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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.

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

This book is a guide for researchers and practitioners to the new frontiers of 3D shape analysis and the complex mathematical tools most methods rely on. The target reader includes students, researchers and professionals with an undergraduate mathematics background, who wish to understand the mathematics behind shape analysis. The authors begin with a quick review of basic concepts in geometry, topology, differential geometry, and proceed to advanced notions of algebraic topology, always keeping an eye on the application of the theory, through examples of shape analysis methods such as 3D segmentation, correspondence, and retrieval. A number of research solutions in the field come from advances in pure and applied mathematics, as well as from the re-reading of classical theories and their adaptation to the discrete setting. In a world where disciplines (fortunately) have blurred boundaries, the authors believe that this guide will help to bridge the distance between theory and practice. Table of Contents: Acknowledgments / Figure Credits / About this Book / 3D Shape Analysis in a Nutshell / Geometry, Topology, and Shape Representation / Differential Geometry and Shape Analysis / Spectral Methods for Shape Analysis / Maps and Distances between Spaces / Algebraic Topology and Topology Invariants / Differential Topology and Shape Analysis / Reeb Graphs / Morse and Morse-Smale Complexes / Topological Persistence / Beyond Geometry and Topology / Resources / Bibliography / Authors' Biographies

Information Theory (IT) tools, widely used in many scientific fields such as engineering, physics, genetics, neuroscience, and many others, are also useful transversal tools in image processing. In this book, we present the basic concepts of IT and how they have been used in the image processing areas of registration, segmentation, video processing, and computational aesthetics. Some of the approaches presented, such as the application of mutual information to registration, are the state of the art in the field. All techniques presented in this book have been previously published in peer-reviewed conference proceedings or international journals. We have stressed here their common aspects, and presented them in an unified way, so to make clear to the reader which problems IT tools can help to solve, which specific tools to use, and how to apply them. The IT basics are presented so as to be self-contained in the book. The intended audiences are students and practitioners of image processing and related areas such as computer graphics and visualization. In addition, students and practitioners of IT will be interested in knowing about these applications. Table of Contents: Preface / Acknowledgments / Information Theory Basics / Image Registration / Image Segmentation / Video Key Frame Selection / Informational Aesthetics Measures / Bibliography / Authors' Biographies

Eye tracking is a process that identifies a specific point in both space and time that is being looked at by the observer. This information can also be used in real-time to control applications using the eyes. Recent innovations in the video game industry include alternative input modalities to provide an enhanced, more immersive user experience. In particular, eye gaze control has recently been explored as an input modality in video games. This book is an introduction for those interested in using eye tracking to control or analyze video games and virtual environments. Key concepts are illustrated through three case studies in which gaze control and voice recognition have been used in combination to control virtual characters and applications. The lessons learned in the case studies are presented and issues relating to incorporating eye tracking in interactive applications are discussed. The reader will be given an introduction to human visual attention, eye movements and eye tracking technologies. Previous work in the field of studying fixation behavior in games and using eye tracking for video game interaction will also be presented. The final chapter discusses ideas for how this field can be developed further to create richer interaction for characters and crowds in virtual environments. Alternative means of interaction in video games are especially important for disabled users for whom traditional techniques, such as mouse and keyboard, may be far from ideal. This book is also relevant for those wishing to use gaze control in applications other than games. Table of Contents: Introduction / The Human Visual System / Eye Tracking / Eye Tracking in Video Games / Gaze and Voice Controlled Video Games: Case Study I and II / Gaze and Voice Controlled Drawing: Case Study III / Conclusion

Quaternion multiplication can be used to rotate vectors in three-dimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is well-known in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this monograph areto provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points;to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane;to derive the formula for quaternion multiplication from first principles;to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection;to show how to apply sandwiching to compute perspective projections. In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolate between two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text. Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography

Information theory (IT) tools, widely used in scientific fields such as engineering, physics, genetics, neuroscience, and many others, are also emerging as useful transversal tools in computer graphics. In this book, we present the basic concepts of IT and how they have been applied to the graphics areas of radiosity, adaptive ray-tracing, shape descriptors, viewpoint selection and saliency, scientific visualization, and geometry simplification. Some of the approaches presented, such as the viewpoint techniques, are now the state of the art in visualization. Almost all of the techniques presented in this book have been previously published in peer-reviewed conference proceedings or international journals. Here, we have stressed their common aspects and presented them in an unified way, so the reader can clearly see which problems IT tools can help solve, which specific tools to use, and how to apply them. A basic level of knowledge in computer graphics is required but basic concepts in IT are presented. The intended audiences are both students and practitioners of the fields above and related areas in computer graphics. In addition, IT practitioners will learn about these applications. Table of Contents: Information Theory Basics / Scene Complexity and Refinement Criteria for Radiosity / Shape Descriptors / Refinement Criteria for Ray-Tracing / Viewpoint Selection and Mesh Saliency / View Selection in Scientific Visualization / Viewpoint-based Geometry Simplification

