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$ \ell_0$ Minimization for wavelet frame based image restoration

Authors: Yong Zhang, Bin Dong and Zhaosong Lu
Journal: Math. Comp. 82 (2013), 995-1015
MSC (2010): Primary 80M50, 90C26, 42C40, 68U10
Published electronically: August 15, 2012
MathSciNet review: 3008846
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Abstract: The theory of (tight) wavelet frames has been extensively studied in the past twenty years and they are currently widely used for image restoration and other image processing and analysis problems. The success of wavelet frame based models, including balanced approach and analysis based approach, is due to their capability of sparsely approximating piecewise smooth functions like images. Motivated by the balanced approach and analysis based approach, we shall propose a wavelet frame based $ \ell _0$ minimization model, where the $ \ell _0$ ``norm'' of the frame coefficients is penalized. We adapt the penalty decomposition (PD) method of Lu and Zhang to solve the proposed optimization problem. Some convergence analysis of the adapted PD method will also be provided. Numerical results showed that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images.

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Additional Information

Yong Zhang
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada.

Bin Dong
Affiliation: Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., Tucson, Arizona, 85721-0089

Zhaosong Lu
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada.

Keywords: $ℓ_{0}$ minimization, hard thresholding, wavelet frame, image restoration.
Received by editor(s): May 17, 2011
Received by editor(s) in revised form: September 2, 2011, and October 6, 2011
Published electronically: August 15, 2012
Additional Notes: The first and third authors were supported in part by NSERC Discovery Grant.
Article copyright: © Copyright 2012 American Mathematical Society

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