Phase Transition of Total Variation Based on Approximate Message Passing Algorithm

In compressed sensing (CS), one seeks to down-sample some high-dimensional signals and recover them accurately by exploiting the sparsity of the signals. However, the traditional sparsity assumption cannot be directly satisfied in most practical applications. Fortunately, many signals-of-interest do at least exhibit a low-complexity representation with respect to a certain transformation. Particularly, total variation (TV) minimization is a notable example when the transformation operator is a difference matrix. Presently, many theoretical properties of total variation have not been completely explored, e.g., how to estimate the precise location of phase transitions and their rigorous understanding is still in its infancy. So far, the performance and robustness of the existing algorithm for phase transition prediction of TV model are not satisfactory. In this paper, we design a new approximate message passing algorithm to solve the above problems, called total variation vector approximate message passing (TV-VAMP) algorithm. To be specific, we first consider the problem from the Bayesian perspective, and formulate it as an optimization problem. Then, the vector factor graph for the TV model is constructed based on the formulized problem. Finally, the TV-VAMP algorithm is derived according to the idea of probabilistic inference in machine learning. Compared with the existing algorithm, our algorithm can be applied to a wider range of measurements distributions, including the non-zero-mean Gaussian distribution measurements matrix and ill-conditioned measurements matrix. Furthermore, in experiments with various settings, including different measurement distribution matrices, signal gradient sparsity, and measurement times, the proposed algorithm can almost reach the target mean squared error (-60 dB) with fewer iterations and better fit the empirical phase transition curve.