# Box Blur

If you ever have heard the terms “Box Blur”, “Boxcar Function”, “Box Filter”, “Boxcar Integrator” or other various combinations of those words, you may have thought it was some advanced concept that is hard to understand and hard to implement. If that’s what you thought, prepare to be surprised!

A box filter is nothing more than taking N samples of data (or NxN samples of data, or NxNxN etc) and averaging them! Yes, that is all there is to it ðŸ˜›

In this post, we are going to implement a box blur by averaging pixels.

## 1D Case

For the case of a 1d box filter, let’s say we wanted every data point to be the result of averaging it with it’s two neighbors. It’d be easy enough to program that by just doing it, but let’s look at it a different way. What weight would we need to multiply each of the three values by (the value and it’s two neighbors) to make it come up with the average?

Yep, you guessed it! For every data value, you multiply it and it’s neighbors by 1/3 to come up with the average value. We could easily increase the size of the filter to 5 pixels, and multiply each pixel by 1/5 instead. We could continue the pattern as high as we wanted.

One thing you might notice is that if we want a buffer with all the results, we can’t just alter the source data as we go, because we want the unaltered source values of the data to use those weights with, to get the correct results. Because of that, we need to make a second buffer to put the results of the filtering into.

Believe it or not, that diagram above is a convolution kernel, and how we talked about applying it is how you do convolution in 1d! It just so happens that this convolution kernel averages three pixels into one, which also happens to provide a low pass filter type effect.

Low pass filtering is what is done before down sampling audio data to prevent aliasing (frequencies higher than the sample rate can handle, which makes audio sound bad).

Surprise… blurring can also be seen as low pass filtering, which is something you can do before scaling an image down in size, to prevent aliasing.

## 2D Case

The 2d case isn’t much more difficult to understand than the 1d case. Instead of only averaging on one axis, we average on two instead:

Something interesting to note is that you can either use this 3×3 2d convolution kernel, or, you could apply the 1d convolution kernel described above on the X axis and then the Y axis. The methods are mathematically equivalent.

Using the 2d convolution kernel would result in 9 multiplications per pixel, but if going with the separated axis X and then Y 1d kernel, you’d only end up doing 6 multiplications per pixel (3 multiplications per axis). In general, if you have a seperable 2d convolution kernel (meaning that you can break it into a per axis 1d convolution), you will end up doing N^2 multiplications when using the 2d kernel, versus N*2 multiplications when using the 1d kernels. You can see that this would add up quickly in favor of using 1d kernels, but unfortunately not all kernels are separable.

Doing two passes does come at a cost though. Since you have to use a temporary buffer for each pass, you end up having to create two temporary buffers instead of one.

You can build 2d kernels from 1d kernels by multiplying them as a row vector, by a column vector. For instance, you can see how multiplying the (1/3,1/3,1/3) kernel by itself as a column vector would create the 2nd kernel, that is 3×3 and has 1/9 in every spot.

The resulting 3×3 matrix is called an outer product, or a tensor product. Something interesting to note is that you don’t have to do the same operation on each axis!

## Examples

Here are some examples of box blurring with different values, using the sample code provided below.

The source image:

Now blurred by a 10×10 box car convolution kernel:

Now blurred by a 100×10 box car convolution kernel:

You can find a shadertoy implementation of box blurring here: Shadertoy:DF Box Blur

