Implementing Gaussian Blur - How to calculate convolution matrix (kernel)

Question

My question is very close to this question: How do I gaussian blur an image without using any in-built gaussian functions?

The answer to this question is very good, but it doesn't give an example of actually calculating a real Gaussian filter kernel. The answer gives an arbitrary kernel and shows how to apply the filter using that kernel but not how to calculate a real kernel itself. I am trying to implement a Gaussian blur in C++ or Matlab from scratch, so I need to know how to calculate the kernel from scratch.

I'd appreciate it if someone could calculate a real Gaussian filter kernel using any small example image matrix.

Answer 1

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