Edward R. Vrscay

On the Mathematical Properties of the Structural Similarity Index

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.

Fractal image denoising

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.

Extraneous fixed points, basin boundaries and chaotic dynamics for Schro¨der and Ko¨nig rational iteration functions

SummaryThe Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newtons method. In both schemes, iteration functionsfm(z) are constructed so that sequenceszn+1=fm(zn) converge locally to a rootz* ofg(z) asO(|zn−z*|m). It is well known that attractive cycles, other than the zerosz*, may exist for Newtons method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsKm(z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z2−1, Cayleys classical result for the basins of attraction of Newtons method is extended for allKm(z). The existence of chaotic {zn} sequences is also demonstrated for these iteration methods.

Iterated fuzzy set systems : a new approach to the inverse problem for fractals and other sets

Abstract Images with grey or colour levels admit a natural representation in terms of fuzzy sets, but without the usual probabilistic interpretation of the latter. We introduce a fuzzy set approach which incorporates, in part, the technique of iterated function systems (IFS) for the construction, analysis, and/or approximation of typically fractal sets and images. The method represents a significant departure from IFS, especially in the interpretation of the resulting image. The introduction of “grey-level maps,” ϑ i : [0, 1] → [0, 1] associated with the contractive maps w i of the IFS affords much greater flexibility in the generation of images as well as in the inverse problem.

Fractal-wavelet image denoising revisited

The essence of fractal image denoising is to predict the fractal code of a noiseless image from its noisy observation. From the predicted fractal code, one can generate an estimate of the original image. We show how well fractal-wavelet denoising predicts parent wavelet subetres of the noiseless image. The performance of various fractal-wavelet denoising schemes (e.g., fixed partitioning, quadtree partitioning) is compared to that of some standard wavelet thresholding methods. We also examine the use of cycle spinning in fractal-based image denoising for the purpose enhancing the denoised estimates. Our experimental results show that these fractal-based image denoising methods are quite competitive with standard wavelet thresholding methods for image denoising. Finally, we compare the performance of the pixel- and wavelet-based fractal denoising schemes

Iterated function systems: theory, applications and the inverse problem

The basic theory and properties of Iterated Function Systems are given, comparing the original approaches of Hutchinson [47] and Barnsley and Demko [8]. Some examples and applications are discussed, along with computational aspects. A generalized recurrent IFS is introduced, with suitably constructed measure space, from which the existence of an invariant measure follows. This yields a Collage Theorem for Measures on generalized RIFS. Finally, the inverse problem of fractal/measure construction is discussed. Some recent applications of Genetic Algorithms as a stochastic optimization method for (i) moment matching and (ii) Collage Theorem for measures are reported.

SSIM-inspired image restoration using sparse representation

Recently, sparse representation based methods have proven to be successful towards solving image restoration problems. The objective of these methods is to use sparsity prior of the underlying signal in terms of some dictionary and achieve optimal performance in terms of mean-squared error, a metric that has been widely criticized in the literature due to its poor performance as a visual quality predictor. In this work, we make one of the first attempts to employ structural similarity (SSIM) index, a more accurate perceptual image measure, by incorporating it into the framework of sparse signal representation and approximation. Specifically, the proposed optimization problem solves for coefficients with minimum ℒ0 norm and maximum SSIM index value. Furthermore, a gradient descent algorithm is developed to achieve SSIM-optimal compromise in combining the input and sparse dictionary reconstructed images. We demonstrate the performance of the proposed method by using image denoising and super-resolution methods as examples. Our experimental results show that the proposed SSIM-based sparse representation algorithm achieves better SSIM performance and better visual quality than the corresponding least square-based method.

The Use of Residuals in Image Denoising

State-of-the-art image denoising algorithms attempt to recover natural image signals from their noisy observations, such that the statistics of the denoised image follow the statistical regularities of natural images. One aspect generally missing in these approaches is that the properties of the residual image (defined as the difference between the noisy observation and the denoised image) have not been well exploited. Here we demonstrate the usefulness of residual images in image denoising. In particular, we show that well-known full-reference image quality measures such as the mean-squared-error and the structural similarity index can be estimated from the residual image without the reference image. We also propose a procedure that has the potential to enhance the image quality of given image denoising algorithms.

A generalized class of fractal-wavelet transforms for image representation and compression

The action of an affine fractal transform or (local) lterated Function System with gray-level Maps (IFSM) on a function f(x) induces a simple mapping on its expansion coefficients, cij, in the Haar wavelet basis. This is the basis of the discrete fractal-wavelet transform, where subtrees of the wavelet coefficient tree are scaled and copied to lower subtrees. Such transforms, which we shall also refer to as IFS on wavelet coefficients (IFSW), were introduced into image processing with other (compactly supported) wavelet basis sets in an attempt to remove the blocking artifacts that plague standard IFS block-encoding algorithms. In this paper a set of generalized 2-D fractal-wavelet transforms is introduced. Their primary difference from usual IFSW transforms lies in treating “horizontal,” “vertical” and “diagonal” quadtrees independently. This approach may seem expensive in terms of coding. However, the added flexibility provided by this method, resulting in a marked improvement in accuracy and low degradation with respect to quantization, makes it quite tractable for image compression. As in the one-dimensional case, the IFSW transforms are equivalent to recurrent IFSM with condensation functions. The net result of an affine IFSW is an extrapolation of high-frequency wavelet coefficients which grow or decay geometrically, according to the magnitudes of fractal scaling parameters αij. This provides a connection between the αij and the regularity/irregularity properties of regions of the image. IFSW extrapolation also makes possible “fractal zooming.” The results of computations, including some simple compression methods, are also presented.

Can One Break the “Collage Barrier” in Fractal Image Coding?

Most fractal image coding methods rely upon collage coding, that is, finding a fractal transform operator T c that sends a target image u as close as possible to itself. The fixed point attractor ū c of T c is generally a good approximation to u. However, it is well known that collage coding does not necessarily yield an optimal attractor, i.e., one for which the approximation error u − ū c is minimized with respect to variations in the fractal transform parameters. A number of studies have employed the “collage attractor” ū c as a starting point from which to obtain better approximations to u. In this paper, we show that attractors ū are differentiate functions of the (affine) fractal parameters. This allows us to use gradient descent methods that search for optimal attractors in fractal parameter space, i.e., local minima of the approximation error u ℒ ū. We report on results of corresponding computer experiments and compare them with those obtained by related (nondifferentiable) methods based on the simplex hill climbing and annealing approaches.