Tiling theory is an elegant branch of mathematics that has applications in several areas of computer science. The most immediate application area is graphics, where tiling theory has been used in the contexts of texture generation, sampling theory, remeshing, and of course the generation of decorative patterns. The combination of a solid theoretical base (complete with tantalizing open problems), practical algorithmic techniques, and exciting applications make tiling theory a worthwhile area of study for practitioners and students in computer science. This synthesis lecture introduces the mathematical and algorithmic foundations of tiling theory to a computer graphics audience. The goal is primarily to introduce concepts and terminology, clear up common misconceptions, and state and apply important results. The book also describes some of the algorithms and data structures that allow several aspects of tiling theory to be used in practice. Table of Contents: Introduction / Tiling Basics / Symmetry / Tilings by Polygons / Isohedral Tilings / Nonperiodic and Aperiodic Tilings / Survey

Irradiance caching is a ray tracing-based technique for computing global illumination on diffuse surfaces. Specifically, it addresses the computation of indirect illumination bouncing off one diffuse object onto another. The sole purpose of irradiance caching is to make this computation reasonably fast. The main idea is to perform the indirect illumination sampling only at a selected set of locations in the scene, store the results in a cache, and reuse the cached value at other points through fast interpolation. This book is for anyone interested in making a production-ready implementation of irradiance caching that reliably renders artifact-free images. Since its invention 20 years ago, the irradiance caching algorithm has been successfully used to accelerate global illumination computation in the Radiance lighting simulation system. Its widespread use had to wait until computers became fast enough to consider global illumination in film production rendering. Since then, its use is ubiquitous. Virtually all commercial and open-source rendering software base the global illumination computation upon irradiance caching. Although elegant and powerful, the algorithm in its basic form often fails to produce artifact-free mages. Unfortunately, practical information on implementing the algorithm is scarce. The main objective of this book is to show the irradiance caching algorithm along with all the details and tricks upon which the success of its practical implementation is dependent. In addition, we discuss some extensions of the basic algorithm, such as a GPU implementation for interactive global illumination computation and temporal caching that exploits temporal coherence to suppress flickering in animations. Our goal is to show the material without being overly theoretical. However, the reader should have some basic understanding of rendering concepts, ray tracing in particular. Familiarity with global illumination is useful but not necessary to read this book. Table of Contents: Introduction to Ray Tracing and Global Illumination / Irradiance Caching Core / Practical Rendering with Irradiance Caching / Irradiance Caching in a Complete Global Illumination / Irradiance Caching on Graphics Hardware / Temporal Irradiance Caching

Many complex signals in computer graphics, such as point distributions and textures, cannot be efficiently synthesized and stored. This book presents tile-based methods based on Wang tiles and corner tiles to solve both these problems. Instead of synthesizing a complex signal when needed, the signal is synthesized beforehand over a small set of Wang tiles or corner tiles. Arbitrary large amounts of that signal can then efficiently be generated when needed by generating a stochastic tiling, and storing only a small set of tiles reduces storage requirements. A tile-based method for generating a complex signal consists of a method for synthesizing the signal over a set of Wang tiles or corner tiles, and a method for generating a stochastic tiling using the set of tiles. The method for generating a stochastic tiling using the set of tiles is independent of the signal. This book covers scanline stochastic tiling algorithms and direct stochastic tiling algorithms for Wang tiles and corner tiles. The method for synthesizing the signal over a set of tiles is dependent on the signal. This book covers tile-based methods for texture synthesis and for generating Poisson disk distributions. This book also explores several applications such as tile-based texture mapping and procedural modeling and texturing. Although the methods for constructing a complex signal over a set of Wang tiles or corner tiles are dependent on the signal, the general idea behind these methods generalizes to other kinds of signals. The methods presented in this book therefore have the potential to make the generation and storage of almost any complex signal efficient. Table of Contents: Introduction / Wang Tiles and Corner Tiles / Tiling Algorithms for Wang Tiles and Corner Tiles / Tile-Based Methods for Texture Synthesis / Tile-Based Methods Generating Poisson Disk Distributions / Applications of Poisson Disk Distributions

