On MAP and MMSE estimators for the co-sparse analysis model

Javier S. Turek, Irad Yavneh, Michael Elad

Research output: Contribution to journalArticlepeer-review


The sparse synthesis model for signals has become very popular in the last decade, leading to improved performance in many signal processing applications. This model assumes that a signal may be described as a linear combination of few columns (atoms) of a given synthesis matrix (dictionary). The Co-Sparse Analysis model is a recently introduced counterpart, whereby signals are assumed to be orthogonal to many rows of a given analysis dictionary. These rows are called the co-support. The Analysis model has already led to a series of contributions that address the pursuit problem: identifying the co-support of a corrupted signal in order to restore it. While all the existing work adopts a deterministic point of view towards the design of such pursuit algorithms, this paper introduces a Bayesian estimation point of view, starting with a random generative model for the co-sparse analysis signals. This is followed by a derivation of Oracle, Minimum-Mean-Squared-Error (MMSE), and Maximum-A-posteriori-Probability (MAP) based estimators. We present a comparison between the deterministic formulations and these estimators, drawing some connections between the two. We develop practical approximations to the MAP and MMSE estimators, and demonstrate the proposed reconstruction algorithms in several synthetic and real image experiments, showing their potential and applicability.

Original languageEnglish
Pages (from-to)57-74
Number of pages18
JournalDigital Signal Processing: A Review Journal
Issue number1
StatePublished - May 2014


  • Analysis model
  • Bayesian estimation
  • Co-sparse signal model
  • MAP
  • MMSE

All Science Journal Classification (ASJC) codes

  • Signal Processing
  • Computer Vision and Pattern Recognition
  • Statistics, Probability and Uncertainty
  • Computational Theory and Mathematics
  • Electrical and Electronic Engineering
  • Artificial Intelligence
  • Applied Mathematics


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