目的 非相关观测是压缩感知(CS)理论中的关键因素。高斯随机矩阵作为一种普适的CS非相关观测矩阵,在压缩感知中得到广泛的研究与应用。但在实际应用中,却存在实际内存占用较多,不适应大规模应用的问题。为寻求降低随机观测矩阵所需的存储空间,提出一种基于半张量积的压缩感知方法,利用该方法可以成倍地降低观测矩阵所需的存储空间。方法 该方法利用半张量积理论,构建降维随机观测矩阵,实现对原始信号的随机观测,并采用l^q(0< q< 1)范数的迭代重加权最小二乘法进行重构,从而得到稀疏信号的估计值。结果 仿真实验分别采用1维稀疏信号和2维图像信号进行了测试,并从重构概率、迭代收敛速度、重构信号的峰值信噪比等角度进行了测试和比较。通过不同大小的随机观测矩阵比较验证表明,采用降维后观测矩阵进行采样和重构,其重构信号质量并没有明显下降,但其观测矩阵所需的存储空间却可大大降低,如降低为通常的1/4,1/16,甚至更低。结论 本文压缩感知方法,可以大大降低观测矩阵所需的存储空间,同时有效降低数据运算复杂度以及内存占用率,有助于压缩感知的应用。

Objective A random measurement matrix plays a critical role for the successful use of compressive sensing (CS) theory and has been widely applied in CS. However, a random measurement matrix requires a large storage space, which is unsuitable for large-scale applications. To reduce the storage space for a random measurement matrix, a method for CS signal reconstruction was proposed based on theory of semi-tensor product. Method We constructed a random measurement matrix, with a dimension smaller than M and N, where M is the length of the sampling vector and N is the length of the signal that we intend to reconstruct. Then, we used the iteratively reweighted least square reconstruction algorithm to obtain the estimated values of sparse coefficients. Result Experiments were conducted using column sparse signals and images with various sizes. During the experiments, the probability of exact reconstruction, error, and peak signal-to-noise ratio (PSNR), of the proposed method were compared with measurement matrices with different dimensions. The proposed algorithm outperformed a smaller storage space with a suitable PSNR performance. Conclusion In this study, we proposed a new method to reduce the storage space of the measurement matrix for CS. The experimental results showed that if we appropriately reduced the dimension of the measurement matrix, then nearly no decline in the PSNR of the reconstruction was observed, but the storage space of the measurement matrix could be reduced by at least 1/4 or 1/16 times. All the results verified the validity of the proposed approach and demonstrated the significant potential for hardware implementation of the proposed sensing framework.