Providing an intuitive modeling system, which would enable us to communicate about any free-form shape we have in mind at least as quickly as with real-world tools, is one of the main challenges of digital shape design. The user should ideally be able to create, deform, and progressively add details to a shape, without being aware of the underlying mathematical representation nor being tied by any constraint on the geometrical or topological nature of the model. This book presents the field of interactive shape design from this perspective. Since interactively creating a shape builds on the humans ability of modeling by gesture, we note that the recent advances in interactive shape design can be classified as those that rely on sculpting as opposed to sketching metaphors. Our synthetic presentation of these strategies enables us to compare the different families of solutions, discuss open issues, and identify directions for future research. Table of Contents: Introduction / Sculpting Metaphors / Sketching Systems / Future Directions: Modeling by Gesture

Interactive display and visualization of large geometric and textured models is becoming a fundamental capability. There are numerous application areas, including games, movies, CAD, virtual prototyping, and scientific visualization. One of observations about geometric models used in interactive applications is that their model complexity continues to increase because of fundamental advances in 3D modeling, simulation, and data capture technologies. As computing power increases, users take advantage of the algorithmic advances and generate even more complex models and data sets. Therefore, there are many cases where we are required to visualize massive models that consist of hundreds of millions of triangles and, even, billions of triangles. However, interactive visualization and handling of such massive models still remains a challenge in computer graphics and visualization. In this monograph we discuss various techniques that enable interactive visualization of massive models. These techniques include visibility computation, simplification, levels-of-detail, and cache-coherent data management.We believe that the combinations of these techniques can make it possible to interactively visualize massive models in commodity hardware. Table of Contents: Introduction / Visibility / Simplification and Levels of Detail / Alternative Representations / Cache-Coherent Data Management / Conclusions / Bibliography

As new displays and cameras offer enhanced color capabilities, there is a need to extend the precision of digital content. High Dynamic Range (HDR) imaging encodes images and video with higher than normal 8 bit-per-color-channel precision, enabling representation of the complete color gamut and the full visible range of luminance.However, to realize transition from the traditional toHDRimaging, it is necessary to develop imaging algorithms that work with the high-precision data. Tomake such algorithms effective and feasible in practice, it is necessary to take advantage of the limitations of the human visual system by aligning the data shortcomings to those of the human eye, thus limiting storage and processing precision. Therefore, human visual perception is the key component of the solutions we discuss in this book. This book presents a complete pipeline forHDR image and video processing fromacquisition, through compression and quality evaluation, to display. At the HDR image and video acquisition stage specialized HDR sensors or multi-exposure techniques suitable for traditional cameras are discussed. Then, we present a practical solution for pixel values calibration in terms of photometric or radiometric quantities, which are required in some technically oriented applications. Also, we cover the problem of efficient image and video compression and encoding either for storage or transmission purposes, including the aspect of backward compatibility with existing formats. Finally, we review existing HDR display technologies and the associated problems of image contrast and brightness adjustment. For this purpose tone mapping is employed to accommodate HDR content to LDR devices. Conversely, the so-called inverse tone mapping is required to upgrade LDR content for displaying on HDR devices. We overview HDR-enabled image and video quality metrics, which are needed to verify algorithms at all stages of the pipeline. Additionally, we cover successful examples of the HDR technology applications, in particular, in computer graphics and computer vision. The goal of this book is to present all discussed components of the HDR pipeline with the main focus on video. For some pipeline stages HDR video solutions are either not well established or do not exist at all, in which case we describe techniques for single HDR images. In such cases we attempt to select the techniques, which can be extended into temporal domain. Whenever needed, relevant background information on human perception is given, which enables better understanding of the design choices behind the discussed algorithms and HDR equipment. Table of Contents: Introduction / Representation of an HDR Image / HDR Image and Video Acquisition / HDR Image Quality / HDR Image, Video, and Texture Compression / Tone Reproduction / HDR Display Devices / LDR2HDR: Recovering Dynamic Range in Legacy Content / HDRI in Computer Graphics / Software

