1D Gaussian filter horizontally and vertically [duplicate]

Question

I am trying to generate Gaussian filter that can be applied on the images, but I want to apply it 1D twice: horizontally and vertically. In addition, I want to apply it to each planes separately.

That mean I want to design 1D Gaussian filter to apply it horizontally in Red, Green, Blue component, then I have the same 1D Gaussian filter to apply it vertically in Red, Green, Blue component.

And I think this operation should equal applying 2D Gaussian filter on the original color image.

I am new in Matlab and in image processing filter.

Answer 1

A convolution with 2D Gaussian can be performed with two 1D Gaussians as:

G(x,y)*I=G(x)*(G(y)*I);

You can do this in MATLAB as follows:

img=im2double(imread('cameraman.tif'));

yourFilterSize=[3 5] %3 rows, 5 columns, can be anything

%two 1D Gaussians
g_x=fspecial('gaussian',[1 yourFilterSize(2)]);
g_y=fspecial('gaussian',[yourFilterSize(1) 1]);

%applying 1D gaussian in X-direction to the original image
img_X=imfilter(img,g_x);

%applying 1D gaussian in Y-direction to img_X
img_XY=imfilter(img_X,g_y);  %DONE

%verifying that the result is correct
g_xy=fspecial('gaussian',yourFilterSize);
img_XY2D=imfilter(img,g_xy);
max(max(abs(img_XY-img_XY2D)))  %this should be very small, 
                                %of the order of machine precision
                                %for the result to be correct

Additional reading:

1. This answer on Stackoverflow.com which shows you how to determine whether a given kernel is separable. For example, Gaussian is separable, while the disk kernel is not.
2. Original article for above mentioned issue.

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Why would you prefer two 1D convolutions instead of one 2D convolution (also given in the link 2 mentioned above):

Suppose you have an image of size MxN and a filter of size PxQ then for a 2D convolution, you need ~ M*P*N*Q multiplications and additions. For two 1D filters (of size P and Q), you only need ~ MNP+MNQ = MN(P+Q) operations. Therefore, you get a speedup of the order of PQ/(P+Q).