Modern Mathematics and Applications in Computer Graphics and Vision

Preface
Brief Contents
Chapter Dependencies
Contents
Symbols and Notations
0 Mathematical Structures
	§1 Branches of Mathematics
	§2 Mathematical Structures
		2.1 Discrete Structures
		2.2 Continuous Structures
		2.3 Mixed Structures
	§3 Axiomatic Systems and Models
Part 1
1 Linear Algebra
	§1 Vectors
		1.1 Vectors and Their Operations
			Def 1. Vectors, vector space
			Def 2. Addition & scalar multiplication of vectors
		1.2 Properties of Vector Spaces
			Th1. (Properties of vector spaces)
	§2 Linear Spaces
		2.1 Linear Spaces
			Def 3. Linear space
		2.2 Linear Independence and Basis
			Def 4. Linear combination
			Def 5. Linear independence
			Th 2. (Linear independence)
			Def 6. Dimension
			Def 7. Span
			Def 8. Basis
			Th 3. (Change of coordinates for vectors)
		2.3 Subspaces, Quotient Spaces and Direct Sums
			Def 9. (Linear) subspace
			Def 10. Quotient (linear) space
			Def 11. Direct sum
	§3 Linear Mappings
		3.1 Linear Mappings
		Def 12. Linear mapping
		linear transformation/operator
		linear function/functional/form
		Def 13. Addition & scalar multiplication of linear mappings
		3.2 Linear Extensions
			Th 5. (Linear extension)
			Def 14. Image, kernel
		3.3 Eigenvalues and Eigenvectors
		3.4 Matrix Representations
	§4 Dual Spaces
		Def 15. Dual space
		Def 16. Dual basis
		affine dual
		Def 17. Adjoint/transpose/dual mapping of a linear mapping
		Euclidean Space
	§5 Inner Product Spaces
		5.1 Inner Products
			Def 18. (Real) Inner/dot product
		5.2 Connection to Dual Spaces
		5.3 Contravariant and Covariant Components of Vectors
			Def 21. Contravariant components, covariant components of a vector
		Def 19. Metric duals, metric dual basis
		Def 20. Reciprocal basis
	§6 Algebras
		Def 22. (Linear) Algebra over a field
	Appendix
		A1. Free Vector Spaces and Free Algebras
			Def 23. Free vector space generated by a set
			3. Free Algebras
2 Tensor Algebra
	§1 Introduction
	§2 Bilinear Mappings
		2.1 Definitions and Examples
			Def 1. Bilinear mapping/form
		2.2 Bilinear Extensions
			Th 1. (Bilinear extension)
			Th 2
			Def 2. Multilinear/p-linear mapping, multilinear/p-linear  function/form
	§3 Tensor Products
		3.1 Definition and Examples
			Definition 3. Tensor product (space)
			tensor product (mapping)
			factor space (of the tensor product space)
			Th 3
			Def 4. (Equivalent Definition) Tensor product
			Th 4
			Bilinear forms
		3.2 Decomposable Tensors
			Def 5. Decomposable tensor
		3.3 Induced Linear Mappings
			Def 6. Induced linear mapping
		3.4 Tensor Product Space of Multiple Linear Spaces
			Def 7. Tensor product space of multiple linear spaces
			Th 5.
	§4 Tensor Spaces
		Def 8. Contravariant, covariant and mixed tensor spaces
		4.2 Change of Basis
			Th 6.
			Th 7. (Change of coordinates for tensors)
			Def 9. Tensor spaces of higher degrees
			Th 8. (Change of coordinates for higher degree tensors)
		4.3 Induced Inner Product
			Def 10. Induced inner product
		4.4 Lowering and Raising Indices
	§5 Tensor Algebras
		5.1 Product of Two Tensors
		5.2 Tensor Algebras
		5.3 Contraction of Tensors
			Def 11. Contraction of a tensor
			Th 9.
			Def 12
	Appendix
		A1. A Brief History of Tensors
		A2. Alternative Definitions of Tensor
			(1) Old-fashioned Definition
			(2) Axiomatic Definition Using the Unique Factorization Property
				Def 13. (Equivalent Definition) Tensor product
			(3) Definition by Construction—Dyadics
				Def 14. (Equivalent Definition) Tensor product
			(4) Definition Using a Model—Bilinear Forms
				Def 15. (Equivalent Definition) Tensor product
		A3. Bilinear Forms and Quadratic Forms
			Def 16. Degenerate, nondegenerate bilinear form
			Def 17. Quadratic form
			Def 18. Positive definite, negative definite, indefinite
3 Exterior Algebra
	§1 Intuition in Geometry
		1.1 Bivectors
			wedge/exterior product
			bivector/2-vector
		1.2 Trivectors
			trivector/3-vector
	§2 Exterior Algebra
		Step 1
			Def 1. Formal wedge product, formal combination, 2-blade, 2-vector
		Step 2.
