Computer vision and image analysis require interdisciplinary collaboration between mathematics and engineering. This book addresses the area of high-accuracy measurements of length, curvature, motion parameters and other geometrical quantities from acquired image data. It is a common problem that these measurements are incomplete or noisy, such that considerable efforts are necessary to regularise the data, to fill in missing information, and to judge the accuracy and reliability of these results. This monograph brings together contributions from researchers in computer vision, engineering and mathematics who are working in this area.

The book can be read both by specialists and graduate students in computer science, electrical engineering or mathematics who take an interest in data evaluations by approximation or interpolation, in particular data obtained in an image analysis context.

Matrix-valued data sets - so-called second order tensor fields - have gained significant importance in scientific visualization and image processing due to recent developments such as diffusion tensor imaging. This book is the first edited volume that presents the state of the art in the visualization and processing of tensor fields. It contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work by leading experts that has not been published before. It serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as as a textbook for specialized classes and seminars for graduate and doctoral students.

Visualisation and Processing of Tensor Fields provides researchers an inspirational look at how to process and visualize complicated 2D and 3D images known as tensor fields. Tensor fields are the natural representation for many physical quantities; they can describe how water moves around in the brain, how gravity varies around the earth, or how materials are stressed and deformed. With its numerous color figures, this book helps the reader understand both the underlying mathematics and the applications of tensor fields. The reader also will learn about the most recent research topics and open research questions.

Matrix-valued data sets - so-called second order tensor fields - have gained significant importance in scientific visualization and image processing due to recent developments such as diffusion tensor imaging. This book is the first edited volume that presents the state of the art in the visualization and processing of tensor fields. It contains some longer chapters dedicated to surveys and tutorials of specific topics, as well as a great deal of original work by leading experts that has not been published before. It serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as as a textbook for specialized classes and seminars for graduate and doctoral students.

Computer vision and image analysis require interdisciplinary collaboration between mathematics and engineering. This book addresses the area of high-accuracy measurements of length, curvature, motion parameters and other geometrical quantities from acquired image data. It is a common problem that these measurements are incomplete or noisy, such that considerable efforts are necessary to regularise the data, to fill in missing information, and to judge the accuracy and reliability of these results. This monograph brings together contributions from researchers in computer vision, engineering and mathematics who are working in this area.

The book can be read both by specialists and graduate students in computer science, electrical engineering or mathematics who take an interest in data evaluations by approximation or interpolation, in particular data obtained in an image analysis context.

ThisbookattemptstocapturesomeoftheexcitementofaninspiringDagstuhl SeminarinJanuary2007. Theauthorsreportonrecentresearchresultsaswell as opining on future directions for the analysis and visualization of tensor ?elds. Topics range from applications of the analysis of tensor ?elds to purer researchintotheirmathematical andanalytical properties. Oneofthegoalsof thisseminarwastobringtogetherresearchersfromalongthatpure-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 David H. Laidlaw Saarbruc ] ken, Saarland, Germany Joachim Weickert August 2008 Contents Part I Models for Di?usion MRI Modelling, Fitting and Sampling in Di?usion MRI Daniel C. Alexander. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Tensors, Polynomials and Models for Directional Data P. G. Batchelor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A Mixture of Wisharts (MOW) Model for Multi?ber Reconstruction ] Bing Jian, Baba C. Vemuri, and Evren Ozarslan. . . . . . . . . . . . . . . . . . . . . 39 The Algebra of Fourth-Order Tensors with Application to Di?usion MRI Maher Moakher. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Part II Higher-Level Analysis of Di?usion Images Structure-Speci?c StatisticalMappingofWhiteMatterTracts Paul A. Yushkevich, Hui Zhang, Tony J. Simon, and James C. Gee. . . . 83 Analysis of Distance/Similarity Measures for Di?usion Tensor Imaging T. H. J. M. Peeters, P. R. Rodrigues, A. Vilanova, and B. M. ter Haar Romeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 XContents Part III Tensor Field Visualization Tensor Glyph Warping: Visualizing Metric Tensor Fields using Riemannian Exponential Maps Anders Brun and Hans Knutsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Interactive Volume Rendering of Di?usion Tensor Data Mario Hlawitschka, Gunther H. Weber, Alfred Anwander, Owen T. Carmichael, Bernd Hamann, and Gerik Scheuermann. . . . . . . . 161 Dense Glyph Sampling for Visualization Louis Feng, Ingrid Hotz, Bernd Hamann, and Kenneth Joy. . . . . . . . . . . ."

This volume constitutes the refereed proceedings of the Second International Conference on Scale-Space Theories in Computer Vision, Scale-Space'99, held in Corfu, Greece, in September 1999.
The 36 revised full papers and the 18 revised posters presented in the book were carefully reviewed and selected from 66 high-quality submissions. The book addresses all current aspects of this young and active field, in particular geometric Image flows, nonlinear diffusion, functional minimization, linear scale-space, etc.

This book constitutes the refereed proceedings of the 35th German Conference on Pattern Recognition, GCPR 2013, held in Saarbrucken, Germany, in September 2013. The 22 revised full papers and 18 revised poster papers were carefully reviewed and selected from 79 submissions. The papers covers topics such as image processing and computer vision, machine learning and pattern recognition, mathematical foundations, statistical data analysis and models, computational photography and confluence of vision and graphics, and applications in natural sciences, engineering, biomedical data analysis, imaging, and industry.