The Equivalence of Half-Quadratic Minimization and the Gradient Linearization Iteration

A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) min...

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Vydáno v:	IEEE transactions on image processing Ročník 16; číslo 6; s. 1623 - 1627
Hlavní autoři:	Nikolova, M., Chan, R.H.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.06.2007 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Algorithms Applied sciences Computer Simulation Cost function Detection, estimation, filtering, equalization, prediction Equivalence Exact sciences and technology Gradient linearization half-quadratic (HQ) regularization Image converters Image Enhancement - methods Image Interpretation, Computer-Assisted - methods Image processing Image reconstruction Image restoration Information, signal and communications theory Inverse problems Iterative methods Linear Models Mathematical models Mathematics Minimization Minimization methods Noise reduction Numerical Analysis, Computer-Assisted Optimization Optimization methods Regularization Signal and communications theory signal and image restoration Signal processing Signal Processing, Computer-Assisted Signal, noise Studies Telecommunications and information theory Tomography variational methods Cost minimization Variational methods Image processing Quasi Newton method Iterative method Image restoration Gradient linearization Optimization Image reconstruction Inverse problem signal and image restoration Signal restoration inverse problems Cost function Gradient method half-quadratic (HQ) regularization Quadratic function Linearization
ISSN:	1057-7149, 1941-0042
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Popis
Shrnutí:	A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularization term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results-including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 ObjectType-Article-2 ObjectType-Undefined-1 ObjectType-Feature-3
ISSN:	1057-7149 1941-0042
DOI:	10.1109/TIP.2007.896622