Convolution theorem for "weighted" convolution?

Question

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 $0<s<1$ this is no longer possible.

Questions:

1. Does the integral I write above have a formal name?
2. Is there something similar to the convolution theorem for such a "weighted" convolution?
3. Is there any other result that may help me perform edge detection on $g$ from knowing only $f$?

Answer 1

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)

Answer 2

It would generally be of advantage to write your weighting function as a function $\bar{s}$ of $\mathbf{r'}$ and $\mathbf{r}-\mathbf{r'}$. That will reveal the properties and symmetries in the coordinates the convolution is symmetric in.

The next step towards a simplification is to expand your weights into a tensorial product series: $$\bar{s}(\mathbf{r'},\mathbf{r}-\mathbf{r'})=\sum_{n=0}^\infty u_n(\mathbf{r'})\times v_n(\mathbf{r}-\mathbf{r'})$$

If this series has only a finite number of terms then you're done. Each term will contribute to a single convolution integral with weighting factors for the two factor functions if you use the linearity of the integral to split the series up.

If the series does requires infinitely many (or simply too many to handle) terms, then you should try to sort the terms by decreasing contribution and introduce a cutoff that allows you to treat it as a finite series. That is not possible in all cases, but for your discrete space there won't be problems.

You can use symmetry properties of your kernel to simplify the series expansion. This is not too hard to work out, so I won't go into the details.

Good luck!