中文摘要：

带有噪声的压缩感知信号重建模型可表示为l_1-范数问题.为了满足使用少量观测值重构出高精度的图像,在设置观测矩阵时需要满足受限等距性（RIP）和非相干性,然而判断一个矩阵的RIP是非常困难的.针对观测矩阵的不确定性,将该模型转化为具有概率约束的随机优化模型,即在约束条件以很大的概率被满足的情况下,求解最小l_1-范数问题.构建了概率约束函数的一个D.C.近似函数,讨论了函数的性质,建立了相应的D.C.近似问题,证明了D.C.近似问题与概率约束优化问题的等价性.

英文摘要：

Compressed sensing signal reconstruction with noise can be expressed as l1-norm problem. In order to reconstruct a high-precision image with a small amount of observations, it is necessary to satisfy the restricted isometric （RIP） and non-coherence when setting the observation matrix. How- ever, it is very difficult to judge the RIP of a matrix. In view of the uncertainty of the observation matrix, the l1-norm problem is transformed into a stochastic optimization model with probability con- straint in this paper. That is, the minimum l1 norm problem is solved when the constraint is satisfied with a large probability. A D. C. approximation function of the probability constraint function is con- structed. The properties of the function are discussed and the corresponding D. C. approximation problem is established. The equivalence between the D. C. approximation and the probability con strained optimization problem is proved.