		Step 3.
			exterior/Grassmann algebra
			Th 1
			Th 2
			Th 3.
			Th 4
	Appendix
		A1. Exterior Forms
			A1.1. Exterior Forms
				Def 2. Linear form
				Def 3. Bilinear form
				Def 4. Multilinear form
				Def 5. 2-form
				Def 6. k-form
				Def 7. Exterior space Λk(V ∗) of degree k
			A1.2. k-Blades
				Def 8. 2-blade
				Def 9. k-blade
				Theorem 5.
				Corollary
			A1.3. Wedge Product of a p-Form and a q-Form
				Def 10. Wedge product of a p-form and a q-form
				Th 6.
				Th 7
		A2. Exterior Algebra as Subalgebra of Tensor Algebra
			Def 11. Antisymmetric tensor
			Def 12. Exterior space of degree p
			Def 13. Antisymmetrizer
			Def 14. Wedge product of two multivectors
			Th 8
		A3. Exterior Algebra as Quotient Algebra of Tensor Algebra
			Def 15. (Equivalent Definition) Exterior algebra
4 Geometric Algebra
	§1 Construction from Exterior Algebra
		geometric product
		Def 1. Geometric/Clifford algebra
		Def 2. Geometric algebra Clp,q(V ) with signature
		Th 3.
	§2 Construction from Tensor Algebra
		Def 3. Clifford algebra
Part 2 Geometry
Ch1 Projective Geometry
	§1 Perspective Drawing
	§2 Projective Planes
		2.1 Extended Euclidean Plane Model
			Def 1. Projective plane—extended Euclidean plane model
			Axioms. (Projective plane)
			Axioms. (Projective plane—alternative form)
			Axiom. (Alternative form of Desargues Axiom )
			Desarguesian/non-Desarguesian geometry
			Axiom. (Desargues)
			Def 2. Dual proposition
			Th 1. (Principle of duality)
			Th 2. (Converse of Desargues Axiom)
		2.2 The Ray Model
			Def 3. Projective plane—the ray model
			Intuition. (The projective plane) The
			depth/projective ambiguity
		2.3 Projective Coordinates for Points
			Def 4. Projective coordinates
		2.4 Projective Frames
			Th 3. (Projective frame)
			projective coordinate system,projective frame
			fundamental points
			unit point
		2.5 Relation to Terminology in Art, Photography and Computer Graphics
			(1) One-point perspective (or parallel perspective)
			(2) Two-point perspective (or angular perspective)
			(3) Three-point perspective (or inclined perspective)
		2.6 Projective Coordinates for Lines
			Th 4. (Equation of a line)
			Def 5. Projective coordinates of a line
			Th 5. (Line passing through two points)
			Th 6. (Principle of duality)
		2.7 Projective Mappings and Projective Transformations
			Def 6. Projective mapping
			Def 7. Collineation
			Th 7. (Fundamental theorem of projective geometry)
			Th 8. (Projective transformation formulas)
		2.8 Perspective Rectification of Images
	§3 Projective Spaces
		3.1 Extended Euclidean Space Model
		3.2 The Ray Model
		3.3 Projective Subspaces
		3.4 Projective Mappings Between Subspaces
			Def 9. Perspective mapping
			Th9
			Th10.
		3.5 Central Projection Revisited
			lossy perspective mapping
			bijective perspective mapping
		3.6 Higher Dimensional Projective Spaces
			Def 10. Projective space P n(V )
Ch2 Differential Geometry
	§1 What is Intrinsic Geometry?
