2012/06/18 by Florent Krzakala, Marc Mézard, François Sausset +3 · 1 citation
Computer Science · Engineering · Mathematics · Medicine · Physics and Astronomy · #Advanced MRI Techniques and Applications #Advanced X-ray Imaging Techniques #Sparse and Compressive Sensing Techniques #cond-mat.stat-mech #cs.IT #math.IT
paper · pdf · doi:10.1088/1742-5468/2012/08/p08009
published as J. Stat. Mech. (2012) P08009 · 42 pages, 37 figures, 3 appendixes
arxiv created 2012/06/18 · openalex publication_date 2012/08/17 · arxiv updated 2012/08/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make fewer measurements than were considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in Krzakala et al (2012 Phys. Rev. X 2 021005) a strategy that allows compressed sensing to be performed at acquisition rates approaching the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation maximization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distributions of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.