Gaussian Blur – Noise Reduction Filter in Image Processing

Gaussian blur is an essential part of many image processing algorithms. It is used to reduce noise in images as well as as a common visual effect in various graphics software.


Online Demo

On our free online demo page you can find a test environment to try out the Gaussian filter. You will find different image analysis algorithms and you can try different parameters.


What is Gaussian blur?

Gaussian blurring is a non-uniform noise reduction low-pass filter (LP filter). The visual effect of this operator is a smooth blurry image.

This filter performs better than other uniform low pass filters such as Average (Box blur) filter.

Gaussian blur filter example with sigma 3
Left – image with some noise, Right – Gaussian blur with sigma = 3.0

How does Gaussian smoothing works?

Gaussian smooth is an essential part of many image analysis algorithms like edge detection and segmentation.

The Gaussian filter is a spatial filter that works by convolving the input image with a 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 such low-pass filter 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 kernel.

One dimensional function

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

Two dimensional function

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

Gaussian function 2D math
Two-dimensional filtering kernel equation (2)

Separable property

In practice, it is better to take advantage of the separable properties of the Gaussian function. This feature allows you to blur the picture in two separate steps.

This saves computing power by using a one-dimensional filtering as two separate operations. As a result, we achieve a fast blur effect by dividing its execution horizontally and vertically.

Features and applications

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

  • Since this is a separable filter, we can save computing power
  • The use of “weighted” masks makes it better for detecting edges than some uniform blurring filters
  • Multiple iterations with a given filter size have the same blur effect as the larger one
  • Useful as a pre-processing step for image size reduction

Discrete Approximations

In many cases it is enough to use an approximation of Gaussian function. Below are examples of popular filtering masks that we often use in computer vision.

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

Note that when converting continuous values to discrete ones, the total sum of the kernel will be different than one. This leads to brightening or darkening of the picture, so in practice we normalize the kernel. We achieve this by dividing each of its elements by the sum of all of them.


Gaussian Filter example code

At the links below you will find different test source code in different languages and technologies:


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Try many DIP algorithms in our free online Computer Vision Studio

References


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