	§2 Parametric Representation of Surfaces
		v-lines
		u-lines
		coordinate curves/lines
		coordinate mesh, curvilinear coordinate system
	§3 Curvature of Plane Curves
		Def 1. Curvature of a plane curve
		Th2. (Curvature of plane curves)
		Th3. (Osculating circle of a curve)
	§4 Curvature of Surfaces—Extrinsic Study
		Def 2. Normal section and normal curvature
		Th4. (Euler)
		Def3. Principal curvatures, principal directions
	§5 Intrinsic Geometry—for Bugs that Don’t Fly
		Def 4. Metric space
		Def5. Intrinsic distance
		Def6. Geodesic line
		Def7. Isometric mapping
		Def8. 1st fundamental form, 1st fundamental quantities
		Remark 5. Intuition — Meaning of the First Fundamental Form
		Th5. (Fundamental theorem of intrinsic geometry of surfaces —Gauss
		Corollary
		Def9. Developable surface
	§6 Extrinsic Geometry—for Bugs that Can Fly
		Def10. 2nd fundamental form, 2nd fundamental quantities
		Remark 8. Intuition — Meaning of the Second Fundamental Form
		Th6. (Normal curvature)
		Corollary. (Normal curvature)
		Th7. (Fundamental theorem of extrinsic geometry of surfaces—Bonnet)
	§7 Curvature of Surfaces—Intrinsic Study
		Def11. Gaussian curvature, mean curvature
		Th8.
		Th9. (Theorema Egregium — Gauss)
		Corollary 1. (Gaussian curvature — Brioschi’s formula)
		Corollary 2. (Gaussian curvature — orthogonal curvilinear coordinates)
		Corollary 3. (Gaussian curvature — Liouville’s formula)
	§8 Meanings of Gaussian Curvature
		8.1 Effects on Triangles—Interior Angle Sum
			Th10. (Gauss-Bonnet)
			Def12. Angle excess, angle defect of a triangle
			Corollary
		8.2 Effects on Circles—Circumference and Area
			Th11
			Th12.
		8.3 Gauss Mapping—Spherical Representation
			Def13. Gauss mapping
			Th13. (Gauss mapping)
		8.4 Effects on Tangent Vectors—Parallel Transport
			Def14. Vector field along a curve
			Def15. Covariant differential
			Def16. Parallel transport
			Def17. Connection coefficients
			Th14. (Connection coefficients)
			Th15. (Connection coefficients)
			Th16. (Properties of covariant differential)
			Th17. (Properties of parallel transport)
			Th18
			Th19.
	§9 Geodesic Lines
		Def18. (Equivalent Definition) — Geodesic line
		Th20.
		Th21. (Equation of geodesic line)
	§10 Look Ahead—Riemannian Geometry
Ch3 Non-Euclidean Geometry
	§1 Axioms of Euclidean Geometry
		Postulates. (Euclid)
		Axioms. (Euclid)
		Remark 1
		Remark 2.
		Th1
		Th2
		Axiom. (Playfair’s axiom of parallels)
		Th3. (Saccheri-Legendre)
		Remark 3
		Th4. (Existence of parallels)
		Corollary
	§2 Hyperbolic Geometry
		Axiom. (Lobachevsky’s axiom of parallels)
		Remark 4
		Def1. Asymptotically parallel line, ultraparallel line
		Remark 5
		Def2. Angle of parallelism
		Th5. (Lobachevsky’s formula) Let b be
		Corollary
		Th6. (Properties of asymptotic parallels)
		Def3. Equidistance curve
		Theorem 7
		Corollary
		Th8. (Saccheri)
		Th9. (Lambert)
		Th10.
		Def4. Angle defect of a triangle
		Th11
		Corollary.
		Th12.
	§3 Models of the Hyperbolic Plane
		Remark 9. Philosophy
		3.1 Beltrami Pseudosphere Model
		3.2 Gans Whole Plane Model
		3.3 Poincar´e Half Plane Model
		3.4 Poincar´e Disk Model
		3.5 Beltrami-Klein Disk Model
		3.6 Weierstrass Hyperboloid Model
		3.7 Models in Riemannian Geometry
	§4 Hyperbolic Spaces
Part 3
Ch1 General Topology
	§1 What is Topology?
	§2 Topology in Euclidean Spaces
		2.1 Euclidean Distance
			Theorem 1
		2.2 Point Sets in Euclidean Spaces
			D1. Interior point, exterior point, boundary point
			D2. Accumulation point, isolated point
			D3. Interior, exterior, boundary, derived set, closure
			Theorem 2
			T3.
			D4. Open set, closed set
			T4.
			T5. (Properties of open sets)
			Corollary
			T6. (Properties of closed sets)
			T7.
			D5. Compact set
			T8. (Heine-Borel)
		2.3 Limits and Continuity
			D6. Convergence, limit
			D7. Continuous function
	§3 Topology in Metric Spaces
		3.1 Metric Spaces
			D8. Metric space
			D9. Isometric mapping
		3.2 Completeness
			D10. Cauchy sequence
			D11. Complete metric space
			T9
			T10
	§4 Topology in Topological Spaces
		4.1 Topological Spaces
			D12. Topological space
			D13. Base, basic open set
			T11
			D14. Closed set
			Axiom. (Hausdorff)
			T12.
