Numerical Methods For Linear Complementarity Problems In Physics Based Animation

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Numerical Methods for Linear Complementarity Problems in Physics-Based Animation

Author : Sarah Niebe
Publisher : Springer Nature
ISBN 13 : 3031795644
Total Pages : 151 pages
Book Rating : 4.0/5 (317 download)

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Book Synopsis Numerical Methods for Linear Complementarity Problems in Physics-Based Animation by : Sarah Niebe

Download or read book Numerical Methods for Linear Complementarity Problems in Physics-Based Animation written by Sarah Niebe and published by Springer Nature. This book was released on 2022-05-31 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: Linear complementarity problems (LCPs) have for many years been used in physics-based animation to model contact forces between rigid bodies in contact. More recently, LCPs have found their way into the realm of fluid dynamics. Here, LCPs are used to model boundary conditions with fluid-wall contacts. LCPs have also started to appear in deformable models and granular simulations. There is an increasing need for numerical methods to solve the resulting LCPs with all these new applications. This book provides a numerical foundation for such methods, especially suited for use in computer graphics. This book is mainly intended for a researcher/Ph.D. student/post-doc/professor who wants to study the algorithms and do more work/research in this area. Programmers might have to invest some time brushing up on math skills, for this we refer to Appendices A and B. The reader should be familiar with linear algebra and differential calculus. We provide pseudo code for all the numerical methods, which should be comprehensible by any computer scientist with rudimentary programming skills. The reader can find an online supplementary code repository, containing Matlab implementations of many of the core methods covered in these notes, as well as a few Python implementations [Erleben, 2011]. Table of Contents: Introduction / Numerical Methods / Guide for Software and Selecting Methods / Bibliography / Authors' Biographies

Finite Element Method Simulation of 3D Deformable Solids

Author : Eftychios Sifakis
Publisher : Springer Nature
ISBN 13 : 3031025857
Total Pages : 57 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Finite Element Method Simulation of 3D Deformable Solids by : Eftychios Sifakis

Download or read book Finite Element Method Simulation of 3D Deformable Solids written by Eftychios Sifakis and published by Springer Nature. This book was released on 2022-06-01 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book serves as a practical guide to simulation of 3D deformable solids using the Finite Element Method (FEM). It reviews a number of topics related to the theory and implementation of FEM approaches: measures of deformation, constitutive laws of nonlinear materials, tetrahedral discretizations, and model reduction techniques for real-time simulation. Simulations of deformable solids are important in many applications in computer graphics, including film special effects, computer games, and virtual surgery. The Finite Element Method has become a popular tool in many such applications. Variants of FEM catering to both offline and real-time simulation have had a mature presence in computer graphics literature. This book is designed for readers familiar with numerical simulation in computer graphics, who would like to obtain a cohesive picture of the various FEM simulation methods available, their strengths and weaknesses, and their applicability in various simulation scenarios. The book is also a practical implementation guide for the visual effects developer, offering a lean yet adequate synopsis of the underlying mathematical theory. Chapter 1 introduces the quantitative descriptions used to capture the deformation of elastic solids, the concept of strain energy, and discusses how force and stress result as a response to deformation. Chapter 2 reviews a number of constitutive models, i.e., analytical laws linking deformation to the resulting force that has successfully been used in various graphics-oriented simulation tasks. Chapter 3 summarizes how deformation and force can be computed discretely on a tetrahedral mesh, and how an implicit integrator can be structured around this discretization. Finally, chapter 4 presents the state of the art in model reduction techniques for real-time FEM solid simulation and discusses which techniques are suitable for which applications. Topics discussed in this chapter include linear modal analysis, modal warping, subspace simulation, and domain decomposition.

