Let two functions $I,J:\{0,1,\ldots,n\}\times \{0,1,\ldots,m\} \mapsto \{0,1,\ldots,p\}$ and $h$ a $v\times v$ matrix ($v\ll n,m$) such that $$ J = I * \underbrace{h*\ldots *h}_{p \text{ times}} $$ where $*$ is the convolution operator.
Given $J$ and $h$, is it possible to retrieve $I$ ?
I tried using the Discrete Fourier Transform (DFT), but I obtain some instability since the DFT of $h$, $\hat h$, may contain small terms (especially when we raise it element-wise at the power $p\simeq 25$ ).
To overcome this instability, I was thinking to solve the optimization problem: $$ \min_{X\in M_{np}(\mathbb{R})} \quad \lVert\hat J- X\hat{\times} (\hat h)^p\lVert^2 $$ where $\hat{\times}$ is the element-wise multiplication.
I wonder if I can derive an analytical solution anf more generally if I there would be an other method to solve this problem.