOTO 2 and Repeat until you have as many sample points as you want.

The algorithm is pretty simple, but there are two other important details that are needed to give you good results:

• When calculating distance between dots, you need to consider wrap around. More info on how to do that here: Calculating the Distance Between Points in “Wrap Around” (Toroidal) Space.
• The number of candidates you generate should scale up with the number of existing sample points. As the original paper describing this technique says, doing that helps ensure that the statistics of your sample points stay constant as the number of sample points changes.

When I first tried to get this algorithm working, I was generating a fixed number of candidates each time. That gave me these pretty terrible results:

However, when I multiplied the number of existing sample points by a constant “m” as the number of sample points to generate, I got much better results, even when m was 1! (Note: m=0 is the same as white noise in this image. I did NumSamples*m+1 candidates each time.)

Related Computer Graphics Stack Exchange Question: Mitchell’s Best Candidate Algorithm

If you store existing sample points in a grid, you can speed up the algorithm since it will be faster to find the closest point to a candidate. In the implementation on this post I didn’t do that.

You may be able to multithread this algorithm but I haven’t tried it. The idea would be if you needed to make and test N candidates, that you could split that among M threads, so long as N was large enough to make that worth while. I didn’t do that in this post.

Lastly, instead of working with distance, you can work with SQUARED distance to avoid many unnecessary square root calculations. The example code here does that optimization.

The 1991 paper that described this technique:
Spectrally optimal sampling for distribution ray tracing

Another interesting link on this algorithm:
Mitchell’s Best-Candidate

This algorithm isn’t that great for making dense sample points, or for use in dithering / stippling. Look for a future blog post about those usage cases, but for now, this is a great resource:
Free Blue Noise Textures (and info / examples on blue noise texture usage)

A physics based approach to blue noise distributed samples:
Electrostatic Half Toning

A neat read on the “void and cluster” method for generating blue noise, and also a good read on what ordered dithering is all about:
The void and cluster method for dither array generation

Source Code

Here is some simple standalone C++ source code which can generate blue noise sample points, and also generated the images used in this post.

It’s also on github (along with the source image) at https://github.com/Atrix256/RandomCode/tree/master/Mitchell

#define _CRT_SECURE_NO_WARNINGS

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

typedef uint8_t uint8;
typedef int64_t int64;

const float c_pi = 3.14159265359f;

//======================================================================================
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;
};

SImageData s_stippleImage;

//======================================================================================
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 DFTImage (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);

// calculate 2d dft (brute force, not using fast fourier transform) multithreadedly
std::atomic<size_t> nextRow(0);
{
[&] ()
{
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
}
}
);
}

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;
}
}
}

//======================================================================================
void GetPhaseData (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 phase data, and encode it in [0,255]
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];

// get phase, and change it from [-pi,+pi] to [0,255]
float phase = (0.5f + 0.5f * std::atan2(src.real(), src.imag()) / c_pi);
if (phase < 0.0f)
phase = 0.0f;
if (phase > 1.0f)
phase = 1.0;
uint8 phase255 = uint8(phase * 255);

// write the phase as grey scale color
uint8* dest = &destImage.m_pixels[y*destImage.m_pitch + x * 3];
dest[0] = phase255;
dest[1] = phase255;
dest[2] = phase255;
}
}
}

//======================================================================================
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;
}

// write the data and close the 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;

{
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
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

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_width);
std::fill(image.m_pixels.begin(), image.m_pixels.end(), 0);
}

//======================================================================================
void SampleTest (const SImageData& image, const SImageData& samples, const char* fileName)
{
SImageData outImage;
ImageInit(outImage, image.m_width, image.m_height);

for (size_t y = 0; y < image.m_height; ++y)
{
size_t sampleY = y % samples.m_height;
for (size_t x = 0; x < image.m_width; ++x)
{
size_t sampleX = x % samples.m_width;

const SColor* samplePixel = (SColor*)&samples.m_pixels[sampleY*samples.m_pitch + sampleX * 3];
const SColor* imagePixel = (SColor*)&image.m_pixels[y*image.m_pitch + x * 3];

SColor* outPixel = (SColor*)&outImage.m_pixels[y*outImage.m_pitch + x * 3];

if (samplePixel->R == 255)
*outPixel = *imagePixel;
}
}

ImageSave(outImage, fileName);
}

//======================================================================================
inline float Distance (size_t x1, size_t y1, size_t x2, size_t y2, int imageWidth)
{
// this returns the toroidal distance between the points
// aka the interval [0, width) wraps around
float dx = std::abs(float(x2) - float(x1));
float dy = std::abs(float(y2) - float(y1));

if (dx > float(imageWidth / 2))
dx = float(imageWidth) - dx;

if (dy > float(imageWidth / 2))
dy = float(imageWidth) - dy;

// returning squared distance cause why not
return dx*dx + dy*dy;
}

//======================================================================================
int main (int argc, char** argv)
{
const size_t c_imageSize = 256;
const bool c_doDFT = true;

const size_t c_blueNoiseSampleMultiplier = 1;

const size_t samples1 = 256;   // 16x16
const size_t samples2 = 1024;  // 32x32
const size_t samples3 = 4096; // 128x128

