As read on https://en.wikipedia.org/wiki/Kernel_(image_processing), kernels on a certain cell perform the sum of neighbor cells weighted by some constant. Are there similar mathematical tools where the kernel doesn't just perform a sum, but some more complicated operations like multiplying the values of 2 adjacent cells.
For example: The following kernel performs the sum of neighbor cells \begin{bmatrix}0&1&0\\1&0&1\\0&1&0\end{bmatrix} How would a kernel that multiplies the values of the neighbor cells, look like?
If there are such extended type of kernels, how are they called and what software/tools can I use to work with them? Or do I have to implement this nonstandard convolution myself?
Such Kernels describe linear transformations on the input. Since a transformation as you described is not linear, you can not describe it as the application of such a kernel.