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Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data

Author: Carl-Fredrik Westin

Publisher: Springer

Published: 2014-07-17

Total Pages: 346

ISBN-13: 3642543014

Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI). A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research. The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.

Visualization and Processing of Higher Order Descriptors for Multi-Valued Data

Author: Ingrid Hotz

Publisher: Springer

Published: 2015-07-03

Total Pages: 378

ISBN-13: 3319150901

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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.

Modeling, Analysis, and Visualization of Anisotropy

Author: Thomas Schultz

Publisher: Springer

Published: 2017-10-14

Total Pages: 407

ISBN-13: 3319613588

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This book focuses on the modeling, processing and visualization of anisotropy, irrespective of the context in which it emerges, using state-of-the-art mathematical tools. As such, it differs substantially from conventional reference works, which are centered on a particular application. It covers the following topics: (i) the geometric structure of tensors, (ii) statistical methods for tensor field processing, (iii) challenges in mapping neural connectivity and structural mechanics, (iv) processing of uncertainty, and (v) visualizing higher-order representations. In addition to original research contributions, it provides insightful reviews. This multidisciplinary book is the sixth in a series that aims to foster scientific exchange between communities employing tensors and other higher-order representations of directionally dependent data. A significant number of the chapters were co-authored by the participants of the workshop titled Multidisciplinary Approaches to Multivalued Data: Modeling, Visualization, Analysis, which was held in Dagstuhl, Germany in April 2016. It offers a valuable resource for those working in the field of multi-directional data, vital inspirations for the development of new models, and essential analysis and visualization techniques, thus furthering the state-of-the-art in studies involving anisotropy.

Information Processing in Medical Imaging

Author: Aasa Feragen

Publisher: Springer Nature

Published: 2021-06-20

Total Pages: 784

ISBN-13: 3030781917

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This book constitutes the proceedings of the 27th International Conference on Information Processing in Medical Imaging, IPMI 2021, which was held online during June 28-30, 2021. The conference was originally planned to take place in Bornholm, Denmark, but changed to a virtual format due to the COVID-19 pandemic. The 59 full papers presented in this volume were carefully reviewed and selected from 200 submissions. They were organized in topical sections as follows: registration; causal models and interpretability; generative modelling; shape; brain connectivity; representation learning; segmentation; sequential modelling; learning with few or low quality labels; uncertainty quantification and generative modelling; and deep learning.

Mathematical Morphology and Its Applications to Signal and Image Processing

Author: Jón Atli Benediktsson

Publisher: Springer

Published: 2015-05-15

Total Pages: 761

ISBN-13: 3319187201

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This book contains the thoroughly refereed proceedings of the 12th International Symposium on Mathematical Morphology, ISMM 2015 held in Reykjavik, Iceland, in May 2015. The 62 revised full papers were carefully reviewed and selected from 72 submissions. The papers are organized in topical sections on evaluations and applications; hierarchies; color, multivalued and orientation fields; optimization, differential calculus and probabilities; topology and discrete geometry; and algorithms and implementation.

Foundations of Data Visualization

Author: Min Chen

Publisher: Springer Nature

Published: 2020-08-11

Total Pages: 395

ISBN-13: 3030344444

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This is the first book that focuses entirely on the fundamental questions in visualization. Unlike other existing books in the field, it contains discussions that go far beyond individual visual representations and individual visualization algorithms. It offers a collection of investigative discourses that probe these questions from different perspectives, including concepts that help frame these questions and their potential answers, mathematical methods that underpin the scientific reasoning of these questions, empirical methods that facilitate the validation and falsification of potential answers, and case studies that stimulate hypotheses about potential answers while providing practical evidence for such hypotheses. Readers are not instructed to follow a specific theory, but their attention is brought to a broad range of schools of thoughts and different ways of investigating fundamental questions. As such, the book represents the by now most significant collective effort for gathering a large collection of discourses on the foundation of data visualization. Data visualization is a relatively young scientific discipline. Over the last three decades, a large collection of computer-supported visualization techniques have been developed, and the merits and benefits of using these techniques have been evidenced by numerous applications in practice. These technical advancements have given rise to the scientific curiosity about some fundamental questions such as why and how visualization works, when it is useful or effective and when it is not, what are the primary factors affecting its usefulness and effectiveness, and so on. This book signifies timely and exciting opportunities to answer such fundamental questions by building on the wealth of knowledge and experience accumulated in developing and deploying visualization technology in practice.

