1-Bit Compressed Sensing via an L1-TV Regularization Method

This paper considers the 1-bit compressed sensing (1-bit CS) of signals that are simultaneously sparse and gradient sparse. Since L1-norm and total variation (TV) penalties are beneficial for reconstructing element-sparse and gradient-sparse signals, respectively, we combine both to propose the L1-TV regularization model for 1-bit CS. We show that the proposed model has a closed-form solution so that one can easily calculate. Despite the apparent simplicity, our theoretical analysis reveals that the solution provides an approximate estimate for the underlying signal. Besides, in the case of introducing a dither vector, we develop an adaptive algorithm to accelerate the decay rate of recovery error. The key idea is that generating the dither for each iteration relying on the last estimate. In addition to theoretical analysis, we conduct a series of experiments on both synthetic and real-world data to show the superiority of our algorithms.