Visualization and Processing of Tensor Fields Advances and Perspecti
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David Laidlaw Joachim Weickert Editors
Visualization and Processing of Tensor Fields Advances and Perspectives With 131 Figures, 92 in Color and 13 Tables
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David Laidlaw
Joachim Weickert
Brown University Department of Computer Science Providence RI 02912 Box 1910 USA [email protected]
Saarland University Faculty of Mathematics and Computer Science Mathematical Image Analysis Group 66041 Saarbrücken Germany [email protected]
ISBN: 978-3-540-88377-7
e-ISBN: 978-3-540-88378-4
Mathematics and Visualization ISSN 1612-3786 Library of Congress Control Number: 2008936143 Mathematics Subject Classification (2000): 68-XX, 70-XX, 15-XX, 76-XX, 92-XX c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper springer.com
We dedicate this book to our families.
Preface
This book attempts to capture some of the excitement of an inspiring Dagstuhl Seminar in January 2007. The authors report on recent research results as well as opining on future directions for the analysis and visualization of tensor fields. Topics range from applications of the analysis of tensor fields to purer research into their mathematical and analytical properties. One of the goals of this seminar was to bring together researchers from along that pure-to-applied disciplinary axis with the hope of fostering new collaborations and research. This book, we hope, will continue to further that goal in a broader context.
Providence, Rhode Island, USA Saarbr¨ ucken, Saarland, Germany August 2008
David H. Laidlaw Joachim Weickert
Contents
Part I Models for Diffusion MRI Modelling, Fitting and Sampling in Diffusion MRI Daniel C. Alexander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Tensors, Polynomials and Models for Directional Data P.G. Batchelor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A Mixture of Wisharts (MOW) Model for Multifiber Reconstruction ¨ Bing Jian, Baba C. Vemuri, and Evren Ozarslan . . . . . . . . . . . . . . . . . . . . . 39 The Algebra of Fourth-Order Tensors with Application to Diffusion MRI Maher Moakher . . . . . . . . . . . . . . . .
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