			D15. Dense
			D16. Separable space
		4.2 Topological Equivalence
			D17. Continuous mapping
			D18. Open mapping, closed mapping
			D19. Homeomorphic mapping
		4.3 Subspaces, Product Spaces and Quotient Spaces
			D20. Topological subspace
			D21. Topological embedding
			D22. Product space
			D23. Quotient space
		4.4 Topological Invariants
			D24. Compact
			T13.
			D25. Connected
			T14
			T15
			D26. Connected component
			T16
			T17
Ch2 Manifolds
	§1 Topological Manifolds
		1.1 Topological Manifolds
			D1. (Topological) manifold
			D2. Coordinate patch, atlas
		1.2 Classification of Curves and Surfaces
			D3. Closed manifold, open manifold
			T1. (Classification of curves)
			D4. Connected sum of two manifolds
			T2. (Classification of surfaces)
			Corollary
	§2 Differentiable Manifolds
		2.1 Differentiable Manifolds
			D5. Compatible patches
			D6. Compatible atlas
			D7. Differentiable/smooth manifold
			D8. Equivalent differential structures
			D9. Differentiable mapping
			D10. Diffeomorphic mapping
		2.2 Tangent Spaces
			D11. Equivalent curves
			D12. Tangent vector
			D13. Tangent space
			D14. Directional derivative of a scalar field
			T3. (Properties of directional derivatives)
			D15. Lie bracket of two vector fields
			D16. (Alternative Definition) Lie bracket of two vector fields
			T4
			T5. (Properties of Lie bracket)
			D17. Differential of a mapping
			T6
			differential form
			T7
		2.3 Tangent Bundles
			D18. Tangent bundle
		2.4 Cotangent Spaces and Differential Forms
			D19. Cotangent vector (differential form), cotangent space
			D20. Differential of a scalar field
			T8
			T9
			D21. Exact form
			D22. Cotangent bundle
		2.5 Submanifolds and Embeddings
			D23. Immersion
			D24. (smooth) Embedding
			D25. Regular embedding and regular submanifold
			D26. Submanifolds
			T10. (Whitney embedding theorem)
	§3 Riemannian Manifolds
		3.1 Curved Spaces
		3.2 Riemannian Metrics
			D27. Riemannian manifold/space
			D28. Length of a curve
			T11
			D29. Geodesic line
			T12. (Equations of a geodesic line)
		3.3 Levi-Civita Parallel Transport
			D30. Parallel transport along a geodesic line on a 2-manifold
			D31. Parallel transport along a geodesic line on an n-manifold
			D32. Parallel transport along arbitrary curve
			D33. Covariant derivative ∇_v Y
			D34. Riemannian connection
			D35. Covariant differential ∇Y
			T13. (Properties of the Riemannian connection)
			D36. Connection coefficients: Γkij
			T14
			Corollary
			T15. (Covariant derivative in local coordinates)
			T16(Coordinate transformation of connection coefficients)
		3.4 Riemann Curvature Tensor
			D37. Curvature operator
			D38. Riemann curvature tensor
			T17.
			T18. (Properties of Riemann curvature tensor)
			Corollary. (Properties of Riemann curvature tensor — component form)
		3.5 Sectional Curvature
			D39. Plane section, geodesic surface
			D40. Sectional curvature
			D41. Isotropic and constant curvature manifold
			T19. (Sectional curvature) The se
			Corollary 1. (Sectional curvature)
			Corollary 2
			T20
			T21
			Corollary
		3.6 Ricci Curvature Tensor and Ricci Scalar Curvature
			D42. Ricci curvature tensor
			T22
			T23. (Geometric meaning of Ricci curvature tensor)
			D43. Ricci scalar curvature
			T24. (Geometric meaning of Ricci scalar curvature)
			T25.
