# The Matrix Form of a 2D Circular Convolution

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.

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 $${H}^{H} {H}$$.