Perceptual uniformity is a highly desirable property of colour spaces or colour difference measures where equal level in colour value difference should result in equal perceptual difference. Designing colour spaces or colour difference measures of perceptual uniformity is a long standing problem in colour science. This has become increasingly important with the growing popularity of high dynamic range (HDR) and wide colour gamut (WCG) cameras, content and displays. We design an efficient testing framework to evaluate perceptual uniformity by subjective just noticeable difference (JND) measurement at a wide range of luminance levels followed by coefficient of variation (CV) computation. We carry out subjective testing on RGB, xyY, L*a*b*, YCbCr, CIECAM02-UCS and ICtCp colour spaces and ΔE2000 metric in ITU-R BT 2020 colour gamut across a wide range of luminance levels (from 0.01 to 500 nits) using a professional HDR/WCG display in a carefully controlled dark testing environment. Our results suggest that on average, the ICtCp space performs the best in the current test, but is still distant from achieving perceptual uniformity.

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.

Perceptual image quality assessment (IQA) and sparse signal representation have recently emerged as high-impact research topics in the field of image processing. Here we make one of the first attempts to incorporate the structural similarity (SSIM) index, a promising IQA measure, into the framework of optimal sparse signal representation and approximation. In particular, we introduce a novel image denoising scheme where a modified orthogonal matching pursuit algorithm is proposed for finding the best sparse coefficient vector in maximum-SSIM sense for a given set of linearly independent atoms. Furthermore, a gradient descent algorithm is developed to achieve SSIM-optimal compromise in combining the input and sparse dictionary reconstructed images. Our experimental results show that the proposed method achieves better SSIM performance and provide better visual quality than least square optimal denoising methods.

We construct a complete metric space $(Y,d_Y)$ of measure-valued images, $\mu : X \to {\cal M}(\mathbb{R}_g)$, where X is the base or pixel space and ${\cal M}(\mathbb{R}_g)$ is the set of probability measures supported on the greyscale range $\mathbb{R}_g$. Such a formalism is well suited to nonlocal (NL) image processing, i.e., the manipulation of the value of an image function $u(x)$ based upon values $u(y_k)$ elsewhere in the image. We then show how the space $(Y,d_Y)$ can be employed with a general model of affine self-similarity of images that includes both same-scale as well as cross-scale similarity. We focus on two particular applications: NL-means denoising (same-scale) and multiparent block fractal image coding (cross-scale). In order to accommodate the latter, a method of fractal transforms is formulated over the metric space $(Y,d_Y)$. Under suitable conditions, a transform $M : Y \to Y$ is contractive, implying the existence of a unique fixed point measure-valued function $\bar \mu = M \bar \mu$. We also show that the pointwise moments of this measure satisfy a set of recursion relations that are generalizations of those satisfied by moments of invariant measures of iterated function systems with probabilities.

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 subtress 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.

The computational complexity of simulated annealing makes it an impractical tool in many applications, particularly for complex, non-local models on very large 2D and 3D domains as desired in many scientific contexts. In particular, it is very difficult to produce large scale structure from a fine, pixellated lattice. Thus a hierarchical approach is intuitively attractive. However, existing approaches are few and limited. Motivated by a current problem in porous media, we develop a hierarchical approach to complex model sampling. In experiments, this approach results in 1-2 orders of magnitude computational gain, and significant gains in convergence as well.

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 propose a simple yet effective fractal-wavelet scheme for edge-preserving smoothing of noisy images. Over the past decade, there has been significant interest in fractal coding for the purpose of image compression. Fractal-wavelet transforms were introduced in an effort to reduce the blockiness and computational complexity that are inherent in fractal image compression. Applications of fractal-based coding to other aspects of image processing, however, have received little attention. The authors proposed a simple yet effective fractal-based image denoising scheme that is applied in the spatial domain of the image. We extend the application of this fractal denoising scheme to the wavelet domain of the image. We find that when the wavelet transform of 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 wavelet transform of the original noise-free image from that of the wavelet transform of the noisy image. The use of the quadtree partitioning scheme for the purpose of fractal-wavelet coding results in 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. The main advantage of the wavelet-based fractal denoising scheme over the standard fractal denoising scheme is that it is computationally less expensive.