Shearlet-based Analysis of Image Inpainting and Convolutional Framelets
Rodriguez Ayllon, Jose Pedro
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The main part of my dissertation deals with image inpainting - a classical problem in image analysis that I analyze from the point of view of microlocal analysis and the theory of sparse approximation. My most important result provides a new set of theoretical performance guarantees for the exact recovery of missing data in images where the information is dominated by curvilinear singularities. In fact, my study shows that a shearlet-based approach for the recovery of missing curvilinear edges in images is provably superior to methods based on conventional wavelets in a precises sense and gives a quantitative assessment on the size of the region that can reliably recovered. As a consequence, this result offers the theoretical underpinning for algorithms based on directional multiscale methods such as shearlets in applications to image inpainting. The arguments in my proofs rely on a new application of the microlocal properties of shearlets and techniques from oscillatory integrals that are inspired in part by a seminal paper by Donoho and Kutyniok, who first introduced methods from microlocal analysis in combination with ideas from sparse representations for problems of image analysis. The second part of my dissertation is a new study of convolutional framelets - a method recently introduced to provide a mathematical framework for the patch-based analysis of images - using tools from tensor analysis. This method gives an alternative approach to analyze framelets and a deeper insight into the mathematical properties of convolutional framelets. The first part of the dissertation follows rather closely some material published by the author in his first journal paper.