基于样本的非线性压缩感知理论及其应用

Compressed sensing's effectiveness depends on the sparsity of signal. In terms of the known concept of sparsity, it mainly refers to the case that a signal possesses sparse representation under certain bases (or dictionary). In other words, if there exists a linear transformation which can make a signal sparse, we say such signal is sparse. However, the definition above can not describe complex and generally incompressible signals, which thus makes the existed compressed sensing theories and methods invalid.In this project, we propose a novel model to characterize the sparsity of signal via nonlinear transformation based on samples, and also make a deep research on existence of such nonlinear transformation, the nonlinear sparse representation theory based on samples, the corresponding complete reconstruction theory, approximation accuracy and the efficient computational methods in nonlinear compressed sensing. Besides, we further test and verify our theoretical results in the access and reconstruction of remote sensing information. The obtained theories and methods have significant value in deepening compressed sensing and its related applications.

压缩感知的有效性以信号的稀疏性为前提。目前已知的信号稀疏性是指存在某个基底(或字典), 使得该信号在此基底下具有稀疏表示, 或等价地说, 存在某个线性变换使信号在该变换下稀疏。这种定义不能描述复杂的、通常意义下难以压缩的信号，从而使得已有的压缩感知理论与方法失效。本项目提出基于样本的经由非线性变换刻画稀疏性的新模型，并深入研究对一般信号使其稀疏的非线性变换的存在性、基于样本的非线性稀疏表示理论、对应非线性压缩感知模型的可计算性、完全可重构性和逼近精度、发展非线性压缩感知的高效实现方法，并在遥感图像处理中应用验证。所发展的理论和方法对推动压缩感知的深化与应用具有重要价值。

压缩感知的有效性以信号的稀疏性为前提. 目前已知的信号稀疏性是指存在某个基底(或字典),使得该信号在此基底下具有稀疏表示, 或等价地说, 存在某个线性变换使信号在该变换下稀疏。这种定义不能描述复杂的、通常意义下难以压缩的信号，从而使得已有的压缩感知理论与方法失效。本项目提出基于样本来经由非线性变换刻画稀疏性的新模型，1-比特压缩感知，张量压缩感知，鲁棒张量1-比特压缩感知、高阶压缩感知模型等，并深入研究1-比特信号，低秩张量信号，低秩矩阵信号，基于样本的非线性稀疏表示理论、对应非线性（高阶）压缩感知模型的可计算性、完全可重构性和逼近精度、发展其高效实现方法，并在遥感信息获取与重建中应用验证。所发展的理论和方法对推动压缩感知的深化与应用具有重要价值。