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A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

Published online by Cambridge University Press:  28 May 2015

Jing Yuan ,
Juan Shi  and
Xue-Cheng Tai
Jing Yuan*
Affiliation:
Computer Science Department, Middlesex College, University of Western Ontario, London, Ontario, N6A 5B7, UK
Juan Shi*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
Xue-Cheng Tai*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore Department of Mathematics, University of Bergen, Norway
*
Corresponding author. Email: cn.yuanjing@gmail.com
Corresponding author. Email: shij0004@e.ntu.edu.sg
Corresponding author. Email: tai@mi.uib.no
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Abstract

We study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function u(x) ∈ {f1,…,fn}, ∀x ∈ Ω. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.

Keywords

Convex optimizationprimal-dual approachtotal-variation regularizationimage processing
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011 , pp. 172 - 186
Copyright
Copyright © Global-Science Press 2011

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