This book presents techniques to render photo-realistic images by programming the Graphics Processing Unit (GPU). We discuss effects such as mirror reflections, refractions, caustics, diffuse or glossy indirect illumination, radiosity, single or multiple scattering in participating media, tone reproduction, glow, and depth of field. The book targets game developers, graphics programmers, and also students with some basic understanding of computer graphics algorithms, rendering APIs like Direct3D or OpenGL, and shader programming. In order to make the book self-contained, the most important concepts of local illumination and global illumination rendering, graphics hardware, and Direct3D/HLSL programming are reviewed in the first chapters. After these introductory chapters we warm up with simple methods including shadow and environment mapping, then we move on toward advanced concepts aiming at global illumination rendering. Since it would have been impossible to give a rigorous review of all approaches proposed in this field, we go into the details of just a few methods solving each particular global illumination effect. However, a short discussion of the state of the art and links to the bibliography are also provided to refer the interested reader to techniques that are not detailed in this book. The implementation of the selected methods is also presented in HLSL, and we discuss their observed performance, merits, and disadvantages. In the last chapter, we also review how these techniques can be integrated in an advanced game engine and present case studies of their exploitation in games. Having gone through this book, the reader will have an overview of the state of the art, will be able to apply and improve these techniques, and most importantly, will be capable of developing brand new GPU algorithms. Table of Contents: Global Illumintation Rendering / Local Illumination Rendering Pipeline of GPUs / Programming and Controlling GPUs / Simple Improvements of the Local Illumination Model / Ray Casting on the GPU / Specular Effects with Rasterization / Diffuse and Glossy Indirect Illumination / Pre-computation Aided Global Illumination / Participating Media Rendering / Fake Global Illumination / Postprocessing Effects / Integrating GI Effects in Games and Virtual Reality Systems / Bibliography

High dynamic range imaging (HDRI) is an emerging field that has the potential to cause a great scientific and technological impact in the near future. Although new, this field is large and complex, with non-trivial relations to many different areas, such as image synthesis, computer vision, video and image processing, digital photography, special effects among others. For the above reasons,HDRI has been extensively researched over the past years and, consequently, the related scientific literature is vast. As an indication that the field is reaching maturity, tutorials and books on HDRI appeared. Moreover, this new resource has already reached interested practitioners in various application areas. In this book, we do not aim at covering the whole field of high dynamic range imaging and its applications, since it is a broad subject that is still evolving. Instead, our intent is to cover the basic principles behind HDRI and focus on one of the currently most important problems, both theoretically and practically. That is, the reconstruction of high dynamic range images from regular low dynamic range pictures. Table of Contents: Introduction / Digital Image / Imaging Devices and Calibration / HDR Reconstruction / HDRI Acquisition and Visualization / Tone Enhancement / References / Biography

The human haptic system, among all senses, provides unique and bidirectional communication between humans and their physical environment. Yet, to date, most human-computer interactive systems have focused primarily on the graphical rendering of visual information and, to a lesser extent, on the display of auditory information. Extending the frontier of visual computing, haptic interfaces, or force feedback devices, have the potential to increase the quality of human-computer interaction by accommodating the sense of touch. They provide an attractive augmentation to visual display and enhance the level of understanding of complex data sets. They have been effectively used for a number of applications including molecular docking, manipulation of nano-materials, surgical training, virtual prototyping, and digital sculpting. Compared with visual and auditory display, haptic rendering has extremely demanding computational requirements. In order to maintain a stable system while displaying smooth and realistic forces and torques, high haptic update rates in the range of 500-1000 Hz or more are typically used. Haptics present many new challenges to researchers and developers in computer graphics and interactive techniques. Some of the critical issues include the development of novel data structures to encode shape and material properties, as well as new techniques for geometry processing, data analysis, physical modeling, and haptic visualization. This synthesis examines some of the latest developments on haptic rendering, while looking forward to exciting future research in this area. It presents novel haptic rendering algorithms that take advantage of the human haptic sensory modality. Specifically it discusses different rendering techniques for various geometric representations (e.g. point-based, polygonal, multiresolution, distance fields, etc), as well as textured surfaces. It also shows how psychophysics of touch can provide the foundational design guidelines for developing perceptually driven force models and concludes with possible applications and issues to consider in future algorithmic design, validating rendering techniques, and evaluating haptic interfaces.

In this lecture, we study Bezier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces that are common in CAD systems and are used to design aircraft and automobiles, as well as in modeling packages used by the computer animation industry. Bezier/B-splines represent polynomials and piecewise polynomials in a geometric manner using sets of control points that define the shape of the surface. The primary analysis tool used in this lecture is blossoming, which gives an elegant labeling of the control points that allows us to analyze their properties geometrically. Blossoming is used to explore both Bzier and B-spline curves, and in particular to investigate continuity properties, change of basis algorithms, forward differencing, B-spline knot multiplicity, and knot insertion algorithms. We also look at triangle diagrams (which are closely related to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.