A Simple Model for Affine Self-similarity of Images and Its Applications

Extensive numerical experiments indicate that images, in general, possess a considerable degree of self-similarity, that is, blocks are well approximated (in the L sense) by a number of other blocks – at the same or different scales – when affine greyscale transformations are employed. This paper outlines a simple model of affine image self-similarity which includes the method of fractal image coding (cross-scale, affine greyscale similarity) and the nonlocal-means denoising method (same-scale, translational similarity) as special cases. Indeed, the general self-similarity of images accounts for the effectiveness of these methods. A complete metric space (Y, dY ) of measure-valued image functions is introduced. Associated with each particular self-similarity model is an operator M : Y → Y . The representation of image functions in this space may be useful in self-similar as well as other nonlocal image processing schemes. Self-similarity is also shown to exist in the wavelet domain, where coefficient quadtrees are approximated by other quadtrees from the same level or higher levels. Finally, the possibility of going beyond L and using other similarity measures to characterize self-similarity is also discussed.