How to Do Convolution a 2D Signal (Image) and Result of Convolution of Two 1D Filters?

Question

Assume a 2-D signal (i.e., some image). Load image and assume it to be signal x. Next assume that instead of having a 2-D filter you have two one D filters

$$f_1[n] =\begin{bmatrix}0.25 && 0.5 && 0.25\end{bmatrix}$$ and $$f_2[n]=\begin{bmatrix}0.25 \\ 0.5 \\ 0.25\end{bmatrix}$$ Assume that the convolution

$$f_1[n]* f_2[n] = f_3[n]= \begin{bmatrix}0.0625 && 0.125 && 0.0625\\ 0.125 && 0.25 && 0.125\\0.0625 && 0.125 && 0.0625\end{bmatrix}$$

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Using this information and output at each stage verify that Associative property holds.

My code:

f3 = [0.0625 0.125 0.0625; 0.125 0.25 0.125; 0.0625 0.125 0.0625]
x=imread('img.jpg')
conv2('x, f3)

But this gives error saying x is not a vector. How do I fix this?

Answer 1

You should show that conv2(f3,x) can be implemented in a separable way. In other words, computing x1 = conv2(f1,x), then applying f2 on the results, x12 = conv2(f2,x1) gives the same result. Or f12 = conv2(f2,f1) then x12 = conv2(f12,x).

You should be aware of the correct "dimensions" for f1 and f2: transpose of each other, so that one works on "rows" and the other on columns.

This is shown in the following Matlab code:

dataImage = zeros(25,25);
% Create a delta-like image
dataImage(13,13)=1;
% Create a random-kernel image
dataImage(11:13,11:13)=rand(3,3);
imagesc(dataImage);colormap gray

f1 = [0.25 0.5 0.25]'; f2 = [0.25 0.5 0.25];
disp(f1*f2);

f3 = [0.0625 0.125 0.0625; 0.125 0.25 0.125; 0.0625 0.125 0.0625];
disp(f3);

dataImagef1f2 = conv2(f2,conv2(f1,dataImage));

dataImagef1f2 = conv2(conv2(f2,f1),dataImage);

dataImagef3 = conv2(f3,dataImage);

subplot(2,2,1)
imagesc(dataImage);
xlabel('data')

subplot(2,2,2)
imagesc(dataImagef1f2);
xlabel('(data*f1)*f2')

subplot(2,2,3)
imagesc(dataImagef1f2);
xlabel('(f1*f2)*data')

subplot(2,2,4)
imagesc(dataImagef3);
xlabel('f3')

Suggestions for tutorials:

• The Matlab Image Processing toolbox guide (900+ pages)
• An Introduction to Digital Image Processing with Matlab