Self-similarity of images in the wavelet domain in terms of ℓ2 and structural similarity (SSIM)

  • Authors:

  • D. Glew;Edward R. Vrscay

  • Affiliations:

  • Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada;Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada

  • Venue:
  • ICIAR'12 Proceedings of the 9th international conference on Image Analysis and Recognition - Volume Part I
  • Year:
  • 2012

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Abstract

Images exhibit a high degree of affine self-similarity with respect to the L2 distance. That is, image subblocks are generally well-approximated in L2 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 L2 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 L2 case. In essence, the greater approximability of flat blocks with respect to L2 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.