I have 3 closely related questions regarding 2d convolutions and how they are represented in matrix form.

1. Miming what happens in 1d, I assume the product of a doubly block circulant matrix $A$ by a (vectorized) image $x$ by can be understood as a circular convolution with a kernel $a$. That is $$ Ax= a \star x. $$ Is this right?

2. When differentiating the cost function $\frac12\|Ax-y\|_2^2$ we obtain $A^T(Ax-y)$. If $A$ is doubly block circulant, how can this last expression be mapped to the circular convolution language? What does $A^TAx$ and $A^Ty$ mean in terms of circular convolutions?

3. What's a good reference (book or paper) on this subject? meaning that I want to interpret 1d/2d convolutions and Fourier transforms using linear algebra.


1 Answer 1

  1. Yes, indeed. You may represent the convolution in a Matrix Form. Pay attention that this form assumes the image is column / row stacked into a vector.

  2. If you're after a circular convolution, you may use DFT matrix to diagonalize the matrix and then simplify the equations. Have a look at Circular Convolution Matrix of $ {H}^{H} {H} $.

  3. I am not aware of books on the subject. But I have written many answers on it in this site:


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