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科学学科中都是至关重要的。稀疏表示近年来引起了信号处理界的广泛关注。基本模型包括考虑自然信号作为冗余字典中极少数原子的组合而允许稀疏分解。最近的结果表明，学习用于图像表示的过完备非参数字典，而不是使用经典的现成字典，显著改善了许多图像和视频处理任务。本研究旨在开发一个全面的理论、计算和实践框架，用于学习用于众多信号分析任务的稀疏表示。首先，研究重点是学习全局和局部鲁棒图像分类任务的稀疏表示，提出了包含稀疏重建和类识别成分的公式，并在字典学习过程中共同优化。多尺度字典学习和研究一些关键的优化挑战也是这个项目不可或缺的组成部分。然后将学习多尺度稀疏表示的框架扩展到多模态数据。与图像和视频的工作一样，多模态的能量建议考虑重构和判别术语，学习给定数据和给定任务的最佳表示。除了图像、视频和音频，研究的其他信号模态包括张量，这在扩散MRI中至关重要，并带来了为非平坦数据开发稀疏表示的额外挑战。提出的工作还扩展到学习感知，其中结合最优字典的学习，研究了最佳线性感知过程的学习。

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.