## Code

Here’s the code I used to blur the example images above:

```#define _CRT_SECURE_NO_WARNINGS

#include <stdio.h>
#include <stdint.h>
#include <array>
#include <vector>
#include <functional>
#include <windows.h>  // for bitmap headers.  Sorry non windows people!

typedef uint8_t uint8;

const float c_pi = 3.14159265359f;

struct SImageData
{
SImageData()
: m_width(0)
, m_height(0)
{ }

long m_width;
long m_height;
long m_pitch;
std::vector<uint8> m_pixels;
};

void WaitForEnter ()
{
printf("Press Enter to quit");
fflush(stdin);
getchar();
}

bool LoadImage (const char *fileName, SImageData& imageData)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "rb");
if (!file)
return false;

// read the headers if we can
header.bfType != 0x4D42 || infoHeader.biBitCount != 24)
{
fclose(file);
return false;
}

// read in our pixel data if we can. Note that it's in BGR order, and width is padded to the next power of 4
if (fread(&imageData.m_pixels[0], imageData.m_pixels.size(), 1, file) != 1)
{
fclose(file);
return false;
}

imageData.m_pitch = imageData.m_width*3;
if (imageData.m_pitch & 3)
{
imageData.m_pitch &= ~3;
imageData.m_pitch += 4;
}

fclose(file);
return true;
}

bool SaveImage (const char *fileName, const SImageData &image)
{
// open the file if we can
FILE *file;
file = fopen(fileName, "wb");
if (!file)
return false;

// make the header info

// write the data and close the file
fwrite(&image.m_pixels[0], infoHeader.biSizeImage, 1, file);
fclose(file);
return true;
}

const uint8* GetPixelOrBlack (const SImageData& image, int x, int y)
{
static const uint8 black[3] = { 0, 0, 0 };
if (x < 0 || x >= image.m_width ||
y < 0 || y >= image.m_height)
{
return black;
}

return &image.m_pixels[(y * image.m_pitch) + x * 3];
}

void BlurImage (const SImageData& srcImage, SImageData &destImage, unsigned int xblur, unsigned int yblur)
{
// allocate space for copying the image for destImage and tmpImage
destImage.m_width = srcImage.m_width;
destImage.m_height = srcImage.m_height;
destImage.m_pitch = srcImage.m_pitch;
destImage.m_pixels.resize(destImage.m_height * destImage.m_pitch);

SImageData tmpImage;
tmpImage.m_width = srcImage.m_width;
tmpImage.m_height = srcImage.m_height;
tmpImage.m_pitch = srcImage.m_pitch;
tmpImage.m_pixels.resize(tmpImage.m_height * tmpImage.m_pitch);

// horizontal blur from srcImage into tmpImage
{
float weight = 1.0f / float(xblur);
int half = xblur / 2;
for (int y = 0; y < tmpImage.m_height; ++y)
{
for (int x = 0; x < tmpImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (int i = -half; i <= half; ++i)
{
const uint8 *pixel = GetPixelOrBlack(srcImage, x + i, y);
blurredPixel[0] += float(pixel[0]) * weight;
blurredPixel[1] += float(pixel[1]) * weight;
blurredPixel[2] += float(pixel[2]) * weight;
}

uint8 *destPixel = &tmpImage.m_pixels[y * tmpImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}

// vertical blur from tmpImage into destImage
{
float weight = 1.0f / float(yblur);
int half = yblur / 2;

for (int y = 0; y < destImage.m_height; ++y)
{
for (int x = 0; x < destImage.m_width; ++x)
{
std::array<float, 3> blurredPixel = { 0.0f, 0.0f, 0.0f };
for (int i = -half; i <= half; ++i)
{
const uint8 *pixel = GetPixelOrBlack(tmpImage, x, y + i);
blurredPixel[0] += float(pixel[0]) * weight;
blurredPixel[1] += float(pixel[1]) * weight;
blurredPixel[2] += float(pixel[2]) * weight;
}

uint8 *destPixel = &destImage.m_pixels[y * destImage.m_pitch + x * 3];

destPixel[0] = uint8(blurredPixel[0]);
destPixel[1] = uint8(blurredPixel[1]);
destPixel[2] = uint8(blurredPixel[2]);
}
}
}
}

int main (int argc, char **argv)
{
int xblur, yblur;

bool showUsage = argc < 5 ||
(sscanf(argv[3], "%i", &xblur) != 1) ||
(sscanf(argv[4], "%i", &yblur) != 1);

char *srcFileName = argv[1];
char *destFileName = argv[2];

if (showUsage)
{
printf("Usage: <source> <dest> <xblur> <yblur>\n\n");
WaitForEnter();
return 1;
}

// make sure blur size is odd
xblur = xblur | 1;
yblur = yblur | 1;

printf("Attempting to blur a 24 bit image.\n");
printf("  Source=%s\n  Dest=%s\n  blur=[%d,%d]\n\n", srcFileName, destFileName, xblur, yblur);

SImageData srcImage;
{
SImageData destImage;
BlurImage(srcImage, destImage, xblur, yblur);
if (SaveImage(destFileName, destImage))
printf("Blurred image saved as %s\n", destFileName);
else
{
printf("Could not save blurred image as %s\n", destFileName);
WaitForEnter();
return 1;
}
}
else
{
printf("could not read 24 bit bmp file %s\n\n", srcFileName);
WaitForEnter();
return 1;
}
return 0;
}
```

## Next Up

Next up will be a Gaussian blur, and I’m nearly done w/ that post but wanted to make this one first as an introductory step!

Before we get there, I wanted to mention that if you do multiple box blurs in a row, it will start to approach Gaussian blurring. I’ve heard that three blurs in a row will make it basically indistinguishable from a Gaussian blur.