MATHEMATICS IN INDUSTRY 10

Editors Hans-Georg Bock Frank de Hoog Avner Friedman Arvind Gupta Helmut Neunzert William R. Pulleyblank Torgeir Rusten Fadil Santosa Anna-Karin Tornberg

THE EUROPEAN CONSORTIUM FOR IN INDUSTRY

SUBSERIES Managing Editor Vincenzo Capasso

Editors Robert Mattheij Helmut Neunzert Otmar Scherzer Otmar Scherzer

Mathematical Models for Registration and Applications to Medical Imaging

With 54 Figures, 12 in Color, and 12 Tables

123 Editor Otmar Scherzer .. Universitat Innsbruck .. Institut fur Informatik, Technikerstr. 21A A -- 6020 Innsbruck, Austria e-mail: [email protected]

Library of Congress Control Number: 2006926829

Mathematics Subject Classification (2000): 65J15, 65F22, 94a08, 94J40, 94K24

ISBN-10 3-540-25029-8 Springer Heidelberg New York ISBN-13 978-3-540-25029-6 Springer Berlin Heidelberg New York

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Image registration is an emerging topic in image processing with many applications in medical imaging, picture and movie processing. The classical problem of image registration is concerned with finding an appropriate transformation between two data sets. This fuzzy definition of registration requires a mathematical modeling and in particular a mathematical specification of the terms appropriate transformations and correlation between data sets. Depending on the type of application, typically Euler, rigid, plastic, elastic deformations are considered. The variety of similarity measures ranges from a simple Lp distance between the pixel values of the data to mutual information or entropy distances. This goal of this book is to highlight by some experts in industry and medicine relevant and emerging image registration applications and to show new emerging mathematical technologies in these areas. Currently, many registration application are solved based on variational princi- ple requiring sophisticated analysis, such as calculus of variations and the theory of partial differential equations, to name but a few. Due to the numerical complex- ity of registration problems efficient numerical realization are required. Concepts like multi-level solver for partial differential equations, non-convex optimization, and so on play an important role. Mathematical and numerical issues in the area of registration are discussed by some of the experts in this volume. Moreover, the importance of registration for industry and medical imaging is discussed from a medical doctor and from a manufacturer point of view. We would like to thank Stephanie Schimkowitsch for a marvelous job in type- setting this manuscript. Moreover, we would like to thank Prof. Vincenzo Capasso for the continuous encouragement and support of this book and I would like to ex- press my thanks to Ute McCrory (Springer) for her patience during the preparation of the manuscript. The work of myself is supported by the FWF, Austria Science Foundation, Projects Y-123INF, FSP 9203-N12 and FSP 9207-N12. Without the support of the FWF for my research this volume would not be possible.

June, 2005 Otmar Scherzer (Innsbruck) Table of Contents

Part I Numerical Methods

A Generalized Image Registration Framework using Incomplete Image Information – with Applications to Lesion Mapping Stefan Henn, Lars Homke,¨ Kristian Witsch...... 3 Medical Image Registration and Interpolation by Optical Flow with Maximal Rigidity Stephen L. Keeling ...... 27 Registration of Histological Serial Sectionings Jan Modersitzki, Oliver Schmitt, and Stefan Wirtz ...... 63 Computational Methods for Nonlinear Image Registration Ulrich Clarenz, Marc Droske, Stefan Henn, Martin Rumpf, Kristian Witsch ... 81 A Survey on Variational Optic Flow Methods for Small Displacements Joachim Weickert, Andres´ Bruhn, Thomas Brox, and Nils Papenberg ...... 103

Part II Applications

Fast Image Matching for Generation of Panorama Ultrasound Armin Schoisswohl ...... 139 Inpainting of Movies Using Optical Flow Harald Grossauer ...... 151

Part III Medical Applications

Multimodality Registration in Daily Clinical Practice Reto Bale ...... 165 Colour Images Clarenz et al., Henn et al., Weickert et al., Bale ...... 185 List of Contributors

Otmar Scherzer 40225 Dusseldorf,¨ Germany University of Innsbruck [email protected] Institute of Computer Science Technikerstraße 21a Lars Homke¨ 6020 Innsbruck, Austria Forschungszentrum Julich¨ GmbH [email protected] Institut fur¨ Medizin Street No. Armin Schoisswohl 52425 Julich,¨ Germany GE Medical Systems [email protected] Kretz Ultrasound Tiefenbach 15 Kristian Witsch 4871 Zipf, Austria Heinrich-Heine University of [email protected] Dusseldorf¨ Lehrstuhl fur¨ Angewandte Mathematik Reto Bale Mathematisches Institut Universitatsklinik¨ fur¨ Radiodiagnostik Universitatsstraße¨ 1 SIP-Labor 40225 Dusseldorf,¨ Germany Anichstraße 35 [email protected] 6020 Innsbruck, Austria [email protected] Stephen L. Keeling Karl-Franzens University of Graz Harald Grossauer Institute of Mathematics University of Innsbruck Heinrichstraße 36 Institute of Computer Science 8010 Graz, Austria Technikerstraße 21a [email protected] 6020 Innsbruck, Austria [email protected] Jan Modersitzki University of Lubeck¨ Stefan Henn Institute of Mathematics Heinrich-Heine University of Wallstraße 40 Dusseldorf¨ D-23560 Lubeck¨ Lehrstuhl fur¨ Mathematische Opti- [email protected] mierung Mathematisches Institut Oliver Schmitt Universitatsstraße¨ 1 University of X List of Contributors

Institute of Anatomy Kristian Witsch Gertrudenstraße 9 Heinrich-Heine University of D-18055 Rostock, Germany Dusseldorf¨ [email protected] Lehrstuhl fur¨ Angewandte Mathematik Universitatsstraße¨ 1 Stefan Wirtz 40225 Dusseldorf,¨ Germany University of Lubeck¨ [email protected] Institute of Mathematics Wallstraße 40 D-23560 Lubeck¨ Joachim Weickert [email protected] Mathematical Image Analysis Group, Faculty of Mathematics and Computer Ulrich Clarenz Science, Gerhard-Mercator University of University, Building 27, Duisburg 66041 Saarbrucken,¨ Germany. Institute of Mathematics [email protected]. Lotharstraße 63/65, 47048 Duisburg, Germany [email protected] Andres´ Bruhn Mathematical Image Analysis Group, Marc Droske Faculty of Mathematics and Computer University of California Science, Math Sciences Department , Building 27, 520 Portola Plaza, 66041 Saarbrucken,¨ Germany. Los Angeles, CA, 90055, USA [email protected]. [email protected]

Stefan Henn Nils Papenberg Heinrich-Heine University of Mathematical Image Analysis Group, Dusseldorf¨ Faculty of Mathematics and Computer Lehrstuhl fur¨ Mathematische Opti- Science, mierung Saarland University, Building 27, Universitatsstraße¨ 1 66041 Saarbrucken,¨ Germany. 40225 Dusseldorf,¨ Germany [email protected]. [email protected]

Martin Rumpf Thomas Brox Rheinische Friedrich-Wilhelms- Mathematical Image Analysis Group, Universitat¨ Bonn Faculty of Mathematics and Computer Institut fur¨ Numerische Simulation Science, Nussallee 15, Saarland University, Building 27, 53115 Bonn, Germany 66041 Saarbrucken,¨ Germany. [email protected] [email protected].