Learning sparse representations for restoration and classification: Theory, Computations, and Applications in Image, Video, and Multimodal Analysis

有效的信号表示在几乎所有的科学学科中都是至关重要的。稀疏表示最近引起了信号处理界的极大关注。基本模型考虑了自然信号的稀疏分解是由一些冗余字典中极少的原子组成的组合。最近的研究结果表明，学习用于图像表示的过度完备的非参数词典，而不是使用经典的现成词典，显著改善了大量的图像和视频处理任务。该研究旨在为众多信号分析任务建立一个全面的学习稀疏表示的理论、计算和实践框架。首先，研究针对全局和局部稳健图像分类任务的学习稀疏表示，提出了在字典学习过程中联合优化的包含稀疏重构和类别区分成分的公式。多尺度词典学习和调查一些关键的优化挑战也是该项目的组成部分。然后将学习多尺度稀疏表示的框架扩展到多模式数据。与图像和视频的工作一样，提出的多通道能量既考虑了重建术语，也考虑了区分术语，学习了给定数据和给定任务的最佳表示。除了图像、视频和音频之外，所研究的其他信号形式还包括张量，这在扩散磁共振成像中是关键的，并且带来了为非平面数据开发稀疏表示的额外挑战。该工作还扩展到感知学习，其中结合最优词典的学习，研究了最佳线性传感过程的学习。

Efficient signal representation is critical in virtually all disciplines of science. Sparse representations have recently drawn much attention from the signal processing community. The basic model consists of considering that natural signals admit a sparse decomposition as a combination of very few atoms in some redundant dictionary. Recent results have shown that learning overcomplete non-parametric dictionaries for image representation, instead of using classical off-the-shelf ones, significantly improves numerous image and video processing tasks. This research aims at developing a comprehensive theoretical, computational, and practical framework for learning sparse representations for numerous signal analysis tasks.First, the research concentrates on learning sparse representations for global and local robust image classification tasks, proposing formulations with both sparse reconstruction and class discrimination components, jointly optimized during dictionary learning. Multiscale dictionary learning and investigating a number of critical optimization challenges are integral components of this project as well. The framework of learning multiscale sparse representations is then extended to multimodal data. As in the work with images and video, the energies proposed for multimodality consider both reconstructive and discriminative terms, learning the optimal representations both for the given data and the given tasks. In addition to image, video, and audio, other signal modalities studied include tensors, which are critical in diffusion MRI, and bring the additional challenge of developing sparse representations for non-flat data. The proposed work is also extended to learning to sense, where combined with the learning of the optimal dictionaries, the learning of the best linear sensing procedures is studied.