半张量积压缩感知模型的l0-范数解

目的 半张量积压缩感知模型是一种可以有效降低压缩感知过程中随机观测矩阵所占存储空间的新方法，利用该模型可以成倍降低观测矩阵所需的存储空间。为寻求基于该模型新的重构方法，同时提升降维后观测矩阵的重构性能，提出一种采用光滑高斯函数拟合l₀-范数方法进行重构。方法 构建降维随机观测矩阵，对原始信号进行采样；构建可微且期望值为零的光滑高斯函数来拟合不连续的l₀-范数，采用最速下降法进行重构，最终得到稀疏信号的估计值。结果 实验分别采用1维稀疏信号和2维图像信号进行测试，并从重构概率、收敛速度、重构信号的峰值信噪比等角度进行了测试和比较。验证结果表明，本文所述算法的重构概率、收敛速度较该模型的l_q-范数（0 <q <1）方法有一定的提升，且当观测矩阵大小降低为通常的1/64，甚至1/256时，仍能保持较高的重构性能。结论 本文所述的重构算法，能在更大程度上降低观测矩阵的大小，同时基本保持重构的精度。

Smooth l0-norm minimization algorithm for compressed sensing with semi-tensor product

Objective The semi-tensor product (STP) approach is an effective way to reduce the storage space of a random measurement matrix for compressed sensing (CS), in which the dimensions of the random measurement matrix can be reduced to a quarter (or a sixteenth, or even less) of the dimensions used for conventional CS. A smooth l₀-norm minimization algorithm for CS with the STP is proposed to improve reconstruction performance. Method We generate a random measurement matrix, in which the matrix dimensions are reduced to 1/4, 1/16, 1/64, or 1/256 of the dimensions used for conventional CS. We then estimate the solutions of the sparse vector with the smooth l₀-norm minimization algorithm. Result Numerical experiments are conducted using column sparse signals and images of various sizes. The probability of exact reconstruction, rate of convergence, and peak signal-to-noise ratio of the reconstruction solutions are compared with the random matrices with different dimensions. Numerical simulation results show that the proposed algorithm can reduce the storage space of the random measurement matrix to at least 1/4 while maintaining reconstruction performance. Conclusion The proposed algorithm can reduce the dimensions of the random measurement matrix to a great extent than the l_q-norm (0 <q <1) minimization algorithm, thereby maintaining the reconstruction quality.