Gaussian Blur – Noise Reduction Filter


The goal of this tutorial is to reveal major details about Gaussian blur and how to use it for image processing and machine vision tasks. FivekoGFX implements a WebGL version of Gaussian filter and uses GPU power by OpenGL shaders (GLSL). Therefore it is suitable for direct use into HTML5 web apps and gives good performance.

What is Gaussian blur?

Gaussian blur is a non-linear noise reduction low-pass filter (LP filter). The visual effect of this filter is a smooth blurry image. This means it reduce intensity variations between adjacent pixels. Gaussian filter performs better than other uniform low pass filters like the Mean filter. Therefore, it is a preferred pre-processing step in machine vision and object recognition algorithms.

flowers gaussian blur filter example
Gaussian blur with sigma = 3.0

How does Gaussian smoothing works?

The Gaussian filter works by convolving the input image with a Gaussian kernel. This process performs a weighted average of the current pixel’s neighborhoods in a way that distant pixels receive lower weight than these at the center. The result of this is a blurry image with better edges than other uniform smoothing algorithms. The math equations below show how to calculate the proper weights of the Gaussian kernel.

Gaussian function 1D

gaussian-blur-1d-shape
One-dimensional Gaussian function
Gaussian function 1D math
Gaussian blur one-dimensional kernel equation (1)

Gaussian function 2D

The equation below (2) shows how to calculate two-dimensional Gaussian function:

Gaussian function 2D math
Gaussian filter two-dimensional kernel equation (2)

Separable Gaussian blur filter

In practice it is better to take advantage of the Gaussian function separable properties. This property allows blur execution in two separate steps. At this way we apply a one dimensional kernel instead of the 2D Gaussian filter. As a result, we achieve a fast blur effect by dividing its execution horizontally and vertically.

To calculate a Gaussian filter parameters, we use the equations mentioned above . If processing uses a separable Gaussian blur then for the kernel math use the first equation. Note that when converting continuous values to discrete ones, the total Gaussian kernel sum will be different than one. This leads to brightening or darkening of the image, so in practice we normalize the kernel. We achieve this by dividing each of its elements by the sum of all of them.

Features

We can summarize some of the Gaussian’s filter features:

  • It is a separable filter – can use it as a separable 1D kernel and process rows and columns separably
  • The use of ‘weighted average’ masks makes it better for edge detection than some uniform blurring filters
  • Multiple iterations with same gaussian filter to an image has the same effect as applying a single, larger gaussian blur
  • Useful as a pre-processing step for image size reduction

Gaussian Discrete Approximations

In many cases it is enough to use an approximation of Gaussian function. Below are listed several famous masks that are frequently used in computer vision. The pictures show the kernel values along with the corresponding scale factor as well as the corresponding one-dimensional separable vectors.

Separable Filter with Size 3×3

$\displaystyle{K}=\frac{1}{{16}}{\left[\begin{matrix}{1}&{2}&{1}\\{2}&{4}&{2}\\{1}&{2}&{1}\end{matrix}\right]}={\left[\begin{matrix}{1}&{2}&{1}\end{matrix}\right]}\frac{1}{{4}}\ast{\left[\begin{matrix}{1}\\{2}\\{1}\end{matrix}\right]}\frac{1}{{4}}$
Discrete approximation of Gaussian filter with kernel size 3×3

Separable Filter with Size 5×5

$\displaystyle{K}=\frac{1}{{256}}{\left[\begin{matrix}{1}&{4}&{6}&{4}&{1}\\{4}&{16}&{24}&{16}&{4}\\{6}&{24}&{36}&{24}&{6}\\{4}&{16}&{24}&{16}&{4}\\{1}&{4}&{6}&{4}&{1}\end{matrix}\right]}=\frac{1}{{16}}{\left[\begin{matrix}{1}&{4}&{6}&{4}&{1}\end{matrix}\right]}\ast\frac{1}{{16}}{\left[\begin{matrix}{1}\\{4}\\{6}\\{4}\\{1}\end{matrix}\right]}$
Discrete approximation of Gaussian filter with kernel size 5×5

How to apply fast Gaussian function by FavkoGFX?

