信号部分支撑集先验下的加权Lp(0<p<=1)正则化重构理论及应用

The Partial Support (PS) prior of signals widely exists in many real applications, such as the audio and video signal processing, and it has been shown that if such kind of signal prior can be well utilized, one can efficiently reconstruct a signal under much weaker theoretical conditions and smaller sampling number. However, there is an undeniable fact that almost all the existing reconstruction theory that utilizes this PS prior is based on the constrained optimal models. In view of the unconstrained optimal models and their convenience in algorithm implementation and effectiveness in signal denoise, this project, under a more general assumption on PS prior, will make a systematic and deep investigation of two representative weighted regularization models, i..e, the convex weighted L1-norm regularization model and the non-convex Lp(0<p<1) regularization model, including the following three major aspects of research: 1) establishing some deterministic (non-stochastic) theoretical frameworks (e.g., reconstruction condition, reconstruction error estimate, sampling number, etc.) for the above two regularization models to guarantee the robust reconstruction of any signals; 2) theoretically analyzing the impact from the PS prior on weight selection and also giving the theoretical basis and implementation plan of the optimal weight selection; 3) designing reasonable algorithms for the above two regularization models and then applying the obtained theoretical results to reconstruct the real signals such as the audio and video signals.

信号的部分支撑集先验广泛地存在于许多实际应用问题中，如音频和视频信号处理等，而已有研究表明：利用好这类先验信息，可以在更弱的理论条件、更低的采样数下实现信号的高效重构。然而一个不容忽视的事实是：现有基于这一先验信息的重构理论研究几乎都是以约束优化模型为基础的。考虑到无约束优化模型在算法设计上的便捷性和噪声抑制上的有效性，本项目在更一般的部分支撑集先验假设下，对两类具有代表性的加权正则化模型，即凸的加权L1范数正则化模型和非凸的加权Lp(0<p<1)正则化模型，展开深入系统的研究，主要有：1) 建立上述两类正则化模型重构任意信号的确定性(非随机性的)理论框架(比如重构条件，误差上界估计、采样次数等)；2) 从理论上分析部分支撑集先验信息对权重选择的影响，并给出最佳权重选择的理论依据和实现方案；3) 为上述两类正则化模型设计合理的求解算法，并将所获理论结果应用到音频和视频等真实信号的重构上。

项目紧密围绕含信号部分支撑集先验信息的鲁棒恢复这一科学问题，通过若干具有代表性的加权正则化模型展开具体研究工作。经过三年的探索和研究，在鲁棒零空间性质的系数估计、相关正则化模型的（充分的）可恢复条件和恢复误差的上界估计、权重参数的最佳取值、正则化参数的最优选择等方面获得了一系列突出的成果，此外，我们还将所获若干理论结果进行了延拓，先后建立起了一系列适宜矩阵数据和张量数据鲁棒恢复的新理论，极大地丰富了项目研究的深度和广度。相关工作共计发表了9篇期刊论文，涵盖了以《Appl. Comput. Harmon. Anal.》、《Inverse Probl.》和《Sci. China Inf. Sci.》等为代表的领域顶尖刊物，以及以《Signal Process.》、《Anal. Appl.》、《IEEE Signal Process. Lett.》、《J. Comput. Appl. Math.》和《J. Inverse Ill-Posed Probl.》等为代表的知名主流刊物，完成了既定的任务。