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.