Applications of Sparse Representation, Low Rank Approximation and Dictionary Learning to Image Processing and Pattern Recognition

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Many problems in image processing and pattern recognition can be modeled by using an energy minimization framework in the context of the variational principle. On the other hand there are other approaches to image processing based on partial differential equations (PDEs) and on the theory of multi-resolution analysis. In the last few years, there has been tremendous progress in the development of the multi-resolution analysis, wavelet transforms, directional filtering and sparse representation. Both the energy minimization and the multi-resolution theories have tremendous successes in image processing and related areas. However, there has been little work on the combination of these theories in a mathematically coherent manner. We plan to study models that can take into account both approaches by forming energy functional that contains the texture component represented by filters, and the cartoon-like components represented the boundary curves. Innovative combination of these models with machine learning methods will lead to new powerful techniques for image processing, pattern recognition and computer vision.