Gaussian Blur

A gaussian blur is a blur where the value of every pixel is the weighted average of all pixels within one of the particular pixel, where the weight is greater, the closer you are to the center.

$$B=\sum \left(G\odot P\right)=\text{Tr}(G^{T}P)$$

• where $P$ is the patch around the current pixel
• where $G$ is the Gaussian Kernel Matrix
• The first expression is us the sum of the Elementwise Product approach. The second expression is the equivalent Linear Algebra approach.

Gaussian Kernel Matrix

$$\begin{array}{rl}{G}_{i,j}& =\frac{1}{V\tau }\cdot \exp \left(-\frac{i^2+j^2}{2V}\right)\\ G_3& =\frac{1}{16}\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right]\end{array}$$