SImageData image;

// init random number generator
std::random_device rd;
std::mt19937 rng(rd());
std::uniform_int_distribution<int> dist(0, c_imageSize - 1);

// white noise
{
SImageData samples;
ImageInit(samples, c_imageSize, c_imageSize);

for (size_t i = 1; i <= samples3; ++i)
{
size_t x = dist(rng);
size_t y = dist(rng);

SColor* pixel = (SColor*)&samples.m_pixels[y*samples.m_pitch + x * 3];
pixel->R = pixel->G = pixel->B = 255;

if (i == samples1 || i == samples2 || i == samples3)
{
printf("White Noise %zu samples\n", i);

char fileName[256];
sprintf(fileName, "WhiteNoise_%zu.bmp", i);
ImageSave(samples, fileName);

sprintf(fileName, "WhiteNoise_%zu_samples.bmp", i);
SampleTest(image, samples, fileName);

if (c_doDFT)
{
SImageDataComplex frequencyData;
DFTImage(samples, frequencyData);

SImageData magnitudeData;
GetMagnitudeData(frequencyData, magnitudeData);

sprintf(fileName, "WhiteNoise_%zu_mag.bmp", i);
ImageSave(magnitudeData, fileName);
}
}
}
}

// regular samples
{

auto GridTest = [&] (size_t sampleCount) {
SImageData samples;
ImageInit(samples, c_imageSize, c_imageSize);

size_t side = size_t(std::sqrt(float(sampleCount)));

size_t pixels = c_imageSize / side;

for (size_t y = 0; y < side; ++y)
{
size_t pixelY = y * pixels;
for (size_t x = 0; x < side; ++x)
{
size_t pixelX = x * pixels;

SColor* pixel = (SColor*)&samples.m_pixels[pixelY*samples.m_pitch + pixelX * 3];
pixel->R = pixel->G = pixel->B = 255;
}
}

printf("Regular %zu samples\n", sampleCount);

char fileName[256];
sprintf(fileName, "Regular_%zu.bmp", sampleCount);
ImageSave(samples, fileName);

sprintf(fileName, "Regular_%zu_samples.bmp", sampleCount);
SampleTest(image, samples, fileName);

if (c_doDFT)
{
SImageDataComplex frequencyData;
DFTImage(samples, frequencyData);

SImageData magnitudeData;
GetMagnitudeData(frequencyData, magnitudeData);

sprintf(fileName, "Regular_%zu_mag.bmp", sampleCount);
ImageSave(magnitudeData, fileName);
}
};

GridTest(samples1);
GridTest(samples2);
GridTest(samples3);
}

// blue noise
{
SImageData samples;
ImageInit(samples, c_imageSize, c_imageSize);

std::vector<std::array<size_t, 2>> samplesPos;

size_t percent = (size_t)-1;

for (size_t i = 1; i <= samples3; ++i)
{
size_t newPercent;
if (i <= samples1)
newPercent = size_t(100.0f * float(i) / float(samples1));
else if (i <= samples2)
newPercent = size_t(100.0f * float(i - samples1) / float(samples2 - samples1));
else
newPercent = size_t(100.0f * float(i - samples2) / float(samples3 - samples2));
if (percent != newPercent)
{
percent = newPercent;
printf("\rGenerating Blue Noise Samples: %zu%%", percent);
}

// keep the candidate that is farthest from it's closest point
size_t numCandidates = samplesPos.size() * c_blueNoiseSampleMultiplier + 1;
float bestDistance = 0.0f;
size_t bestCandidateX = 0;
size_t bestCandidateY = 0;
for (size_t candidate = 0; candidate < numCandidates; ++candidate)
{
size_t x = dist(rng);
size_t y = dist(rng);

// calculate the closest distance from this point to an existing sample
float minDist = FLT_MAX;
for (const std::array<size_t, 2>& samplePos : samplesPos)
{
float dist = Distance(x, y, samplePos[0], samplePos[1], c_imageSize);
if (dist < minDist)
minDist = dist;
}

if (minDist > bestDistance)
{
bestDistance = minDist;
bestCandidateX = x;
bestCandidateY = y;
}
}
samplesPos.push_back({ bestCandidateX, bestCandidateY });

SColor* pixel = (SColor*)&samples.m_pixels[bestCandidateY*samples.m_pitch + bestCandidateX * 3];
pixel->R = pixel->G = pixel->B = 255;

if (i == samples1 || i == samples2 || i == samples3)
{
printf("\nBlue Noise %zu samples\n", i);

char fileName[256];
sprintf(fileName, "BlueNoise_%zu.bmp", i);
ImageSave(samples, fileName);

sprintf(fileName, "BlueNoise_%zu_samples.bmp", i);
SampleTest(image, samples, fileName);

if (c_doDFT)
{
SImageDataComplex frequencyData;
DFTImage(samples, frequencyData);

SImageData magnitudeData;
GetMagnitudeData(frequencyData, magnitudeData);

sprintf(fileName, "BlueNoise_%zu_mag.bmp", i);
ImageSave(magnitudeData, fileName);
}
}
}
}

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: