Suppose I want to solve a linear inverse problem. In this example we take a convolution with the kernel: $$\frac{1}{(y^2+z^2)^{3/2}}$$ We only take a fixed $z$ for the computation and convolve with respect to $y$. The associated matrix $A$ is dependent on $z$, denoted as $A_z$.

The output vector $b$ is (spatially) located at the same $y$ values, i.e. measurements done along $y$. Furthermore, we impose a nonnegativity constraint on the sources $x$.

We want to estimate the sources $x$ via minimizing the cost function $$||A_zx-b||^2+\lambda\mathcal{R}(x) \,\,\,\,\,\,s.t.\,\,\,x\geq0$$ With $\mathcal{R}(x)$ being the regularizer.

My question: If I do not know $z$ exactly, am I somehow able to both correctly estimate $x$ and $z$ simultaneously? I could impose that the solution $x$ lies on some ball via the regularizer, such that $z$ could (maybe) be estimated correctly. But is this necessary, or can I somehow use less information than knowing the norm of $x$ beforehand?

I also think this may be dependent on the spatial frequencies present in the vector $b$. That is, if $b$ has high spatial frequencies with strong amplitude present, then we must (I think) have a minimum value for $z$, because too large $z$ can not result in $b$ having these high frequencies amplitudes.



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