Now after we defined some of the Gaussian filter kernels it’s time to pick up one and create a blurry image. The example below uses one of the masks above to smooth the image quickly with a simple separable convolution. The code uses our web based graphics library FivekoGFX. Therefore you can combine it with your own HTML5 scripts and do processing directly into your web browser.

var fivekogfx = new FivekoGFX();
// Load source image from Canvas object
fivekogfx.load(canvas);
// Perform separable convolution with 1D Gaussian function 
// In this case we use 5x5 Gaussian kernel size 
fivekogfx.conv1d([1,4, 6, 4, 1]);
// Display the resulting image
fivekogfx.draw(canvas);

 

Gaussian Filter source code

Gaussian kernel calculation

As a first step, we need to fill in the Gaussian kernel values using the sigma input parameter. The kernel size is calculated using the following code snippet as described into Gaussian function 1D:


const GAUSSKERN = 2.0;
var dim = parseInt(Math.max(3.0, GAUSSKERN * sigma + 1.0));

Next we calculate the kernel values by the following JavaScript code snippet.


function makeGaussKernel(sigma){
	const GAUSSKERN = 2.0;
	var dim = parseInt(Math.max(3.0, GAUSSKERN * sigma + 1.0));
	var sqrtSigmaPi2 = Math.sqrt(Math.PI*2.0)*sigma;
	var sum = 0.0;

	if (dim % 2 == 0)
		dim++;

	var s2 = 2.0*sigma * sigma;
	var i, j, c = parseInt(dim / 2);
	var kernel = new Float32Array(dim);
	for (j = 0, i = -c; i <= c; i++, j++) 
	{
		kernel[j] = Math.exp(-(i*i)/(s2)) / sqrtSigmaPi2;
		sum += kernel[j];
	}

	// Normalize the gaussian kernel to prevent image drakening/brightening
	for (i = 0; i < dim; i++) 
	{
		kernel[i] /= sum;
	}
	return kernel;
}

Note that we normalize the kernel values to prevent image darkening/brightening at convolution stage.

Gausian blur convolution

At this stage the image is processed by the already calculated blurring kernel. For better performance we take advantage of Gaussian’s separable property. Therefore we perform the Gaussian convolution in two steps – once for rows and once for columns.


var kernel = makeGaussKernel(sigma);
var d = pixels.data;
var w = pixels.width;
var h = pixels.height;
var buff = new Float32Array(w*h); 
var i, j, k, ch = 0; // Color channel (0 - Red, 1 - Green ...)
var kl = kernel.length;
var c = parseInt(kernel.length / 2);

// First step process columns
for (j = 0; j < h; j++) 
{
	var hw = j*w;
	for (i = 0; i < w; i++)
	{
		var p = 0;
		for (k = 0; k < kl; k++)
		{
			var col = i + (k - c);
			col = (col < 0) ? 0 : ((col >= w) ? w - 1 : col);
			p += d[(hw + col)*4 + ch]*kernel[k];
		}
		buff[hw + i] = p;
	}
}

We process the rows in a similar manner as in the next code segment:


// Second step process rows
for (j = 0; j < h; j++) 
{
	for (i = 0; i < w; i++)
	{
		var p = 0;
		for (k = 0; k < kl; k++)
		{
			var row = j + (k - c);
			row = (row < 0) ? 0 : ((row >= h) ? h - 1 : row);
			
			p += buff[(row*w + i)]*kernel[k];
		}
		var off = (j*w + i)*4;
		(colors == 3) ? d[off + ch] = p : d[off] = d[off + 1] = d[off + 2] = p;
	}
}

Gaussian Blur by OpenGL

The OpenGL GLSL (fragment shader) source code below represents a single direction Gaussian blur algorithm. The uniform parameter u_direction selects trace direction – rows or columns. The variable u_kernel holds the normalized kernel values for convolution with Gaussian function.


precision mediump float;

// our texture
uniform sampler2D u_image;

#define KERNEL_SIZE 15
uniform vec2 u_textureSize;
uniform int u_direction;
uniform float u_kernel[KERNEL_SIZE];
void main() {
	vec2 textCoord = gl_FragCoord.xy / u_textureSize;
	vec2 onePixel = ((u_direction == 0) ? vec2(1.0, 0.0) : vec2(0.0, 1.0)) / u_textureSize;
	vec4 meanColor = vec4(0);
	int ms = KERNEL_SIZE / 2;
	for (int i = 0; i < KERNEL_SIZE; i++)
	{
		meanColor += texture2D(u_image, textCoord  + onePixel*vec2(i - ms))*u_kernel[i];
	}
	gl_FragColor = meanColor;
}

Visit FivekoGFX on GitHub for more image processing examples


Gaussian Filter Demo

 
Sigma: 3

References


Related Articles