Mathematical Foundations Of Scientific Visualization Computer Graphics And Massive Data Exploration

The goal of visualization is the accurate, interactive, and intuitive presentation of data. Complex numerical simulations, high-resolution imaging devices and incre- ingly common environment-embedded sensors are the primary generators of m- sive data sets. Being able to derive scienti?c insight from data increasingly depends on having mathematical and perceptual models to provide the necessary foundation for effective data analysis and comprehension. The peer-reviewed state-of-the-art research papers included in this book focus on continuous data models, such as is common in medical imaging or computational modeling. From the viewpoint of a visualization scientist, we typically collaborate with an application scientist or engineer who needs to visually explore or study an object which is given by a set of sample points, which originally may or may not have been connected by a mesh. At some point, one generally employs low-order piecewise polynomial approximationsof an object, using one or several dependent functions. In order to have an understanding of a higher-dimensional geometrical “object” or function, ef?cient algorithms supporting real-time analysis and manipulation (- tation, zooming) are needed. Often, the data represents 3D or even time-varying 3D phenomena (such as medical data), and the access to different layers (slices) and structures (the underlying topology) comprising such data is needed.

The goal of visualization is the accurate, interactive, and intuitive presentation of data. Complex numerical simulations, high-resolution imaging devices and incre- ingly common environment-embedded sensors are the primary generators of m- sive data sets. Being able to derive scienti?c insight from data increasingly depends on having mathematical and perceptual models to provide the necessary foundation for effective data analysis and comprehension. The peer-reviewed state-of-the-art research papers included in this book focus on continuous data models, such as is common in medical imaging or computational modeling. From the viewpoint of a visualization scientist, we typically collaborate with an application scientist or engineer who needs to visually explore or study an object which is given by a set of sample points, which originally may or may not have been connected by a mesh. At some point, one generally employs low-order piecewise polynomial approximationsof an object, using one or several dependent functions. In order to have an understanding of a higher-dimensional geometrical “object” or function, ef?cient algorithms supporting real-time analysis and manipulation (- tation, zooming) are needed. Often, the data represents 3D or even time-varying 3D phenomena (such as medical data), and the access to different layers (slices) and structures (the underlying topology) comprising such data is needed.

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.

Data visualization is currently a very active and vital area of research, teaching and development. The term unites the established field of scientific visualization and the more recent field of information visualization. The success of data visualization is due to the soundness of the basic idea behind it: the use of computer-generated images to gain insight and knowledge from data and its inherent patterns and relationships. A second premise is the utilization of the broad bandwidth of the human sensory system in steering and interpreting complex processes, and simulations involving data sets from diverse scientific disciplines and large collections of abstract data from many sources. These concepts are extremely important and have a profound and widespread impact on the methodology of computational science and engineering, as well as on management and administration. The interplay between various application areas and their specific problem solving visualization techniques is emphasized in this book. Reflecting the heterogeneous structure of Data Visualization, emphasis was placed on these topics: -Visualization Algorithms and Techniques; -Volume Visualization; -Information Visualization; -Multiresolution Techniques; -Interactive Data Exploration. Data Visualization: The State of the Art presents the state of the art in scientific and information visualization techniques by experts in this field. It can serve as an overview for the inquiring scientist, and as a basic foundation for developers. This edited volume contains chapters dedicated to surveys of specific topics, and a great deal of original work not previously published illustrated by examples from a wealth of applications. The book will also provide basic material for teaching the state of the art techniques in data visualization. Data Visualization: The State of the Art is designed to meet the needs of practitioners and researchers in scientific and information visualization. This book is also suitable as a secondary text for graduate level students in computer science and engineering.

This non-traditional introduction to the mathematics of scientific computation describes the principles behind the major methods, from statistics, applied mathematics, scientific visualization, and elsewhere, in a way that is accessible to a large part of the scientific community. Introductory material includes computational basics, a review of coordinate systems, an introduction to facets (planes and triangle meshes) and an introduction to computer graphics. The scientific computing part of the book covers topics in numerical linear algebra (basics, solving linear system, eigen-problems, SVD, and PCA) and numerical calculus (basics, data fitting, dynamic processes, root finding, and multivariate functions). The visualization component of the book is separated into three parts: empirical data, scalar values over 2D data, and volumes.

Scientific visualization has always been an integral part of discovery, starting first with simplified drawings of the pre-Enlightenment and progressing to present day. Mathematical formalism often supersedes visual methods, but their use is at the core of the mental process. As historical examples, a spatial description of flow led to electromagnetic theory, and without visualization of crystals, structural chemistry would not exist. With the advent of computer graphics technology, visualization has become a driving force in modern computing. A Concise Introduction to Scientific Visualization – Past, Present, and Future serves as a primer to visualization without assuming prior knowledge. It discusses both the history of visualization in scientific endeavour, and how scientific visualization is currently shaping the progress of science as a multi-disciplinary domain.

Numerical simulations of global warming, Mars observation data, and aircraft design are but a few of the topics where the use of human visual perception for data understanding are considered essential. Ten years agoa handful of pioneers professed the value of visualization to skeptical audiences. Today, with supercomputers and sensors producing ever-increasing amounts of data, scientific visualization is accepted throughout much of science and engineering as the fundamental tool for data analysis. Written by a world-wide panel of visualization experts, Scientific Visualization: Advances and Challenges presents astute coverage of prevailing trends, issues, and practice of scientific visualization. From algorithmic topics such as volume graphics and the modeling and visualization of large data sets, to foundations, perception, and interface technology (including virtual reality), this book provides the latest advances in the area. The book demonstrates new techniques, examines diverse application areas, and discusses current limitations and upcoming requirements. Scientific Visualization:Advances and Challenges $> presents readers with a unique opportunity to examine expert thinking and current practice, and to obtain a vision of potential future directions. It will be essential reading for scientific and engineering practitioners and visualization researchers alike. Offers extremely topical and timely coverage of a rapidly evolving area Includes contributions from an international panel of visualization experts in one accessible volume Provides scientific and engineering practitioners as well as visualization researchers with an essential guide to the literature

The nature of the physical Universe has been increasingly better understood in recent years, and cosmological concepts have undergone a rapid evolution (see, e.g., [11], [2],or [5]). Although there are alternate theories, it is generally believed that the large-scale relationships and homogeneities that we see can only be explainedby having the universe expand suddenlyin a very early “in?ationary” period. Subsequent evolution of the Universe is described by the Hubble expansion, the observation that the galaxies are ?ying away from each other. We can attribute di?erent rates of this expansion to domination of di?erent cosmological processes, beginning with radiation, evolving to matter domination, and, relatively recently, to vacuum domination (the Cosmological Constant term)[4]. We assume throughout that we will be relying as much as possible on observational data, with simulations used only for limited purposes, e.g., the appearance of the Milky Wayfrom nearbyintergalactic viewpoints. The visualization of large-scale astronomical data sets using?xed, non-interactive animations has a long history. Several books and ?lms exist, ranging from “Cosmic View: The Universe in Forty Jumps” [3] by Kees Boeke to “Powers of 10” [6,13] by Charles and Ray Eames, and the recent Imax ?lm “Cosmic Voyage” [15]. We have added our own contribution [9], “Cosmic Clock,” which is an animation based entirely on the concepts and implementation described in this paper.