An Introduction to Verification of Visualization Techniques

Author : Tiago Etiene
Publisher : Springer Nature
ISBN 13 : 3031025873
Total Pages : 80 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis An Introduction to Verification of Visualization Techniques by : Tiago Etiene

Download or read book An Introduction to Verification of Visualization Techniques written by Tiago Etiene and published by Springer Nature. This book was released on 2022-06-01 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt: As we increase our reliance on computer-generated information, often using it as part of our decision-making process, we must devise tools to assess the correctness of that information. Consider, for example, software embedded on vehicles, used for simulating aircraft performance, or used in medical imaging. In those cases, software correctness is of paramount importance as there's little room for error. Software verification is one of the tools available to attain such goals. Verification is a well known and widely studied subfield of computer science and computational science and the goal is to help us increase confidence in the software implementation by verifying that the software does what it is supposed to do. The goal of this book is to introduce the reader to software verification in the context of visualization. In the same way we became more dependent on commercial software, we have also increased our reliance on visualization software. The reason is simple: visualization is the lens through which users can understand complex data, and as such it must be verified. The explosion in our ability to amass data requires tools not only to store and analyze data, but also to visualize it. This book is comprised of six chapters. After an introduction to the goals of the book, we present a brief description of both worlds of visualization (Chapter 2) and verification (Chapter 3). We then proceed to illustrate the main steps of the verification pipeline for visualization algorithms. We focus on two classic volume visualization techniques, namely, Isosurface Extraction (Chapter 4) and Direct Volume Rendering (Chapter 5). We explain how to verify implementations of those techniques and report the latest results in the field of verification of visualization techniques. The last chapter concludes the book and highlights new research topics for the future.

Virtual Crowds

Author : Mubbasir Kapadia
Publisher : Springer Nature
ISBN 13 : 3031025865
Total Pages : 248 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Virtual Crowds by : Mubbasir Kapadia

Download or read book Virtual Crowds written by Mubbasir Kapadia and published by Springer Nature. This book was released on 2022-05-31 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents novel computational models for representing digital humans and their interactions with other virtual characters and meaningful environments. In this context, we describe efficient algorithms to animate, control, and author human-like agents having their own set of unique capabilities, personalities, and desires. We begin with the lowest level of footstep determination to steer agents in collision-free paths. Steering choices are controlled by navigation in complex environments, including multi-domain planning with dynamically changing situations. Virtual agents are given perceptual capabilities analogous to those of real people, including sound perception, multi-sense attention, and understanding of environment semantics which affect their behavior choices. The roles and impacts of individual attributes, such as memory and personality are explored. The animation challenges of integrating a number of simultaneous behavior and movement demands on an agent are addressed through an open source software system. Finally, the creation of stories and narratives with groups of agents subject to planning and environmental constraints culminates the presentation.

Geometric and Discrete Path Planning for Interactive Virtual Worlds

Author : Marcelo Kallmann
Publisher : Springer Nature
ISBN 13 : 3031025881
Total Pages : 181 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Geometric and Discrete Path Planning for Interactive Virtual Worlds by : Marcelo Kallmann

Download or read book Geometric and Discrete Path Planning for Interactive Virtual Worlds written by Marcelo Kallmann and published by Springer Nature. This book was released on 2022-05-31 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: Path planning and navigation are indispensable components for controlling autonomous agents in interactive virtual worlds. Given the growing demands on the size and complexity of modern virtual worlds, a number of new techniques have been developed for achieving intelligent navigation for the next generation of interactive multi-agent simulations. This book reviews the evolution of several related techniques, starting from classical planning and computational geometry techniques and then gradually moving toward more advanced topics with focus on recent developments from the work of the authors. The covered topics range from discrete search and geometric representations to planning under different types of constraints and harnessing the power of graphics hardware in order to address Euclidean shortest paths and discrete search for multiple agents under limited time budgets. The use of planning algorithms beyond path planning is also discussed in the areas of crowd animation and whole-body motion planning for virtual characters.

Efficient Quadrature Rules for Illumination Integrals

Author : Ricardo Marques
Publisher : Springer Nature
ISBN 13 : 3031795679
Total Pages : 82 pages
Book Rating : 4.0/5 (317 download)

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Book Synopsis Efficient Quadrature Rules for Illumination Integrals by : Ricardo Marques

