Computing convolutions on mesh-like structures

Although the computation of two dimensional convolutions is one of the basic computational tools for the processing of digitized images, and although it is general knowledge that convolutions can be efficiently computed sequentially with the aid of Fourier transforms, previous work on the parallel computation of convolutions has not been based on Fourier transforms. This is probably due to the fact that the fast Fourier transform cannot be implemented efficiently on simple structures such as the mesh, the mesh with broadcasting the mesh of trees, or the pyramid computer. It is shown that it makes sense to use the Fourier transform, even on such simpler structures, obtaining nearly optimal algorithms for the computation of convolutions on the parallel structures listed above. As an application, an algorithm is given that computes the digitized configuration space of a robot with translation only in the plane.<>