Could you clarify what you are trying to do and please define how you are using the terms 'isotropic kernel', 'convex', and 'monotonic'. A kernel normally implies a set of all elements of one group which are mapped to a identity elements of another group. Are you trying to modify the pixel values of an image based upon a mathematical function with the proposed kernels (i.e. : a custom filter)?
After a little research, I found the following link that describes a method of using a Gaussian isotropic kernel for low pass image filtering (refer to section 3.5). Perhaps this will help.
http://www.imm.dtu.dk/~jmc/02501/exercises/filtering/filtering.pdf
