Visualization and Processing of Higher Order Descriptors for Multi Valued Data Modern imaging techniques and computational simulations yield complex multi valued data that require higher order mathematical descriptors. This book addresses topics of importance when dealing with such data including frameworks for image processing visualization and statistical analysis of higher order descriptors. It also provides examples of the successful use of higher order descriptors in specific applications and a glimpse of the next generation of diffusion MRI. To do so it combines contributions on new developments current challenges in this area and state of the art surveys. Compared to the increasing importance of higher order descriptors in a range of applications tools for analysis and processing are still relatively hard to come by. Even though application areas such as medical imaging fluid dynamics and structural mechanics are very different in nature they face many shared challenges. This book provides an interdisciplinary perspective on this topic with contributions from key researchers in disciplines ranging from visualization and image processing to applications. It is based on the 5th Dagstuhl seminar on Visualization and Processing of Higher Order Descriptors for Multi Valued Data. This book will appeal to scientists who are working to develop new analysis methods in the areas of image processing and visualization as well as those who work with applications that generate higher order data or could benefit from higher order models and are searching for novel analytical tools Part I: Mathematical Foundations: 1 Cem Yolcu and Evren Özarslan: Diffusion Weighted Magnetic Resonance Signal for General Gradient Waveforms: Multiple Correlation Function Framework Path Integrals and Parallels Between Them 2 T.C.J. Dela Haije A. Fuster and L.M.J. Florack: Finslerian Diffusion and the Bloch Torrey Equation 3 M. Moakher and P. J. Basser: Fiber Orientation Distribution Functions and Orientation Tensors for Different Material Symmetries 4 Yue Zhang Jonathan Palacios and Eugene Zhang: Topology of 3D Linear Symmetric Tensor Fields 5 Carmeliza Navasca and Deonnia N. Pompey: Random Projections for Low Multilinear Rank Tensors Part II: Processing Filtering and Interpolation: 6 Jasper J. van de Gronde Mikola Lysenko and Jos B.T.M. Roerdink: Path Based Mathematical Morphology on Tensor Fields 7 Andreas Kleefeld and Bernhard Burgeth: Processing Multispectral Images via Mathematical Morphology 8 Luc Florack Tom Dela Haije and Andrea Fuster: Direction Controlled DTI Interpolation 9 Daniel Jörgens and Rodrigo Moreno: Tensor Voting: Current State Challenges and New Trends in the Context of Medical Image Analysis Part III: Visualization: 10 Olivier Vaillancourt Maxime Chamberland Jean Christophe Houde and Maxime Descoteaux: Visualization of Diffusion Propagator and Multiple Parameter Diffusion Signal 11 Sujal Bista Jiachen Zhuo Rao P. Gullapalli and Amitabh Varshney: Visual Knowledge Discovery for Diffusion Kurtosis Datasets of the Human Brain 12 Tobias Isenberg: A Survey of Illustrative Visualization Techniques for Diffusion Weighted MRI Tractography 13 Valentin Zobel Jan Reininghaus and Ingrid Hotz: Visualizing Symmetric Indefinite 2D Tensor Fields sing the Heat Kernel Signature Part IV: Statistical Analysis: 14 Maxime Taquet Benoit Scherrer and Simon K. Warfield: A Framework for the Analysis of Diffusion Compartment Imaging (DCT) 15 Lauren J. O’Donnell and Thomas Schultz: Statistical and Machine Learning Methods for Neuroimaging: Examples Challenges and Extensions to Diffusion Imaging Data Part V: Applications: 16 A. aries T. Luciani P.H. Pisciuneri M.B. Nik S.L. Yilmaz P. Givi and G.E. Marai A Clustering Method for Identifying Regions of Interest in Turbulent Combustion Tensor Fields 17 Mark Schöneich Andrea Kratz Valentin Zobel Gerik Scheuermann Markus Stommel and Ingrid Hotz: Tensor Lines in Engineering Success Failure and Open Questions 18 Vesna Prchkovska Magí Andorrà Pablo Villoslada Eloy Martinez Heras Remco Duits David Fortin Paulo Rodrigues and Maxime Descoteaux: Contextual Diffusion Image Post Processing Aids Clinical Applications