Analysis versus synthesis in signal priors

The concept of prior probability for signals plays a key role in the successful solution of many inverse problems. Much of the literature on this topic can be divided between analysis-based and synthesis-based priors. Analysis-based priors assign probability to a signal through various forward measurements of it, while synthesis-based priors seek a reconstruction of the signal as a combination of atom signals. In this paper we describe these two prior classes, focusing on the distinction between them. We show that although when reducing to the complete and under-complete formulations the two become equivalent, in their more interesting overcomplete formulation the two types depart. Focusing on the ℓ1 denoising case, we present several ways of comparing the two types of priors, establishing the existence of an unbridgeable gap between them.

[1]  D. Donoho,et al.  Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) , 2005 .

[2]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[3]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

[4]  Minh N. Do,et al.  Ieee Transactions on Image Processing the Contourlet Transform: an Efficient Directional Multiresolution Image Representation , 2022 .

[5]  Michael Elad,et al.  On the bilateral filter and ways to improve it , 2002 .

[6]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[7]  Tony F. Chan,et al.  Color TV: total variation methods for restoration of vector-valued images , 1998, IEEE Trans. Image Process..

[8]  David J. Field,et al.  Emergence of simple-cell receptive field properties by learning a sparse code for natural images , 1996, Nature.

[9]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[10]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[11]  A. Bruckstein,et al.  K-SVD : An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation , 2005 .

[12]  M. Elad,et al.  $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation , 2006, IEEE Transactions on Signal Processing.

[13]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Michael Elad,et al.  Analysis versus synthesis in signal priors , 2006 .

[15]  E. Candès,et al.  Ridgelets: a key to higher-dimensional intermittency? , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[17]  Eero P. Simoncelli.  Bayesian Denoising of Visual Images in the Wavelet Domain , 1999 .

[18]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[19]  Robert L. Stevenson,et al.  A Bayesian approach to image expansion for improved definitio , 1994, IEEE Trans. Image Process..

[20]  Ken D. Sauer,et al.  A generalized Gaussian image model for edge-preserving MAP estimation , 1993, IEEE Trans. Image Process..

[21]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[22]  Ronald J. Jaszczak,et al.  Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data , 1997, IEEE Transactions on Medical Imaging.

[23]  Michael Elad,et al.  On the origin of the bilateral filter and ways to improve it , 2002, IEEE Trans. Image Process..

[24]  Jia Jie.  Bayesian denoising of visual images in the wavelet domain , 2003 .

[25]  Michael Elad,et al.  Multiframe demosaicing and super-resolution of color images , 2006, IEEE Transactions on Image Processing.

[26]  David L. Donoho,et al.  High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension , 2006, Discret. Comput. Geom..

[27]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .