Self-similarity of Images in the Wavelet Domain in Terms of ℓ² and Structural Similarity (SSIM)

Abstract

Images exhibit a high degree of affine self-similarity with respect to the L ² distance. That is, image subblocks are generally well-approximated in L ² by a number of other (affine greyscale modified) image subblocks. This is due, at least in part, to the large number of flatter blocks that comprise such images. These blocks are more easily approximated in the L ² sense, especially when affine greyscale transformations are employed. In this paper, we show that wavelet coefficient quadtrees also demonstrate a high degree of self-similarity under various affine transformations in terms of the ℓ² distance. We also show that the approximability of a wavelet coefficient quadtree is determined by the lowness of its energy (ℓ² norm).

In terms of the structural similarity (SSIM) index, however, the degree of self-similarity of natural images in the pixel domain is not as high as in the L ² case. In essence, the greater approximability of flat blocks with respect to L ² distance is taken into consideration by the SSIM measure. We derive a new form for the SSIM index in terms of wavelet quadtrees and show that wavelet quadtrees are also not as self-similar with respect to SSIM. In an analgous way, the greater approximability of low-energy quadtrees is taken into consideration by the wavelet-based SSIM measure.