基于奇异值分解的可分离压缩成像方法

可分离压缩传感可以通过一定比例的额外测量有效地解决压缩成像问题中面临的测量矩阵维数过大的瓶颈. 但是现有可分离压缩传感(separable compressive sensing, SCS)方法需要2个可分离的测量矩阵都必须是行归一化后的正交随机矩阵，其显著地限制了该方法的应用范围. 将奇异值分解(singular value decomposition, SVD)方法引入可分离可压缩传感测量过程，可以有效地实现测量矩阵和重建矩阵的分离：在感知阶段可以更多地考虑测量矩阵物理易于实现的性质，如Toeplitz或Circulant等确定性结构的矩阵；在重建阶段，更多地考虑测量矩阵的优化.通过引入奇异值分解对重建阶段的测量矩阵进行优化，可以有效地改善重建性能，尤其是Toeplitz或Circulant矩阵在大尺度图像的压缩重建情形.数值实验结果验证了该方法的有效性.

Separable Compressive Imaging Method Based on Singular Value Decomposition

When facing the compressive imaging problem that the measurement matrix has too large dimension, separable compressive sensing (SCS) can effectively achieve this problem at a cost of a certain percentage of additional measurements. However, the both separable measurement matrices in existing separable compressive sensing method should be row-normalized orthogonal random matrix, which limits its application significantly. In this paper, the method of singular value decomposition (SVD) is introduced into separable compressive sensing measurement process, which can effectively achieve the separation of measurement matrix and reconstruction matrix: the design of the measurement matrix in sensing stage is more to consider the physical properties for easy implementations, such as the deterministic structure of Toeplitz or Circulant matrices and etc; in the reconstruction stage, it is more to consider the optimization of reconstruction matrix. Through the introduction of singular value decomposition method to optimize the measurement matrix in reconstruction stage, the reconstruction performance can be effectively facilitated, especially for Toeplitz and Circulant matrix in large-scale image compressive reconstruction. Numerical results demonstrate the validity of our proposed method.