5
$\begingroup$

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.

$\endgroup$

1 Answer 1

5
$\begingroup$
  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:

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.