Approximating filtered scale-variant signals

Abstract

-We develop theorems that place limits on the point wise approximation of the responses of filters, both linear shift-invariant (LSI) and linear shift-variant (LSV), to input signals and images that are LSV in the following sense: they can be expressed as the outputs of systems with LSV impulse responses, where the shift-variance is with respect to the filter scale of a single prototype filter. The approximations take the form of LSI approximations to the responses. We develop tight bounds on the approximation errors expressed in terms of filter durations and derivative (Sobolev) norms. Finally, we find application of the developed theory to defoveation of images, deblurring of shift-variant blurs, and shift-variant edge detection. _____________________________________________________Manuscript received _______ ; revised _______. A.C. Bovik is with the Laboratory for Image and Video Engineering