Modern Mathematics and Applications in Computer Graphics and Vision

Hongyu Guo

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814449359
• Topic: Computer Science
• Subtopic: Computer Graphics
• Index: Computer Science

This book presents a concise exposition of modern mathematical concepts, models and methods with applications in computer graphics, vision and machine learning. The compendium is organized in four parts — Algebra, Geometry, Topology, and Applications. One of the features is a unique treatment of tensor and manifold topics to make them easier for the students. All proofs are omitted to give an emphasis on the exposition of the concepts. Effort is made to help students to build intuition and avoid parrot-like learning.

There is minimal inter-chapter dependency. Each chapter can be used as an independent crash course and the reader can start reading from any chapter — almost. This book is intended for upper level undergraduate students, graduate students and researchers in computer graphics, geometric modeling, computer vision, pattern recognition and machine learning. It can be used as a reference book, or a textbook for a selected topics course with the instructor's choice of any of the topics.

Contents:

• • Mathematical Structures
• Algebra:
  • Linear Algebra
  • Tensor Algebra
  • Exterior Algebra
  • Geometric Algebra
• Geometry:
  • Projective Geometry
  • Differential Geometry
  • Non-Euclidean Geometry
• Topology and More:
  • General Topology
  • Manifolds
  • Hilbert Spaces
  • Measure Spaces and Probability Spaces
• Applications:
  • Color Spaces
  • Perspective Analysis of Images
  • Quaternions and 3-D Rotations
  • Support Vector Machines and Reproducing Kernel Hilbert Spaces
  • Manifold Learning in Machine Learning

Readership: Upper level undergraduate students, graduate students and researchers in computer graphics, geometric modeling, computer vision and machine learning.

PART I
ALGEBRA

William R. Hamilton
(1805 – 1865)

J. Willard Gibbs
(1839 – 1903)

Giuseppe Peano
(1858 – 1932)

G. Ricci-Curbastro
(1853 – 1925)

Hermann Grassmann
(1809 – 1877)

William K. Clifford
(1845 – 1879)

Chapter 1

Linear Algebra

Professor: “Give me an example of a vector space.” Student: “V.”

— name unknown

§1.Vectors

1.1Vectors and Their Operations

1.2Properties of Vector Spaces

§2.Linear Spaces

2.1Linear Spaces

2.2Linear Independence and Basis

2.3Subspaces, Quotient Spaces and Direct Sums

§3.Linear Mappings

3.1Linear Mappings

3.2Linear Extensions

3.3Eigenvalues and Eigen vectors

3.4Matrix Representations

§4.Dual Spaces

§5.Inner Product Spaces

5.1Inner Products

5.2Connection to Dual Spaces

5.3Contravariant and Covariant Components of Vectors

§6.Algebras

Exercises

Appendix

A1.Free Vector Spaces and Free Algebras

Reading Guide. In case you did not read the preface, the preface chased you down here. That is, I copied a couple of sentences from there to here about how to read this book: This book is not intended to be read from the first page to the last page in the sequential order. You can start with any part, any chapter and feel free to skip around. See the chapter dependency chart in the front of the book. If you are familiar with linear algebra, you may skip this chapter and move on to read other chapters of your interest. You may come back to this chapter for reference on some concepts. This chapter discusses the fundamental concepts of linear spaces and linear mappings. We give a more in-depth discussion of contravariant vectors and covariant vectors in Sections 4 and 5, which will be beneficial to understand the concepts of contravariant tensors and covariant tensors in the next chapter.

§1 Vectors

1.1 Vectors and Their Operations

The concept of vectors comes from quantities which have both a magnitude and a direction, like forces and velocities. They are represented by a line segment with an arrow. The length of the segment represents the magnitude while the arrow represents the direction. When we set up a coordinate system, such a vector can be represented by three components (x₁, x₂, x₃) projected on the three axes, which are called a 3-tuple.

This can be generalized to n ordered numbers, (x₁, x₂, . . . , x_n), which are called n-tuples. An n-tuple is called a vector, or n-dimensional vector. We will see the meaning of dimension shortly.

Definition 1. Vectors and vector space

Let v = (x₁, . . . , x_n) be an n-tuple from a field F. We call v a vector. The set V of all vectors is called the vector space over F, denoted by Fⁿ. An element a ∈ F is called a scalar.

Remark 1. Historical Note — Vectors and Scalars

W. R. Hamilton introduced the terms scalar and vector in 3 dimensions in the context of quaternions.

Most often the field F is the field ℝ of all real nu...