稀疏模型被广泛应用于图像及信号处理，模式识别等诸多领域。将稀疏建模应用于工程中的热门领域“盲去卷积”问题，系统研究其理论及算法，有重要的理论和实际意义。本项目利用逼近论、小波分析、优化理论、随机矩阵理论、代数学和组合数论等研究工具，研究稀疏信号的盲去卷积恢复问题。历史文献中盲去卷积的理论分析基于信号在随机字典下具有稀疏表示的假设。该假设在实际问题中具有很强的局限性。申请人将基于确定型或构造型字典，如离散的小波框架，分析信号恢复所需的条件。进一步地，通过引入新的稀疏优化模型，分析盲去卷积问题的信号恢复稳定性，逼近性及算法收敛性。此外，针对多通道融合的盲去卷积问题，进行信号分离的理论及算法分析。本项目将稀疏模型、小波分析和图像处理等热门领域结合，建立更深刻的联系，旨在为盲去卷积问题提供更完善的数学支撑，克服现有技术和分析的欠缺，推动理论和实际应用的发展。

Sparse models are widely used in the fields such as image and signal processing, pattern recognition and so on. Applying sparse modeling to “blind deconvolution”, which is one of the important problems in engineering, and systematically studying its theory and algorithms has important theoretical and practical implications. The project will use the methods of approximation theory, wavelet analysis, optimization theory, random matrix theory, algebra and combinational number theory to study sparse signal recovery in blind deconvolution problem. The theoretical analysis in literature was based on the assumption that the signal is sparse under some random dictionary. This assumption is strongly limited in real applications. The applicant will analyze the conditions required for signal recovery under some deterministic or constructed dictionary, such as discrete wavelet frames. Furthermore, by introducing some new sparse optimization model, the project will analyze the stability and approximation result of blind deconvolution problem, and also the convergence theory of the related algorithm. In addition, for multichannel demixing problem in blind deconvolution, the theory and algorithms are analyzed. The project combines sparse model, wavelet analysis and image processing to establish a deeper connection. It aims to provide more complete mathematical support for blind deconvolution problem, overcome the shortcomings of the prior art, and promote the development of both theory and real applications.