Spherical Harmonics and Approximations on the Unit Sphere: An Introduction


Book Description

These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.







Spherical Harmonic Analysis


Book Description




Approximation Theory and Harmonic Analysis on Spheres and Balls


Book Description

This monograph records progress in approximation theory and harmonic analysis on balls and spheres, and presents contemporary material that will be useful to analysts in this area. While the first part of the book contains mainstream material on the subject, the second and the third parts deal with more specialized topics, such as analysis in weight spaces with reflection invariant weight functions, and analysis on balls and simplexes. The last part of the book features several applications, including cubature formulas, distribution of points on the sphere, and the reconstruction algorithm in computerized tomography. This book is directed at researchers and advanced graduate students in analysis. Mathematicians who are familiar with Fourier analysis and harmonic analysis will understand many of the concepts that appear in this manuscript: spherical harmonics, the Hardy-Littlewood maximal function, the Marcinkiewicz multiplier theorem, the Riesz transform, and doubling weights are all familiar tools to researchers in this area.




Ellipsoidal Harmonics


Book Description

The sphere is what might be called a perfect shape. Unfortunately nature is imperfect and many bodies are better represented by an ellipsoid. The theory of ellipsoidal harmonics, originated in the nineteenth century, could only be seriously applied with the kind of computational power available in recent years. This, therefore, is the first book devoted to ellipsoidal harmonics. Topics are drawn from geometry, physics, biosciences and inverse problems. It contains classical results as well as new material, including ellipsoidal bi-harmonic functions, the theory of images in ellipsoidal geometry and vector surface ellipsoidal harmonics, which exhibit an interesting analytical structure. Extended appendices provide everything one needs to solve formally boundary value problems. End-of-chapter problems complement the theory and test the reader's understanding. The book serves as a comprehensive reference for applied mathematicians, physicists, engineers and for anyone who needs to know the current state of the art in this fascinating subject.







Handbook of Geomathematics


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Harmonic Function Theory


Book Description

This book is about harmonic functions in Euclidean space. This new edition contains a completely rewritten chapter on spherical harmonics, a new section on extensions of Bochers Theorem, new exercises and proofs, as well as revisions throughout to improve the text. A unique software package supplements the text for readers who wish to explore harmonic function theory on a computer.







Face Processing: Advanced Modeling and Methods


Book Description

Major strides have been made in face processing in the last ten years due to the fast growing need for security in various locations around the globe. A human eye can discern the details of a specific face with relative ease. It is this level of detail that researchers are striving to create with ever evolving computer technologies that will become our perfect mechanical eyes. The difficulty that confronts researchers stems from turning a 3D object into a 2D image. That subject is covered in depth from several different perspectives in this volume. Face Processing: Advanced Modeling and Methods begins with a comprehensive introductory chapter for those who are new to the field. A compendium of articles follows that is divided into three sections. The first covers basic aspects of face processing from human to computer. The second deals with face modeling from computational and physiological points of view. The third tackles the advanced methods, which include illumination, pose, expression, and more. Editors Zhao and Chellappa have compiled a concise and necessary text for industrial research scientists, students, and professionals working in the area of image and signal processing. - Contributions from over 35 leading experts in face detection, recognition and image processing - Over 150 informative images with 16 images in FULL COLOR illustrate and offer insight into the most up-to-date advanced face processing methods and techniques - Extensive detail makes this a need-to-own book for all involved with image and signal processing