# Implementing Gaussian Blur – How to calculate convolution matrix (kernel)

You can create a Gaussian kernel from scratch as noted in MATLAB documentation of fspecial. Please read the Gaussian kernel creation formula in the algorithms part in that page and follow the code below. The code is to create an m-by-n matrix with sigma = 1.

m = 5; n = 5;
sigma = 1;
[h1, h2] = meshgrid(-(m-1)/2:(m-1)/2, -(n-1)/2:(n-1)/2);
hg = exp(- (h1.^2+h2.^2) / (2*sigma^2));
h = hg ./ sum(hg(:));

h =

0.0030    0.0133    0.0219    0.0133    0.0030
0.0133    0.0596    0.0983    0.0596    0.0133
0.0219    0.0983    0.1621    0.0983    0.0219
0.0133    0.0596    0.0983    0.0596    0.0133
0.0030    0.0133    0.0219    0.0133    0.0030


Observe that this can be done by the built-in fspecial as follows:

fspecial('gaussian', [m n], sigma)
ans =

0.0030    0.0133    0.0219    0.0133    0.0030
0.0133    0.0596    0.0983    0.0596    0.0133
0.0219    0.0983    0.1621    0.0983    0.0219
0.0133    0.0596    0.0983    0.0596    0.0133
0.0030    0.0133    0.0219    0.0133    0.0030


I think it is straightforward to implement this in any language you like.

EDIT: Let me also add the values of h1 and h2 for the given case, since you may be unfamiliar with meshgrid if you code in C++.

h1 =

-2    -1     0     1     2
-2    -1     0     1     2
-2    -1     0     1     2
-2    -1     0     1     2
-2    -1     0     1     2

h2 =

-2    -2    -2    -2    -2
-1    -1    -1    -1    -1
0     0     0     0     0
1     1     1     1     1
2     2     2     2     2