It is possible to filter an image by using a 2d DFT. This will result in periodic boundary conditions. Now, is it possible to use the DFT to filter an image, while maintaining predefined boundary conditions (either Dirichlet or Neumann type).
I could implement a filter directly in the spatial domain. For example, solving the diffusion equation and pinnig pixel values, is equivalent to apply gaussian blur (strength depends on the number of iterations). However, this technique does not generalize very well, because one needs to find a PDE (and discretize it) whose Green's function has the desired behaviour in the frequency domain. Another approach would be to implement a convolution, and not touch the boundary itself but let it affect the surrounding pixels. However, this will becomes computationally expensive for larger images or filter sizes.