A Levenberg--Marquardt scheme for nonlinear image registration
type: Item
instance of: scholarly article
title: A Levenberg--Marquardt scheme for nonlinear image registration
zbMATH Open document ID: 1047.65048
DOI: 10.1023/B:BITN.0000009940.58397.98
author: Stefan Henn
published in: BIT
publication date: 2004-03-15T00:00:00Z
review text: Two images are represented by compactly supported functionals \(T,R:\Omega\to \mathbb{R}\) where \(\Omega\subset\mathbb{R}^2\). A displacement vector field \(u:\Omega\to\mathbb{R}^2\) is considered which matches the two images, \(R(x)\approx T(x- u(x))\). This leads to the optimization problem \[ D[u]:= \int_\Omega (T(x- u(x))- R(x))^2 \,dx\to\min!.\tag{1} \] The images are encoded as two-dimensional arrays of pixel values, and the problem (1) is replaced by its discrete approximation. The Levenberg-Marquardt (LM) method is applied which uses the linearization of the functional \(D\) around a current approximation. The resulting quadratic minimization problem is ill-posed. To regularize the problem, an auxiliary Lagrange term is used which is borrowed from linear elasticity theory. Numerical experiments show that: a) the LM scheme usually converges in 5 to 7 iterations, b) the method outperforms the steepest descent methods, currently used for image registration, see \textit{M. Bro-Nielsen} and \textit{C. Gramkow} [Fast fluid registrationof medical images. Lecture Notes Comput. Sci. 1131, 267--276 (1996)], and c) the approach works well when the images are contaminated by small errors.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 49J10
Mathematics Subject Classification ID: 49M25
Mathematics Subject Classification ID: 94A08
zbMATH DE Number: 2057840
zbMATH Keywords: Levenberg-Marquardt method
zbMATH Keywords: nonlinear least squares
zbMATH Keywords: image processing
zbMATH Keywords: physical regularization
zbMATH Keywords: ill-posed problem
zbMATH Keywords: discrete approximation
zbMATH Keywords: quadratic minimization problem
zbMATH Keywords: numerical experiments
zbMATH Keywords: steepest descent methods
zbMATH Keywords: image registration
reviewed by: A. Kukush
MaRDI profile type: Publication
full work available at URL: b:bitn.0000009940.58397.98
OpenAlex ID: W1998598480
description: scientific article; zbMATH DE number 2057840