Download or read book Efficient Quadrature Rules for Illumination Integrals written by Ricardo Marques and published by Springer Nature. This book was released on 2022-06-01 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: Rendering photorealistic images is a costly process which can take up to several days in the case of high quality images. In most cases, the task of sampling the incident radiance function to evaluate the illumination integral is responsible for an important share of the computation time. Therefore, to reach acceptable rendering times, the illumination integral must be evaluated using a limited set of samples. Such a restriction raises the question of how to obtain the most accurate approximation possible with such a limited set of samples. One must thus ensure that sampling produces the highest amount of information possible by carefully placing and weighting the limited set of samples. Furthermore, the integral evaluation should take into account not only the information brought by sampling but also possible information available prior to sampling, such as the integrand smoothness. This idea of sparse information and the need to fully exploit the little information available is present throughout this book. The presented methods correspond to the state-of-the-art solutions in computer graphics, and take into account information which had so far been underexploited (or even neglected) by the previous approaches. The intended audiences are Ph.D. students and researchers in the field of realistic image synthesis or global illumination algorithms, or any person with a solid background in graphics and numerical techniques.

Mathematical Basics of Motion and Deformation in Computer Graphics

Author : Ken Anjyo
Publisher : Morgan & Claypool Publishers
ISBN 13 : 1627059849
Total Pages : 97 pages
Book Rating : 4.6/5 (27 download)

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Book Synopsis Mathematical Basics of Motion and Deformation in Computer Graphics by : Ken Anjyo

Download or read book Mathematical Basics of Motion and Deformation in Computer Graphics written by Ken Anjyo and published by Morgan & Claypool Publishers. This book was released on 2017-04-13 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.

Design, Representations, and Processing for Additive Manufacturing

Author : Marco Attene
Publisher : Springer Nature
ISBN 13 : 3031025962
Total Pages : 136 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Design, Representations, and Processing for Additive Manufacturing by : Marco Attene

Download or read book Design, Representations, and Processing for Additive Manufacturing written by Marco Attene and published by Springer Nature. This book was released on 2022-06-01 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: The wide diffusion of 3D printing technologies continuously calls for effective solutions for designing and fabricating objects of increasing complexity. The so called "computational fabrication" pipeline comprises all the steps necessary to turn a design idea into a physical object, and this book describes the most recent advancements in the two fundamental phases along this pipeline: design and process planning. We examine recent systems in the computer graphics community that allow us to take a design idea from conception to a digital model, and classify algorithms that are necessary to turn such a digital model into an appropriate sequence of machining instructions.

Geometric Continuity of Curves and Surfaces

Author : Przemysław Kiciak
Publisher : Springer Nature
ISBN 13 : 3031025903
Total Pages : 233 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Geometric Continuity of Curves and Surfaces by : Przemysław Kiciak

Download or read book Geometric Continuity of Curves and Surfaces written by Przemysław Kiciak and published by Springer Nature. This book was released on 2022-05-31 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is written for students, CAD system users and software developers who are interested in geometric continuity—a notion needed in everyday practice of Computer-Aided Design and also a hot subject of research. It contains a description of the classical geometric spline curves and a solid theoretical basis for various constructions of smooth surfaces. Textbooks on computer graphics usually cover the most basic and necessary information about spline curves and surfaces in order to explain simple algorithms. In textbooks on geometric design, one can find more details, more algorithms and more theory. This book teaches how various parts of the theory can be gathered together and turned into constructions of smooth curves and smooth surfaces of arbitrary topology. The mathematical background needed to understand this book is similar to what is necessary to read other textbooks on geometric design; most of it is basic linear algebra and analysis. More advanced mathematical material is introduced using elementary explanations. Reading Geometric Continuity of Curves and Surfaces provides an excellent opportunity to recall and exercise necessary mathematical notions and it may be your next step towards better practice and higher understanding of design principles.

Digital Heritage Reconstruction Using Super-resolution and Inpainting

Author : Milind G. Padalkar
Publisher : Springer Nature
ISBN 13 : 3031025911
Total Pages : 150 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Digital Heritage Reconstruction Using Super-resolution and Inpainting by : Milind G. Padalkar

