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Author: Kenichi Kanatani
Publisher: CRC Press
ISBN: 1000071650
Category : Computers
Languages : en
Pages : 192
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Book Description
3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.
Author: Kenichi Kanatani
Publisher: CRC Press
ISBN: 1000071650
Category : Computers
Languages : en
Pages : 192
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Book Description
3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.
Author: Kenichi Kanatani
Publisher: CRC Press
ISBN: 1000070662
Category : Computers
Languages : en
Pages : 171
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Book Description
3D rotation analysis is widely encountered in everyday problems thanks to the development of computers. Sensing 3D using cameras and sensors, analyzing and modeling 3D for computer vision and computer graphics, and controlling and simulating robot motion all require 3D rotation computation. This book focuses on the computational analysis of 3D rotation, rather than classical motion analysis. It regards noise as random variables and models their probability distributions. It also pursues statistically optimal computation for maximizing the expected accuracy, as is typical of nonlinear optimization. All concepts are illustrated using computer vision applications as examples. Mathematically, the set of all 3D rotations forms a group denoted by SO(3). Exploiting this group property, we obtain an optimal solution analytical or numerically, depending on the problem. Our numerical scheme, which we call the "Lie algebra method," is based on the Lie group structure of SO(3). This book also proposes computing projects for readers who want to code the theories presented in this book, describing necessary 3D simulation setting as well as providing real GPS 3D measurement data. To help readers not very familiar with abstract mathematics, a brief overview of quaternion algebra, matrix analysis, Lie groups, and Lie algebras is provided as Appendix at the end of the volume.
Author: J. B. Kuipers
Publisher: Princeton University Press
ISBN: 0691211701
Category : Mathematics
Languages : en
Pages : 396
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Book Description
Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations. The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.
Author: Jan Flusser
Publisher: John Wiley & Sons
ISBN: 1119039363
Category : Technology & Engineering
Languages : en
Pages : 776
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Book Description
Presents recent significant and rapid development in the field of 2D and 3D image analysis 2D and 3D Image Analysis by Moments, is a unique compendium of moment-based image analysis which includes traditional methods and also reflects the latest development of the field. The book presents a survey of 2D and 3D moment invariants with respect to similarity and affine spatial transformations and to image blurring and smoothing by various filters. The book comprehensively describes the mathematical background and theorems about the invariants but a large part is also devoted to practical usage of moments. Applications from various fields of computer vision, remote sensing, medical imaging, image retrieval, watermarking, and forensic analysis are demonstrated. Attention is also paid to efficient algorithms of moment computation. Key features: Presents a systematic overview of moment-based features used in 2D and 3D image analysis. Demonstrates invariant properties of moments with respect to various spatial and intensity transformations. Reviews and compares several orthogonal polynomials and respective moments. Describes efficient numerical algorithms for moment computation. It is a "classroom ready" textbook with a self-contained introduction to classifier design. The accompanying website contains around 300 lecture slides, Matlab codes, complete lists of the invariants, test images, and other supplementary material. 2D and 3D Image Analysis by Moments, is ideal for mathematicians, computer scientists, engineers, software developers, and Ph.D students involved in image analysis and recognition. Due to the addition of two introductory chapters on classifier design, the book may also serve as a self-contained textbook for graduate university courses on object recognition.
Author: Simon L. Altmann
Publisher: Courier Corporation
ISBN: 0486317730
Category : Mathematics
Languages : en
Pages : 315
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Book Description
This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.
Author: Steven J. Gortler
Publisher: MIT Press
ISBN: 0262304813
Category : Computers
Languages : en
Pages : 291
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Book Description
An introduction to the basic concepts of 3D computer graphics that offers a careful mathematical exposition within a modern computer graphics application programming interface. Computer graphics technology is an amazing success story. Today, all of our PCs are capable of producing high-quality computer-generated images, mostly in the form of video games and virtual-life environments; every summer blockbuster movie includes jaw-dropping computer generated special effects. This book explains the fundamental concepts of 3D computer graphics. It introduces the basic algorithmic technology needed to produce 3D computer graphics, and covers such topics as understanding and manipulating 3D geometric transformations, camera transformations, the image-rendering process, and materials and texture mapping. It also touches on advanced topics including color representations, light simulation, dealing with geometric representations, and producing animated computer graphics. The book takes special care to develop an original exposition that is accessible and concise but also offers a clear explanation of the more difficult and subtle mathematical issues. The topics are organized around a modern shader-based version of OpenGL, a widely used computer graphics application programming interface that provides a real-time “rasterization-based” rendering environment. Each chapter concludes with exercises. The book is suitable for a rigorous one-semester introductory course in computer graphics for upper-level undergraduates or as a professional reference. Readers should be moderately competent programmers and have had some experience with linear algebra. After mastering the material presented, they will be on the path to expertise in an exciting and challenging field.
Author: Andrew J. Hanson
Publisher: Elsevier
ISBN: 0080474772
Category : Mathematics
Languages : en
Pages : 530
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Book Description
Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions. - Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. - Covers both non-mathematical and mathematical approaches to quaternions.
Author: Ernesto Olguín Díaz
Publisher: Springer
ISBN: 3030042758
Category : Technology & Engineering
Languages : en
Pages : 488
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Book Description
This book offers an excellent complementary text for an advanced course on the modelling and dynamic analysis of multi-body mechanical systems, and provides readers an in-depth understanding of the modelling and control of robots. While the Lagrangian formulation is well suited to multi-body systems, its physical meaning becomes paradoxically complicated for single rigid bodies. Yet the most advanced numerical methods rely on the physics of these single rigid bodies, whose dynamic is then given among multiple formulations by the set of the Newton–Euler equations in any of their multiple expression forms. This book presents a range of simple tools to express in succinct form the dynamic equation for the motion of a single rigid body, either free motion (6-dimension), such as that of any free space navigation robot or constrained motion (less than 6-dimension), such as that of ground or surface vehicles. In the process, the book also explains the equivalences of (and differences between) the different formulations.
Author: Fletcher Dunn
Publisher: CRC Press
ISBN: 1498759890
Category : Computers
Languages : en
Pages : 848
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Book Description
This engaging book presents the essential mathematics needed to describe, simulate, and render a 3D world. Reflecting both academic and in-the-trenches practical experience, the authors teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. The text provides an introduction to mathematics for
Author: John Vince
Publisher: Springer Science & Business Media
ISBN: 0857291548
Category : Computers
Languages : en
Pages : 240
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Book Description
Rotation transforms are used everywhere in computer graphics from rotating pictures in editing software, to providing an arbitrary view of a 3D virtual environment. Although the former is a trivial operation, the latter can be a challenging task. Rotation Transforms for Computer Graphics covers a wide range of mathematical techniques used for rotating points and frames of reference in the plane and 3D space. It includes many worked examples and over 100 illustrations that make it essential reading for students, academics, researchers and professional practitioners. The book includes introductory chapters on complex numbers, matrices, quaternions and geometric algebra, and further chapters on how these techniques are employed in 2D and 3D computer graphics. In particular, matrix and bivector transforms are developed and evaluated to rotate points in a fixed frame of reference, and vice versa.
