Patch-based nonlocal imaging methods rely on the assumption that natural images contain a large number of mutually similar patches at different locations within the image. Patch similarity is typically assessed through the Euclidean distance of the pixel intensities and therefore depends on the patch size: while large patches guarantee stability with respect to degradations such as noise, the mutual similarity that can be verified between pairs of patches tends to reduce as the patch size grows. Thus, a windowed Euclidean distance is commonly used to balance these two conflicting aspects, assigning lower weights to pixels far from the patch center. We propose patch foveation as an alternative to windowing in nonlocal imaging. Foveation is performed by a spatially variant blur operator, characterized by point-spread functions having bandwidth decreasing with the spatial distance from the patch center. Patch similarity is thus assessed by the Euclidean distance of foveated patches, leading to the concept of foveated self-similarity. In contrast with the conventional windowing, which is only spatially selective and attenuates sharp details and smooth areas in equal way, patch foveation is selective in both space and frequency. In particular, we present an explicit construction of a patch-foveation operator that, given an arbitrary windowing kernel, replaces the corresponding windowing operator providing equivalent attenuation of i.i.d. Gaussian noise, yet giving full weights to flat regions. Examples of this special form of self-similarity are shown for a number of imaging applications, with particular emphasis on image filtering, for which we demonstrate that foveated self-similarity is a more effective regularity assumption than the windowed self-similarity in assessing the patch similarity in nonlocal means denoising..
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