Download or read book Digital Heritage Reconstruction Using Super-resolution and Inpainting written by Milind G. Padalkar and published by Springer Nature. This book was released on 2022-06-01 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: Heritage sites across the world have witnessed a number of natural calamities, sabotage and damage from visitors, resulting in their present ruined condition. Many sites are now restricted to reduce the risk of further damage. Yet these masterpieces are significant cultural icons and critical markers of past civilizations that future generations need to see. A digitally reconstructed heritage site could diminish further harm by using immersive navigation or walkthrough systems for virtual environments. An exciting key element for the viewer is observing fine details of the historic work and viewing monuments in their undamaged form. This book presents image super-resolution methods and techniques for automatically detecting and inpainting damaged regions in heritage monuments, in order to provide an enhanced visual experience. The book presents techniques to obtain higher resolution photographs of the digitally reconstructed monuments, and the resulting images can serve as input to immersive walkthrough systems. It begins with the discussion of two novel techniques for image super-resolution and an approach for inpainting a user-supplied region in the given image, followed by a technique to simultaneously perform super-resolution and inpainting of given missing regions. It then introduces a method for automatically detecting and repairing the damage to dominant facial regions in statues, followed by a few approaches for automatic crack repair in images of heritage scenes. This book is a giant step toward ensuring that the iconic sites of our past are always available, and will never be truly lost.

Mathematical Basics of Motion and Deformation in Computer Graphics, Second Edition

Author : Ken Anjyo
Publisher : Springer Nature
ISBN 13 : 303102592X
Total Pages : 79 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Mathematical Basics of Motion and Deformation in Computer Graphics, Second Edition by : Ken Anjyo

Download or read book Mathematical Basics of Motion and Deformation in Computer Graphics, Second Edition written by Ken Anjyo and published by Springer Nature. This book was released on 2022-06-01 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.

Heterogeneous Spatial Data

Author : Giuseppe Patanè
Publisher : Springer Nature
ISBN 13 : 303102589X
Total Pages : 129 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Heterogeneous Spatial Data by : Giuseppe Patanè

Download or read book Heterogeneous Spatial Data written by Giuseppe Patanè and published by Springer Nature. This book was released on 2022-05-31 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: New data acquisition techniques are emerging and are providing fast and efficient means for multidimensional spatial data collection. Airborne LIDAR surveys, SAR satellites, stereo-photogrammetry and mobile mapping systems are increasingly used for the digital reconstruction of the environment. All these systems provide extremely high volumes of raw data, often enriched with other sensor data (e.g., beam intensity). Improving methods to process and visually analyze this massive amount of geospatial and user-generated data is crucial to increase the efficiency of organizations and to better manage societal challenges. Within this context, this book proposes an up-to-date view of computational methods and tools for spatio-temporal data fusion, multivariate surface generation, and feature extraction, along with their main applications for surface approximation and rainfall analysis. The book is intended to attract interest from different fields, such as computer vision, computer graphics, geomatics, and remote sensing, working on the common goal of processing 3D data. To this end, it presents and compares methods that process and analyze the massive amount of geospatial data in order to support better management of societal challenges through more timely and better decision making, independent of a specific data modeling paradigm (e.g., 2D vector data, regular grids or 3D point clouds). We also show how current research is developing from the traditional layered approach, adopted by most GIS softwares, to intelligent methods for integrating existing data sets that might contain important information on a geographical area and environmental phenomenon. These services combine traditional map-oriented visualization with fully 3D visual decision support methods and exploit semantics-oriented information (e.g., a-priori knowledge, annotations, segmentations) when processing, merging, and integrating big pre-existing data sets.

Stochastic Partial Differential Equations for Computer Vision with Uncertain Data

Author : Tobias Preusser
Publisher : Springer Nature
ISBN 13 : 3031025946
Total Pages : 150 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Stochastic Partial Differential Equations for Computer Vision with Uncertain Data by : Tobias Preusser

Download or read book Stochastic Partial Differential Equations for Computer Vision with Uncertain Data written by Tobias Preusser and published by Springer Nature. This book was released on 2022-06-01 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be—and more and more frequently are—taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate. The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness. This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.

