Gaussian blur algorithm is common used in image processing filed. In this tutorial, we will introduce how to use this algorithm to blur an image for beginners.

**Formula of Gaussian distribution**

As to 1-demension data **x**, Gaussian distribution is:

where **σ** is variance of** x**, mean of **x** is **0**.

The distribution is:

As to 2-demenison data **x**, Gaussian distribution will be:

The distribution of it is:

**How to blur an image with Gaussian distribution?**

The key idea is to adjust the center pixel data by its near pixels.

For example, see image below.

The center pixel value is **2**. In order to blur it, we can average its near pixels data to replace its value.

If the **raduis = 1**. the value of center pixel is:** (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) / 8 = 1**.

Blur image above, we will get an image below:

However, averaging all near pixels data is not a good solution, we should assign different weights to them.

**Assign different weights to near pixels**

Seeing image below, the data of each pixel is:

Here, the center pixel data is 25, radius = 1. The coordination of these pixels are:

To assign different weights to near pixels, we can use Gaussian Distribution.

In image proccessing, we will use 2-D gaussian distribution.

Suppose σ = 1.5, and mean = 0

As to center pixel, the coordination is (0, 0) , the probability is:

p(0,0) = 1 / (2πσ^{2}) = 0.0707355

p(-1,1) = 1 / (2πσ^{2}) × e^{-(1+1)/(2σ2)}= 0.0453542

Different probabilities of pixels are:

However, sum all probabilities, we can find:

0.0453542 + 0.0566406 + …… + 0.0453542 = 0.4787147, which is not equivlent to 1. So we have to normalize them.

**Normalize all probabilities**

Normalize all probabilities, we will get weight of each pixel.

As to center pixel, the weight of it will be:

w(0,0) = 0.0707355 / 0.4787147 = 0.147761

The weight of each pixel is below:

Then we can calculate the blured value of each pixel by its weight.

The computed value is:

We blur this image successfully by Gaussian Distribution.

Here we should notice:

To use gaussian distribution to blur an image, these variables will affect the blured effect.

mean: we usually set it to 0.

σ: we usually set it to 0 or 1.5.

radius: it means we will use how many near pixels to blur center pixel.

If radius = 1: number of near pixels is 8.

If radius = 2: the number is 24.

The more bigger of radius , the image will be more blurred. Here is an example.

We will find when radius = 10, the image will be more blurred than radius = 3.