Kristian Witsch

Iterative Multigrid Regularization Techniques for Image Matching

In this paper, we consider the problem of matching images, i.e., to find a deformation u, which transforms a digital image into another such that the images have nearly equal gray values in every image element. The difference of the two images is measured by their L2-difference, which should be minimized. This yields a nonlinear ill conditioned inverse problem for u, so the numerical solution is quite difficult. A Tikhonov regularization method is considered to rule out discontinuous and irregular solutions to the minimization problem. An important problem is a proper choice of the regularization parameter

Spectral collocation methods and polar coordinate singularities

\alpha

A multigrid approach for minimizing a nonlinear functional for digital image matching

. For the practical choice of

Computational methods for nonlinear image registration

\alpha,

Elastische Anpassung in der digitalen Bildverarbeitung auf mehreren Auflösungsstufen mit Hilfe von Mehrgitterverfahren

we use iterative regularization methods based on multigrid techniques. To obtain a suitable initial guess, we use an approach similar to the full multigrid (FMG) developed by Brandt [Math. Comp., 31 (1977), pp. 333--390]. The algorithms have optimal complexity: the amount of work is proportional to the number of picture elements. Finally, we present some experimental results for synthetic and real images.

Multimodal Image Registration Using a Variational Approach

This paper considers the numerical solution of elliptic differential equations on the unit disk. Using polar coordinates, the disk is mapped onto a rectangle. The resulting transformed problem is solved by a method related to collocation. Since the origin is a coordinate singularity, some natural trial functions are singular there and a special technique is applied to use zero as a collocation point. For Poisson and Helmholtz equations, a fast algorithm with an operation count of O(N2 log N) is presented. Numerical results show the different stability and convergence properties of the algorithms.

Image Registration Based on Multiscale Energy Information

Abstract In this paper, we consider a multigrid application in digital image processing. Here, the problem is to find a map, which transforms an image T into another image R such that the grey level of the different images are nearly equal in every picture-element. This problem arises in the investigation of human brains. The complete inverse problem is ill posed in the sense of Hadamard and nonlinear, so the numerical solution is quite difficult. We solve the inverse problem by a Landweber iteration. In each minimization step an approximate solution for the linearized problem is computed with a multigrid method as an inner iteration. Finally, we present some experimental results for synthetic and real images.

Lesion Preserving Image Registration with Applications to Human Brains

1 Institut fur Mathematik, Gerhard-Mercator Universitat Duisburg, Lotharstrase 63/65, 47048 Duisburg, Germany {clarenz|droske|rumpf}@math.uni-duisburg.de. 2 Lehrstuhl fur Mathematische Optimierung, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. henn@am.uni-duesseldorf.de 3 Lehrstuhl fur Angewandte Mathematik, Mathematisches Institut, Heinrich-Heine Universitat Dusseldorf, Universitatsstrase 1, D-40225 Dusseldorf, Germany. witsch@am.uni-duesseldorf.de Summary. Image registration is the process of the alignment of two or more data sets recorded with the same or different imaging machineries. Especially nonlinear image registration techniques allow the alignment of data sets that are mismatched in a nonuniform manner. Mathematically, this yields a nonlinear ill–conditioned inverse problem. In this presentation, we introduce several computational methods based on variational PDE approaches to obtain an approximate solution of the nonlinear registration problem. In each approach we have to solve a sequence of subproblems. Each subproblem has to be well-posed and should be efficiently solvable.

A variational image registration approach based on curvature scale space

Es wird ein neues Verfahren zur elastischen Anpassung eines Ausgangsvolumens an ein Referenzvolumen beschrieben, welches auf mehreren Auflosungsstufen arbeitet und standardmasig parallelisierbar ist. Dabei wird auf jeder Auflosungsstufe durch Variation der potentiellen Energie einer elastischen Verschiebung und der quadratischen Abweichung zweier Bildvolumina eine Bedingung hergeleitet, die eine pixelgenaue Anpassung der Bilder ermoglicht. Die Anpassung erfolgt durch einen iterativen Prozes, der sukzessive das elastische Potential des elastisch modellierten Ausgangsvolumens reduziert. Die Berechnung der Verschiebungsvektoren erfolgt hierbei auf jeder Auflosungsstufe durch ein Mehrgitterverfahren mit einem asymptotischen Aufwand O(N) proportional zur Anzahl N der Bildpunkte.

A Generalized Image Registration Framework using Incomplete Image Information – with Applications to Lesion Mapping

This paper presents an approach to obtain a deformation which matches two images acquired from different medical imaging modalities. This problem arises in the investigation of human brains. Two distance functionals for the images are proposed with different pros and cons. These functionals are to be minimized. We add a smoothing term to the minimization problem which retains certain desired elastic features in the solution. At each minimization step an approximate solution for the linearized problem is computed with a multigrid method as an inner iteration. Furthermore, we use a multiresolution minimization approach to obtain a suitable initial guess. Finally, we present some experimental results for registration problems of synthetic images and for a real computer tomography (CT)--magnetic resonance imaging (MRI) registration.