From Algebraic Structures to Tensors

Author: Gérard Favier

Publisher: John Wiley & Sons

Published: 2020-01-02

Total Pages: 324

ISBN-13: 1786301547

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Nowadays, tensors play a central role for the representation, mining, analysis, and fusion of multidimensional, multimodal, and heterogeneous big data in numerous fields. This set on Matrices and Tensors in Signal Processing aims at giving a self-contained and comprehensive presentation of various concepts and methods, starting from fundamental algebraic structures to advanced tensor-based applications, including recently developed tensor models and efficient algorithms for dimensionality reduction and parameter estimation. Although its title suggests an orientation towards signal processing, the results presented in this set will also be of use to readers interested in other disciplines. This first book provides an introduction to matrices and tensors of higher-order based on the structures of vector space and tensor space. Some standard algebraic structures are first described, with a focus on the hilbertian approach for signal representation, and function approximation based on Fourier series and orthogonal polynomial series. Matrices and hypermatrices associated with linear, bilinear and multilinear maps are more particularly studied. Some basic results are presented for block matrices. The notions of decomposition, rank, eigenvalue, singular value, and unfolding of a tensor are introduced, by emphasizing similarities and differences between matrices and tensors of higher-order.

Computer Analysis of Images and Patterns

Author: George Azzopardi

Publisher: Springer

Published: 2015-08-25

Total Pages: 821

ISBN-13: 3319231170

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The two volume set LNCS 9256 and 9257 constitutes the refereed proceedings of the 16th International Conference on Computer Analysis of Images and Patterns, CAIP 2015, held in Valletta, Malta, in September 2015. The 138 papers presented were carefully reviewed and selected from numerous submissions. CAIP 2015 is the sixteenth in the CAIP series of biennial international conferences devoted to all aspects of computer vision, image analysis and processing, pattern recognition, and related fields.

Computational Diffusion MRI

Author: Andrea Fuster

Publisher: Springer

Published: 2016-04-08

Total Pages: 236

ISBN-13: 3319285882

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These Proceedings of the 2015 MICCAI Workshop “Computational Diffusion MRI” offer a snapshot of the current state of the art on a broad range of topics within the highly active and growing field of diffusion MRI. The topics vary from fundamental theoretical work on mathematical modeling, to the development and evaluation of robust algorithms, new computational methods applied to diffusion magnetic resonance imaging data, and applications in neuroscientific studies and clinical practice. Over the last decade interest in diffusion MRI has exploded. The technique provides unique insights into the microstructure of living tissue and enables in-vivo connectivity mapping of the brain. Computational techniques are key to the continued success and development of diffusion MRI and to its widespread transfer into clinical practice. New processing methods are essential for addressing issues at each stage of the diffusion MRI pipeline: acquisition, reconstruction, modeling and model fitting, image processing, fiber tracking, connectivity mapping, visualization, group studies and inference. This volume, which includes both careful mathematical derivations and a wealth of rich, full-color visualizations and biologically or clinically relevant results, offers a valuable starting point for anyone interested in learning about computational diffusion MRI and mathematical methods for mapping brain connectivity, as well as new perspectives and insights on current research challenges for those currently working in the field. It will be of interest to researchers and practitioners in the fields of computer science, MR physics, and applied mathematics.

Neuromathematics of Vision

Author: Giovanna Citti

Publisher: Springer Science & Business Media

Published: 2014-02-08

Total Pages: 378

ISBN-13: 3642344445

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This book is devoted to the study of the functional architecture of the visual cortex. Its geometrical structure is the differential geometry of the connectivity between neural cells. This connectivity is building and shaping the hidden brain structures underlying visual perception. The story of the problem runs over the last 30 years, since the discovery of Hubel and Wiesel of the modular structure of the primary visual cortex, and slowly cams towards a theoretical understanding of the experimental data on what we now know as functional architecture of the primary visual cortex. Experimental data comes from several domains: neurophysiology, phenomenology of perception and neurocognitive imaging. Imaging techniques like functional MRI and diffusion tensor MRI allow to deepen the study of cortical structures. Due to this variety of experimental data, neuromathematematics deals with modelling both cortical structures and perceptual spaces. From the mathematical point of view, neuromathematical call for new instruments of pure mathematics: sub-Riemannian geometry models horizontal connectivity, harmonic analysis in non commutative groups allows to understand pinwheels structure, as well as non-linear dimensionality reduction is at the base of many neural morphologies and possibly of the emergence of perceptual units. But at the center of the neurogeometry is the problem of harmonizing contemporary mathematical instruments with neurophysiological findings and phenomenological experiments in an unitary science of vision. The contributions to this book come from the very founders of the discipline.