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...

Full description

Saved in:
Bibliographic Details
Published in:IEEE transactions on image processing Vol. 16; no. 6; pp. 1623 - 1627
Main Authors: Nikolova, M., Chan, R.H.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.06.2007
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
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
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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
Bibliography: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