Sparse and Redundant Representations_From Theory to Applications...

This textbook introduces sparse and redundant representations with a focus on applications in signal and image processing. The theoretical and numerical foundations are tackled before the applications are discussed. Mathematical modeling for signal sources is discussed along with how to use the proper model for tasks such as denoising, restoration, separation, interpolation and extrapolation, compression, sampling, analysis and synthesis, detection, recognition, and more. The presentation is elegant and engaging. Sparse and Redundant Representations is intended for graduate students in applied mathematics and electrical engineering, as well as applied mathematicians, engineers, and researchers who are active in the fields of signal and image processing. * Introduces theoretical and numerical foundations before tackling applications * Discusses how to use the proper model for various situations * Introduces sparse and redundant representations * Focuses on applications in signal and image processing The field of sparse and redundant representation modeling has gone through a major revolution in the past two decades. This started with a series of algorithms for approximating the sparsest solutions of linear systems of equations, later to be followed by surprising theoretical results that guarantee these algorithms’ performance. With these contributions in place, major barriers in making this model practical and applicable were removed, and sparsity and redundancy became central, leading to state-of-the-art results in various disciplines. One of the main beneficiaries of this progress is the field of image processing, where this model has been shown to lead to unprecedented performance in various applications. This book provides a comprehensive view of the topic of sparse and redundant representation modeling, and its use in signal and image processing. It offers a systematic and ordered exposure to the theoretical foundations of this data model, the numerical aspects of the involved algorithms, and the signal and image processing applications that benefit from these advancements. The book is well-written, presenting clearly the flow of the ideas that brought this field of research to its current achievements. It avoids a succession of theorems and proofs by providing an informal description of the analysis goals and building this way the path to the proofs. The applications described help the reader to better understand advanced and up-to-date concepts in signal and image processing. Written as a text-book for a graduate course for engineering students, this book can also be used as an easy entry point for readers interested in stepping into this field, and for others already active in this area that are interested in expanding their understanding and knowledge. ### 稀疏与冗余表示：从理论到信号与图像处理的应用 #### 理解稀疏与冗余表示的基础概念 **稀疏表示**（Sparse Representation）是指一种信号或图像可以被表示为少量非零系数的方式。这种表示方法在处理大量数据时尤其有用，因为它能够将复杂的数据简化为易于管理和分析的形式。例如，在图像处理中，通过识别出图像中的关键特征（如边缘），我们可以使用较少的非零系数来近似整个图像，从而实现数据压缩的目的。 **冗余表示**（Redundant Representation）则涉及到多个不同的表示方式能够描述同一信号或图像的情况。这种冗余性提高了表示的灵活性，允许我们在多种不同的情况下选择最适合的表示方法。例如，在噪声环境下恢复原始信号时，可以通过选择对噪声具有较好鲁棒性的表示方法来提高恢复的质量。 #### 理论与数值基础 稀疏与冗余表示领域的理论研究主要围绕着如何有效地找到信号或图像的最稀疏表示。这包括了一系列算法的发展，如匹配追踪（Matching Pursuit）、正交匹配追踪（Orthogonal Matching Pursuit）等，这些算法能够从过完备字典中选择最佳的基向量来逼近原始信号。此外，还有重要的理论成果支持了这些算法的有效性，如压缩感知理论（Compressive Sensing），它揭示了即使在数据采样率远低于奈奎斯特速率的情况下，也可以准确地恢复出稀疏信号。 数值方面，稀疏表示通常涉及解决优化问题，例如最小化系数的L1范数来找到最稀疏的表示。这需要高效的优化算法和技术，如梯度下降、坐标下降以及交替方向乘子法（ADMM）等。 #### 在信号与图像处理中的应用 稀疏与冗余表示模型在信号与图像处理领域有着广泛的应用： 1. **去噪（Denoising）**：通过对图像进行稀疏表示，可以有效地分离信号和噪声部分，从而去除图像中的噪声。 2. **恢复（Restoration）**：对于退化的图像，如模糊图像，可以利用稀疏表示来恢复其清晰度。 3. **分离（Separation）**：当信号源混合在一起时，稀疏表示可以帮助我们分离出各个源信号。 4. **插值与外推（Interpolation and Extrapolation）**：稀疏表示可用于填补图像中的缺失部分，或是预测超出现有数据范围的信息。 5. **压缩（Compression）**：利用稀疏表示的特性可以实现高效的数据压缩，减少存储空间和传输带宽的需求。 6. **采样（Sampling）**：压缩感知理论提供了一种新的采样方法，可以在远低于传统采样率的情况下捕获信号信息。 7. **分析与合成（Analysis and Synthesis）**：通过对信号进行分析，可以提取出其内在结构，而合成过程则是根据分析结果重构信号。 8. **检测与识别（Detection and Recognition）**：稀疏表示有助于在复杂的背景中识别特定的目标或模式。 #### 教材特点与读者对象 该书《稀疏与冗余表示：从理论到信号与图像处理的应用》不仅系统地介绍了稀疏表示的理论基础和数值方法，还深入探讨了这些理论在实际信号与图像处理任务中的应用。作者通过清晰的叙述和实例，使读者能够轻松理解这一复杂主题。此外，书中还提供了大量的练习和案例研究，帮助读者巩固所学知识。 这本书适合研究生级别的工程学生作为教材使用，同时也非常适合希望进入该领域的研究人员以及已经在该领域工作但希望扩展自己知识面的专业人士阅读。无论是作为学习材料还是参考书籍，该书都是一个宝贵资源。