Analysis of Half-Quadratic Minimization Methods for Signal and Image Recovery

We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive half-quadratic reformulation of the original cost-function hav...

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Veröffentlicht in:SIAM journal on scientific computing Jg. 27; H. 3; S. 937 - 966
Hauptverfasser: Nikolova, Mila, Ng, Michael K.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Schlagworte:
Applied mathematics
Calculus of variations and optimal control
Exact sciences and technology
Inverse problems
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Partial differential equations
Sciences and techniques of general use
Variational methods
Optimization method
optimization
A posteriori estimation
inverse problems
Convergence rate
65K10
Ill posed problem
Signal reconstruction
Quadratic form
94A12
49N45
Mathematical programming
35A15
Convergence acceleration
Numerical linear algebra
half-quadratic regularization
65Y20
Image restoration
Inverse problem
Half quadratic regularization
Regularization method
Upper bound
signal and image restoration
Numerical analysis
Scientific computation
convergence analysis
65F22
94A08
maximum a posteriori estimation
Preconditioning
ISSN:1064-8275, 1095-7197
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Zusammenfassung:We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive half-quadratic reformulation of the original cost-function have been pioneered in Geman and Reynolds [IEEE Trans. Pattern Anal. Machine Intelligence, 14 (1992), pp. 367--383] and Geman and Yang IEEE Trans. Image Process., 4 (1995), pp. 932--946]. The alternate minimization of the resultant (augmented) cost-functions has a simple explicit form. The goal of this paper is to provide a systematic analysis of the convergence rate achieved by these methods. For the multiplicative and additive half-quadratic regularizations, we determine their upper bounds for their root-convergence factors. The bound for the multiplicative form is seen to be always smaller than the bound for the additive form. Experiments show that the number of iterations required for convergence for the multiplicative form is always less than that for the additive form. However, the computational cost of each iteration is much higher for the multiplicative form than for the additive form. The global assessment is that minimization using the additive form of half-quadratic regularization is faster than using the multiplicative form. When the additive form is applicable, it is hence recommended. Extensive experiments demonstrate that in our MATLAB implementation, both methods are substantially faster (in terms of computational times) than the standard MATLAB Optimization Toolbox routines used in our comparison study.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1064-8275
1095-7197
DOI:10.1137/030600862