Inexact Alternating Direction Methods for Image Recovery

In the image processing community, there have recently been many restoration and reconstruction problems that can be reformulated into linearly constrained convex programming models whose objective functions have separable structures. These favorable reformulations have promoted impressive applicati...

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Bibliographic Details
Published in:SIAM journal on scientific computing Vol. 33; no. 4; pp. 1643 - 1668
Main Authors: Ng, Michael K., Wang, Fan, Yuan, Xiaoming
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Subjects:
Algorithms
Applied mathematics
Decomposition
Exact sciences and technology
Fourier transforms
Mathematics
Methods
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Sciences and techniques of general use
Alternating direction method
68U10
65K15
Image processing
Iteration
Convex programming
Convergence
Exact solution
inexact
Numerical analysis
Scientific computation
image restoration
65F22
94A08
65J22
65K10
construction
Convex function
image re
65J20
compression
ISSN:1064-8275, 1095-7197
Online Access:Get full text
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Summary:In the image processing community, there have recently been many restoration and reconstruction problems that can be reformulated into linearly constrained convex programming models whose objective functions have separable structures. These favorable reformulations have promoted impressive applications of the alternating direction method (ADM) in the field of image processing. At each iteration, the computation of ADM is dominated by solving two subproblems exactly. However, in many restoration and reconstruction applications, it is either impossible or extremely expensive to obtain exact solutions of these ADM subproblems. This fact urges the development on inexact versions of ADM, which allow the generated ADM subproblems to be solved approximately subject to certain inexactness criteria. In this paper, we develop some truly implementable inexact ADMs whose inexactness criteria controlling the accuracy of the ADM subproblems are easily implementable. The convergence of the new inexact ADMs will be proved. Numerical results on several image processing problems will be given to illustrate the effectiveness of the proposed inexact ADMs.
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ISSN:1064-8275
1095-7197
DOI:10.1137/100807697