Convolution theorem for "weighted" convolution? Consider what I call a "weighted" convolution of a two-dimensional signal (image) with itself:

$f({\bf r}) = \int d {\bf r}' \,  g({\bf r}') s({\bf r},{\bf r}') g({\bf r}-{\bf r}')$

where $s$ is a weighting function, with $s({\bf r},{\bf r}') = s({\bf r}+{\bf r}',-{\bf r}')$. My ultimate goal is to perform edge detection on $g$, i.e., find discontinuities in $g$, from knowing only $f$.

If $s=1$, I believe I can obtain $g$ from $f$ using the convolution theorem and a deconvolution scheme, but with a general $0&lt;s&lt;1$ this is no longer possible.

Questions:


Does the integral I write above have a formal name?
Is there something similar to the convolution theorem for such a "weighted" convolution?
Is there any other result that may help me perform edge detection on $g$ from knowing only $f$?
1) The terms autoconvolution or selfconvolution can be found in the literature, without the kernel $s$. So you would probably have a weighed, kernel or windowed self-convolution.
However, i foresee two potential caveats on your formulation. First, i do not understand why you require a two-dimensional periodic signal (and nothing on the kernel $s$?). This could result in infinite values of your integrals. Second, don't you assume some kind of symmetry in $s(r,r′)$?

2) Suppose you can solve your system with $s=1$, then with appropriate symmetry on $s$ you may be able to factorize $s$ and slit it over $g(r')$ and $g(r-r')$. Unfortunately the convolution theorem is likely to give you a family of solutions $g$ knowing $f$ only.

3) You may find useful to explore a multiscale form of your weighted self-correlation operator.

Affine invariant pattern recognition using Multiscale Autoconvolution
Rahtu E., Salo M., Heikkilä J.
IEEE Trans Pattern Anal Mach Intell. 2005 Jun;27(6):908-18.

An iterative deautoconvolution algorithm for nonnegative functions
Kerkil Choi and Aaron D Lanterman 
2005 Inverse Problems 21 981

Autoconvolution and panorama: Augmenting second-order signal analysis 
Douglas, S.C. and  Mandic, D.P., 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)