To ensure smoothly space-variant filtering, we overlap neighboring patches, process each patch individually, and interpolate the resulting patch vectors. This patching operation may look ad hoc: Shouldn't we apply a spatially varying operator instead? In the case of a local-stationary filtering operation, the interpolation of filters and the interpolation of data patches approximately yield the same result, if the interpolation weights vary slowly with respect to the filter size. Under such circumstances, the sequence of convolution and interpolation can be commuted. Similar commutations should be permissible for other stationary operators.
I will demonstrate the equivalence of data and filter interpolation on a simple deconvolution example. The local-stationary data has two stationary components. We find a normalized, smoothly varying, weighting function that isolates the two stationary components. If we define the deconvolution filters for the two stationary components as and , we can compute two filtered images
If, however, is slowly varying compared to the size of the filters f1 and f2, the weight function can be assumed to be locally constant. Under this assumption, multiplication and convolution commute. Now, the filter - not the data - is interpolated:
In terms of the two scales involved, the filter f varies smoothly on every scale, while the filter components f1 and f2 are locally constant.