Cloth Simulation for Computer Graphics

Author : Tuur Stuyck
Publisher : Springer Nature
ISBN 13 : 3031025970
Total Pages : 110 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Cloth Simulation for Computer Graphics by : Tuur Stuyck

Download or read book Cloth Simulation for Computer Graphics written by Tuur Stuyck and published by Springer Nature. This book was released on 2022-06-01 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: Physics-based animation is commonplace in animated feature films and even special effects for live-action movies. Think about a recent movie and there will be some sort of special effects such as explosions or virtual worlds. Cloth simulation is no different and is ubiquitous because most virtual characters (hopefully!) wear some sort of clothing. The focus of this book is physics-based cloth simulation. We start by providing background information and discuss a range of applications. This book provides explanations of multiple cloth simulation techniques. More specifically, we start with the most simple explicitly integrated mass-spring model and gradually work our way up to more complex and commonly used implicitly integrated continuum techniques in state-of-the-art implementations. We give an intuitive explanation of the techniques and give additional information on how to efficiently implement them on a computer. This book discusses explicit and implicit integration schemes for cloth simulation modeled with mass-spring systems. In addition to this simple model, we explain the more advanced continuum-inspired cloth model introduced in the seminal work of Baraff and Witkin [1998]. This method is commonly used in industry. We also explain recent work by Liu et al. [2013] that provides a technique to obtain fast simulations. In addition to these simulation approaches, we discuss how cloth simulations can be art directed for stylized animations based on the work of Wojan et al. [2016]. Controllability is an essential component of a feature animation film production pipeline. We conclude by pointing the reader to more advanced techniques.

Virtual Material Acquisition and Representation for Computer Graphics

Author : Dar'ya Guarnera
Publisher : Springer Nature
ISBN 13 : 3031025954
Total Pages : 93 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Virtual Material Acquisition and Representation for Computer Graphics by : Dar'ya Guarnera

Download or read book Virtual Material Acquisition and Representation for Computer Graphics written by Dar'ya Guarnera and published by Springer Nature. This book was released on 2022-05-31 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides beginners in computer graphics and related fields a guide to the concepts, models, and technologies for realistic rendering of material appearance. It provides a complete and thorough overview of reflectance models and acquisition setups, along with providing a selection of the available tools to explore, visualize, and render the reflectance data. Reflectance models are under continuous development, since there is still no straightforward solution for general material representations. Every reflectance model is specific to a class of materials. Hence, each has strengths and weaknesses, which the book highlights in order to help the reader choose the most suitable model for any purpose. The overview of the acquisition setups will provide guidance to a reader who needs to acquire virtual materials and will help them to understand which measurement setup can be useful for a particular purpose, while taking into account the performance and the expected cost derived from the required components. The book also describes several recent open source software solutions, useful for visualizing and manipulating a wide variety of reflectance models and data.

An Introduction to Laplacian Spectral Distances and Kernels

Author : Giuseppe Patanè
Publisher : Springer Nature
ISBN 13 : 3031025938
Total Pages : 120 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis An Introduction to Laplacian Spectral Distances and Kernels by : Giuseppe Patanè

Download or read book An Introduction to Laplacian Spectral Distances and Kernels written by Giuseppe Patanè and published by Springer Nature. This book was released on 2022-05-31 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noisy scalar functions. All the reviewed numerical schemes are discussed and compared in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate with respect to shape representation, computational resources, and target application.

GPU Ray Tracing in Non-Euclidean Spaces

Author : Tiago Novello
Publisher : Morgan & Claypool Publishers
ISBN 13 : 1636393276
Total Pages : 129 pages
Book Rating : 4.6/5 (363 download)

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Book Synopsis GPU Ray Tracing in Non-Euclidean Spaces by : Tiago Novello

Download or read book GPU Ray Tracing in Non-Euclidean Spaces written by Tiago Novello and published by Morgan & Claypool Publishers. This book was released on 2022-03-21 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the visualization of three-dimensional non-Euclidean spaces using raytracing techniques in Graphics Processing Unit (GPU). This is a trending topic in mathematical visualization that combines the mathematics areas of geometry and topology, with visualization concepts of computer graphics. Several conditions made this a special moment for such topic. On one hand, the development of mathematical research, computer graphics, and algorithms have provided the necessary theoretical framework. On the other hand, the evolution of the technologies and media allows us to be immersed in three-dimensional spaces using Virtual Reality. The content of this book serves both experts in the areas and students. Although this is a short book, it is self-contained since it considers all the ideas, motivations, references, and intuitive explanations of the required fundamental concepts.