Since its introduction in 2004, the structural similarity (SSIM) index has gained widespread popularity as a tool to assess the quality of images and to evaluate the performance of image processing algorithms and systems. There has been also a growing interest of using SSIM as an objective function in optimization problems in a variety of image processing applications. One major issue that could strongly impede the progress of such efforts is the lack of understanding of the mathematical properties of the SSIM measure. For example, some highly desirable properties such as convexity and triangular inequality that are possessed by the mean squared error may not hold. In this paper, we first construct a series of normalized and generalized (vector-valued) metrics based on the important ingredients of SSIM. We then show that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations. The groundwork laid here extends the potentials of SSIM in both theoretical development and practical applications.

Summary. The Schr6der and K6nig iteration schemes to find the zeros of a (polynomial) function g(z) represent generalizations of Newton's method. In both schemes, iteration functions fro(Z) are constructed so that sequences z,+~=f,,(z,) converge locally to a root z* of g(z) as O(Iz,--z*f"). It is well known that attractive cycles, other than the zeros z*, may exist for Newton's method (m = 2). As m increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The K6nig functions K,,(z) appear to minimize such perturbations. In the case of two roots, e.g. g(z)= z 2-1, Cayley's classical result for the basins of attraction of Newton's method is extended for all Kin(z).The existence of chaotic {z,} sequences is also demonstrated for these iteration methods.

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Over the past decade, there has been significant interest in fractal coding for the purpose of image compression. However, applications of fractal-based coding to other aspects of image processing have received little attention. We propose a fractal-based method to enhance and restore a noisy image. If the noisy image is simply fractally coded, a significant amount of the noise is suppressed. However, one can go a step further and estimate the fractal code of the original noise-free image from that of the noisy image, based upon a knowledge (or estimate) of the variance of the noise, assumed to be zero-mean, stationary and Gaussian. The resulting fractal code yields a significantly enhanced and restored representation of the original noisy image. The enhancement is consistent with the human visual system where extra smoothing is performed in flat and low activity regions and a lower degree of smoothing is performed near high frequency components, e.g., edges, of the image. We find that, for significant noise variance (sigma > or = 20), the fractal-based scheme yields results that are generally better than those obtained by the Lee filter which uses a localized first order filtering process similar to fractal schemes. We also show that the Lee filter and the fractal method are closely related.

We consider the use of Banach's fixed point theorem for contraction maps to solve the following class of inverse problems for ordinary differential equations (ODEs): given a function x(t) (which may be an interpolation of a set of experimental data points (x i , t i)), find an ODĖ x(t) = f (x, t) that admits x(t) as an exact or approximate solution, where f is restricted to a class of functional forms, for example affine, quadratic. Borrowing from 'fractal-based' methods of approximation, the method described here represents a systematic procedure for finding optimal vector fields f such that the associated contractive Picard operators T map the target solution x as close as possible to itself.