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

图像处理和模式识别中的许多问题都可以在变分原理的背景下使用能量最小化框架来建模。另一方面，还有基于偏微分方程和多分辨率分析理论的其他图像处理方法。近年来，在多分辨率分析、小波变换、方向滤波和稀疏表示等方面取得了巨大进展。能量最小化理论和多分辨率理论在图像处理及相关领域都取得了巨大的成功。然而，很少有人以数学上连贯的方式将这些理论结合起来。我们计划通过形成能量函数来研究可以同时考虑这两种方法的模型，其中包含由过滤器表示的纹理成分，以及代表边界曲线的卡通成分。这些模型与机器学习方法的创新结合将为图像处理、模式识别和计算机视觉带来新的强大技术。

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.