Convex approaches to model wavelet sparsity patterns

Statistical dependencies among wavelet coefficients are commonly represented by graphical models such as hidden Markov trees (HMTs). However, in linear inverse problems such as deconvolution, tomography, and compressed sensing, the presence of a sensing or observation matrix produces a linear mixing...

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Bibliographic Details
Main Authors: Rao, N. S., Nowak, R. D., Wright, S. J., Kingsbury, N. G.
Format: Conference Proceeding
Language:English
Subjects:
Compressed sensing
Convex functions
Deconvolution
Hidden Markov models
Image reconstruction
Noise reduction
wavelet modeling
Wavelet transforms
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Summary:Statistical dependencies among wavelet coefficients are commonly represented by graphical models such as hidden Markov trees (HMTs). However, in linear inverse problems such as deconvolution, tomography, and compressed sensing, the presence of a sensing or observation matrix produces a linear mixing of the simple Markovian dependency structure. This leads to reconstruction problems that are non-convex optimizations. Past work has dealt with this issue by resorting to greedy or suboptimal iterative reconstruction methods. In this paper, we propose new modeling approaches based on group-sparsity penalties that leads to convex optimizations that can be solved exactly and efficiently. We show that the methods we develop perform significantly better in de-convolution and compressed sensing applications, while being as computationally efficient as standard coefficient-wise approaches such as lasso.
ISSN:1522-4880
2381-8549
DOI:10.1109/ICIP.2011.6115845