Compressed Sensing (CS) is an advanced theory on signal sampling and reconstruction. In CS theory, the reconstruction condition of signal is an important theory problem, and spark is a good index to study this problem. But the computation of spark is NP hard. In this paper, we study the problem of computing spark. For some special matrixes, for example, the Gaussian random matrix and 0-1 random matrix, we obtain some conclusions. Furthermore, for Gaussian random matrix with fewer rows than columns, we prove that its spark equals to the number of its rows plus one with probability 1. For general matrix, two methods are given to compute its spark. One is the method of directly searching and the other is the method of dual-tree searching. By simulating 24 Gaussian random matrixes and 18 0-1 random matrixes, we tested the computation time of these two methods. Numerical results showed that the dual-tree searching method had higher efficiency than directly searching, especially for those matrixes which has as much as rows and columns.
The problem of image restoration has been extensively studied for its practical importance and theoretical interest. This paper mainly discusses the problem of image restoration with partly unknown kernel. In this model, the degraded kernel function is known but its parameters are unknown. With this model, we should estimate the parameters in Gaussian kernel and the real image simultaneity. For this new problem, a total variation restoration model is put out and an intersect direction iteration algorithm is designed. Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measurement (SSIM) are used to measure the performance of the method. Numerical results show that we can estimate the parameters in kernel accurately, and the new method has both much higher PSNR and much higher SSIM than the expectation maximization (EM) method in many cases. In addition, the accuracy of estimation is not sensitive to noise. Furthermore, even though the support of the kernel is unknown, we can also use this method to get accurate estimation.