Amelia Kunze| Applied Mathematics, University of Waterloo
An investigation of the use of gradients in imaging, including best approximation and the Structural Similarity image quality measure
The L^2-based mean squared error (MSE) and its variations continue to be the most widely employed metrics in image processing. This is most probably due to the fact that (1) the MSE is simple to compute and (2) it possesses a number of convenient mathematical properties, including differentiability and convexity. It is well known, however, that these L^2-based measures perform poorly in terms of measuring the visual quality of images. Their failure is partially due to the fact that the L^2 metric does not capture spatial relationships between pixels. This was a motivation for the introduction of the so-called Structural Similarity (SSIM) image quality measure  which, along with is variations, continues to be one of the most effective measure of visual quality. The SSIM index measures the similarity between two images by combining three components of the human visual system--luminance, contrast, and structure. It is our belief that the structure term, which measures the correlation between images, is the most important component of the SSIM.
A considerable portion of this thesis focusses on adapting the L^2 distance for image processing applications. Our first approach involves inserting an intensity-dependent weight function into the integral such that it conforms to generalized Weber's model of perception. We solve the associated best approximation problem and discuss examples in both one- and two-dimensions. Motivated by the success of the SSIM, we also solve the Weberized best approximation problem with an added regularization term involving the correlation.
Another approach we take towards adapting the MSE for image processing involves introducing gradient information into the metric. Specifically, we study the traditional L^2 best approximation problem with an added regularization term involving the L^2 distance between gradients. We use orthonormal functions to solve the best approximation problem in both the continuous and discrete setting. In both cases, we prove that the Fourier coefficients remain optimal provided certain assumptions on the orthonormal basis hold.
Our final best approximation problem to be considered involves maximizing the correlation between gradients. We obtain the relevant stationarity conditions and show that an infinity of solutions exists. A unique solution can be obtained using two assumptions adapted from . We demonstrate that this problem is equivalent to maximizing the entire SSIM function between gradients. During this work, we prove that the discrete derivatives of the DCT and DFT basis functions form an orthogonal set, a result which has not appeared in the literature to the best of our knowledge.
Our study of gradients is not limited to best approximation problems. A second major focus of this thesis concerns the development of gradient-based image quality measures. This was based on the idea that the human visual system may also be sensitive to distortions in the magnitudes and/or direction of variations in greyscale or colour intensities--in other words, their gradients. Indeed, as we show in a persuasive simple example, the use of the L^2 distances between image gradients already yields a significant improvement over the MSE. One naturally wonders whether a measure of the correlation between image gradients could yield even better results--in fact, possibly "better" than the SSIM itself! (We will define what we mean by "better" in this thesis.) For this reason, we pursue many possible forms of a "gradient-based SSIM".
First, however, we must address the question of how to define the correlation between the gradient vectors of two images. We formulate and compare many novel gradient similarity measures. Among those, we justify our selection of a preferred measure which, although simple-minded, we show to be highly correlated with the "rigorous" canonical correlation method. We then present many attempts at incorporating our gradient similarity measure into the SSIM. We finally arrive at a novel gradient-based SSIM, our so-called "gradSSIM1", which we argue does, in fact, improve the SSIM. The novelty of our approach lies in its use of SSIM-dependent exponents, which allow us to seamlessly blend our measure of gradient correlation and the traditional SSIM.
To compare two image quality measures, e.g., the SSIM and our "gradSSIM1", we require use of the LIVE image database . This database contains numerous distorted images, each of which is associated with a single score indicating visual quality. We suggest that these scores be considered as the independent variable, an understanding that does not appear to be have been adopted elsewhere in the literature. This work also necessitated a detailed analysis of the SSIM, including the roles of its individual components and the effect of varying its stability constants. It appears that such analysis has not been performed elsewhere in the literature.
 Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli. Image Quality Assessment: From Error Visibility to Structural Similarity. IEEE Transactions on Image Processing, 13(4):600-612, 2004.
 P. Bendevis and E.R. Vrscay. Structural Similarity-Based Approximation over Orthogonal Bases: Investigating the Use of Individual Components Functions S_k(x,y). In Aurelio Campilho and Mohamed Kamel, editors, Image Analysis and Recognition - 11th International COnference, ICIAR 2014, Vilamoura, Portugal, October 22-24, 2014, Proceedings, Part 1, volume 8814 of Lecture Notes in Computer Science, pages 55-64, 2014.
 H.R. Sheikh, M.F. Sabir, and A.C. Bovik. A Statistical Evaluation of Recent Full Reference Image Quality Assessment Algorithms. IEEE Transactions on Image Processing, 15(11):3440-2451, November 2006.