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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Vydané v:	SIAM journal on scientific computing Ročník 27; číslo 3; s. 937 - 966
Hlavní autori:	Nikolova, Mila, Ng, Michael K.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Predmet:	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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Abstract	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.
AbstractList	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.
Author	Nikolova, Mila Ng, Michael K.
Author_xml	– sequence: 1 givenname: Mila surname: Nikolova fullname: Nikolova, Mila – sequence: 2 givenname: Michael K. surname: Ng fullname: Ng, Michael K.
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Copyright	2006 INIST-CNRS [Copyright] © 2005 Society for Industrial and Applied Mathematics
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Keywords	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
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References	Tikhonov Andrey (R31) 1977 R21 R20 R23 R25 R24 R26 R28 R1 R3 R4 R5 R6 R8 R30 Aubert Gilles (R2) 2002 R10 R12 R11 R14 R13 R15 R17 R19 Rockafellar R. (R27) 1970
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Snippet	We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and...
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SubjectTerms	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
Title	Analysis of Half-Quadratic Minimization Methods for Signal and Image Recovery
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