Structural Similarity-Based Approximation of Signals and Images Using Orthogonal Bases

The structural similarity (SSIM) index has been shown to be an useful tool in a wide variety of applications that involve the assessment of image quality and similarity. However, in-depth studies are still lacking on how to incorporate it for signal representation and approximation problems, where minimal mean squared error is still the dominant optimization criterion. Here we examine the problem of best approximation of signals and images by maximizing the SSIM between them. In the case of a decomposition of a signal in terms of an orthonormal basis, the optimal SSIM-based coefficients are determined with a surprisingly simple approach, namely, a scaling of the optimal L2 coefficients. We then examine a very simple algorithm to maximize SSIM with a constrained number of basis functions. The algorithm is applied to the DCT approximation of images.