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.

Measuring information gain for frequency-encoded super-resolution MRI.

Super-resolution (SR)-based methods that implement multiple data sets related by spatial translations in the frequency-encoding (FE) direction have recently been proposed for resolution enhancement in MRI. This approach, however, was initially received with controversy. It was suggested that when the shifts are applied in the FE direction, no new information was acquired after the first image. Recent developments suggest, however, that shifting the object between image acquisitions can introduce new information to each data set. For this reason, SRMRI may be possible in the FE direction. In this article, we point out that the presence of new information in each acquisition is not sufficient for an SR algorithm to be practical. Indeed, there are situations where the amount of new information is relatively small and possibly not even measurable in the presence of noise. We explore the question of how much new information can be present in each acquisition for FE SRMRI. In particular, we investigate the extent to which the effect of the spatial shift - applied before the object is imaged - can be undone using simple image-processing techniques. Visual comparisons and numerical measures are used to characterize the amount of new information that is acquired in each data set. It is shown that since the amount of new information can be relatively small, each image can be approximated by applying simple signal-processing techniques to a single data set. Ultimately, our research suggests that little progress may be possible by using this approach to perform resolution enhancement in the FE direction.

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 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.

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.