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This paper proposes a fast approximate message-passing (AMP) algorithm forsolving compressed sensing (CS) recovery problems with 1D-finite-differencesparsity in term of MMSE estimation. The proposed algorithm, named ssAMP-BGFD,is low-computational with its fast convergence and cheap per-iteration cost,providing phase transition nearly approaching to the state-of-the-art. Theproposed algorithm is originated from a sum-product message-passing rule,applying a Bernoulli-Gaussian (BG) prior, seeking an MMSE solution. Thealgorithm construction includes not only the conventional AMP technique for themeasurement fidelity, but also suggests a simplified message-passing method topromote the signal sparsity in finite-difference. Furthermore, we provide anEM-tuning methodology to learn the BG prior parameters, suggesting how to usesome practical measurement matrices satisfying the RIP requirement under thessAMP-BGFD recovery. Extensive empirical results confirms performance of theproposed algorithm, in phase transition, convergence speed, and CPU runtime,compared to the recent algorithms.

Bernoulli-Gaussian Approximate Message-Passing Algorithm for Compressed Sensing with 1D-Finite-Difference Sparsity