Fundamentals of statistical signal processing: estimation theory
Fundamentals of statistical signal processing: estimation theory
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
A Non-Local Algorithm for Image Denoising
CVPR '05 Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) - Volume 2 - Volume 02
International Journal of Computer Vision
Exploring Strategies for Training Deep Neural Networks
The Journal of Machine Learning Research
Learning Deep Architectures for AI
Foundations and Trends® in Machine Learning
A plurality of sparse representations is better than the sparsest one alone
IEEE Transactions on Information Theory
Image restoration through L0 analysis-based sparse optimization in tight frames
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity
Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity
IEEE Transactions on Signal Processing
Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing
Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing
Linear Regression With a Sparse Parameter Vector
IEEE Transactions on Signal Processing
-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
IEEE Transactions on Signal Processing
Matching pursuits with time-frequency dictionaries
IEEE Transactions on Signal Processing
A Theory for Sampling Signals From a Union of Subspaces
IEEE Transactions on Signal Processing
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
IEEE Transactions on Pattern Analysis and Machine Intelligence
Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering
IEEE Transactions on Image Processing
On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
IEEE Transactions on Signal Processing
Representation Learning: A Review and New Perspectives
IEEE Transactions on Pattern Analysis and Machine Intelligence
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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.