Relation between diagonalization of 2D convolution and convolution kernel

Question

Let $c$ be a 2d convolution kernel, we can rewrite the convolution operation as a matrix multiplication: $c\star(\cdot) = A(\cdot)$ where $A$ is a doubly block circulant matrix. This matrix $A$ has a nice diagonalization property that $A = F^*DF$ where $D$ is a block diagonal matrix. My question is, how is $D$ related to $c$?

Answer 1

As shown in here, the eigenvalue of $A$ can be expressed as $F^TcF$, where $c$ is the convolution kernel, and $F$ is the fourier basis. Furthermore, $A$ can be diagonalized by $Q^*AQ$ where $Q=\frac{1}{n}(F\otimes F)$, one can effectively write $A=Qdiag(vec(F^TcF))Q^*$.