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Mathematical basics of motion and deformation in computer graphics (2nd ed.)
Anjyo K., Ochiai H., Morgan & Claypool Publishers, San Rafael, CA, 2017. 96 pp. Type: Book (978-1-627056-97-7)
Date Reviewed: Apr 18 2018

Computer graphics is a branch of computer science involved in studying and devising methods to generate and manage digital graphical representations. Nowadays computer graphics is ubiquitous, found in basic or complicated devices, and used in a variety of fields requiring visual rendering, including gaming, medical investigation, education, and statistics. Computer graphics started to evolve in the 1960s and has continued to evolve into a distinct scientific field, tightly intertwined with several areas of mathematics.

This book is a contribution to the computer graphics domain, presenting a coherent theoretical framework to model and explain motion and morphing aspects of computer graphics. The mathematical background presented throughout is beyond the expected typical geometric operations; it is situated in more abstract spheres of high-level algebra and differential geometry, used to disguise the intuitive notions. More precisely, it is about how to use algebraic structures and especially Lie theory to facilitate generation of motion and deformation in 2D and 3D computer animation. The book is an extension of a first edition, based on SIGGRAPH ASIA 2013 and 2014 courses, and includes some new references and an appendix to investigate some 3D formulas. It also slightly modifies the organization of some chapters and adds clarifications of certain concepts. As the authors claim in the foreword, the book is intended as a “guide for students in geometric modeling and animation.” It is also recommended and challenging reading for experienced animation programmers. I would also suggest it to computer graphics teachers, to provide more theoretical insight in their classes, as well as to high-level algebra lecturers to inspire more application-specific lessons. It is not about shading or 3D perspective; it is solely about motion and morphing.

Beyond the introductory chapters, the book is organized as follows.

Chapter 2 reviews possible transforms on rigid bodies and relates their mathematical formulas to the subjacent algebraic structures and meanings. These are translations, rotations, and 2D reflections (flip). Unavoidable concepts such as Euler angles, quaternions, and dual quaternions are explained and used to generate equivalent expressions of 3D rotations, in addition to the axis angle formula. In the appendix, 3D rotation by axis angle (Rodrigues formulas) and its variants are detailed.

The third chapter represents the group of affine transforms as a special group of matrices and demonstrates its properties. It investigates the way composition and decomposition of basic motion transforms can be operated by matrices. Some important matrix decomposition methods are succinctly explained.

Chapter 4, “Exponential and Logarithm of Matrices,” goes beyond these two definitions. It presents the theoretical context for Lie groups and the associated Lie algebras, and further, the way matrix exponentials can be used to reverse this mapping, to relate a Lie algebra to its corresponding Lie group. Several groups of motions are investigated, and their Lie algebras are deduced. The reader will find other interesting related issues such as an uncommon expression for Euler angles, as canonical coordinates of second order, or how to handle interpolation and blending issues by exploiting exponential transforms on certain motion matrices.

Chapter 5 dwells on simple interpolation issues because interpolation is an important tool in figure motion and morphing. The authors introduce the so-called Log-Exp interpolation, which takes advantage of the results on Lie groups and algebras demonstrated in the previous chapters. They investigate and compare this interpolation method to two others by applying them on triangles. These two methods are linear and rotation-based interpolation. The performance of the three methods is discussed and illustrated graphically. This edition introduces two schemes describing the interpolation process of these methods; one is referred to as ARAP (this term is not mentioned previously, although it undoubtedly refers to the second rotation-based interpolation).

The sixth chapter takes a step forward to deal with the interpolation of global 2D shapes. The primary assumption is shape triangulation. The authors formalize the global interpolation by emphasizing three distinct stages to perform the transition from local to global. The specific issues discussed in the chapter are the definition of an error function for the global interpolation and selection of a constraint function to model the intermediary loci.

Chapter 7 discusses some implementation issues regarding 3D deformer applications of Lie algebras and affine transforms, and proposes a Lie algebra-based solution to overcome the limitations of former parameterization solutions simply based on Euler angles and quaternions. The chapter covers some related topics such as parameterization of Poisson mesh, an approach used for editing the graphics geometry by altering boundary conditions.

A last chapter discusses material meant to enlighten the reader in some areas such as differential geometry, Lie theory, and its applications. It also contains a reference to the 2016 SIGGRAPH conference course and a link to the accompanying video. The video arranges the concepts presented throughout the book, rendering them in an animated and more intuitive fashion.

The ideas in this interesting book are motivating; passionate computer graphics developers will need to understand them. The video is helpful and amusing; it is a model of well-done animation and an example of a successful tutorial.

Reviewer:  Svetlana Segarceanu Review #: CR145982 (1807-0374)
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Computational Geometry And Object Modeling (I.3.5 )
 
 
Mathematics And Statistics (J.2 ... )
 
 
Methodology And Techniques (I.3.6 )
 
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