			Corollary
		3.7 Embedding of Riemannian Manifolds
			Theorem 26. (Nash embedding theorem)
	§4 Affinely-Connected Manifolds
		4.1 Curvature Tensors and Torsion Tensors
			D44. Affinely-connected manifold
			D45. Covariant differential
			T27. (Affine connections)
			D46. Connection coefficients
			T28. (Covariant derivative in local coordinates)
			D47. Parallel vector field on a curve, parallel transport along a curve
			D48. Geodesic line
			T29. (Equations of a geodesic line)
			D49. Curvature tensor
			D50. Torsion tensor
			T30
		4.2 Metrizability
Ch3 Hilbert Spaces
	§1 Hilbert Spaces
		D1. Inner product
		Example 3. (Sequence space l2)
		Example 5. (L2[a, b])
		D2. Length, distance
		T1
		D3. Hilbert space
		D4. Orthogonal set
		D5. Orthogonal basis
		T2. (Orthogonal dimension)
		D6. Hamel basis
		T3
		T4
		T5
		Corollary
		T6. (Riesz representation theorem)
	§2 Reproducing Kernel Hilbert Spaces
		D7. Kernel
		D8. Reproducing kernel
		D9. Reproducing kernel Hilbert space (RKHS)
		T7
		T8. (Positive definiteness of kernels)
		T9. (Mercer)
	§3 Banach Spaces
		D10. Normed linear space
		Example 13. (Lebesgue spaces Lp[a, b])
		D11. Banach space
		D12. Schauder basis
		T10. (Parallelogram equality
		T11. (Jordan-von Neumann)
		Corollary
Ch4 Measure Spaces and Probability Spaces
	§1 Length, Area and Volume
		Axioms. (Area of polygons)
		T1. (Wallace-Bolyai-Gerwien)
	§2 Jordan Measure
		D1. Jordan outer measure
		D2. Jordan inner measure
		D3. Jordan measurable set, Jordan measure
		T2.
		Intuition. (Jordan measurable sets)
		T3
		Corollary
		T4
		T5
		Remark 2
		T6. (Properties of Jordan measure)
		T7. (Properties of Jordan measurable sets)
	§3 Lebesgue Measure
		3.1 Lebesgue Measure
			D4. Lebesgue outer measure
			D5. Lebesgue inner measure
			Intuition. (Jordan outer/inner and Lebesgue outer/inner measures)
			D6. Lebesgue measurable set, Lebesgue measure
			T8
			T9
			T10. (Properties of Lebesgue measure)
			completely/countably/σ- additive
			Corollary
			T11. (Properties of Lebesgue measurable sets)
		3.2 σ-algebras
			D7. σ-algebra
			D8. (Equivalent Definition) σ-algebra
			D9. Algebra
			D10. σ-algebra generated by a family of sets F
			T12
			T13
			T14
			T15
	§4 Measure Spaces
		D11. Measurable space
		D12. Measure space
		D13. Borel measure in a topological space
	§5 Probability Spaces
		D14. Probability space, probability measure
Part 4
Ch1 Color Spaces
	§1 Some Questions and Mysteries about Colors
	§2 Light, Colors and Human Visual Anatomy
	§3 Color Matching Experiments and Grassmann’s Law
		D1. Independent colors
		Grassmann’s Law
	§4 Primary Colors and Color Gamut
		D2. Primary colors
		D3. Color gamut
		D4. Primary colors by design choice
	§5 CIE RGB Primaries and XYZ Coordinates
	6 Color Temperatures
	§7 White Point and White Balance
	§8 Color Spaces
	§9 Hue, Saturation, Brightness and HSV, HSL Color Spaces
Ch2 Perspective Analysis of Images
	§1 Geometric Model of the Camera
		Perspective Projection
	§2 Images Captured From Different Angles
		2.1 2-D Scenes
		2.2 3-D Scenes
	§3 Images Captured From Different Distances
		3.1 2-D Scenes
		3.2 3-D Scenes
	§4 Perspective Depth Inference
		4.1 One-Point Perspective
		4.2 Two-Point Perspective
	§5 Perspective Diminution and Foreshortening
		5.1 Perspective Diminution Factor
			D1. Perspective diminution factor
			T1.
		5.2 Perspective Foreshortening Factor
			D2. Perspective foreshortening factor
			T2.
	§6 “Perspective Distortion”
Ch3 Quaternions and 3-D Rotations
	§1 Complex Numbers and 2-D Rotations
		1.1 Addition and Multiplication
			D1. Addition of two complex numbers
			D2. Square of imaginary unit
		1.2 Conjugate, Modulus and Inverse
			D3. Conjugate of a complex number
			D4. Modulus of a complex number
			T1. (Properties of complex conjugate and modulus)
		1.3 Polar Representation
		1.4 Unit-Modulus Complex Numbers as 2-D Rotation Operators
	§2 Quaternions and 3-D Rotations
		2.1 Addition and Multiplication
			D5. Addition of two quaternions
			D6. Multiplication of imaginary units
		2.2 Conjugate, Modulus and Inverse
			D7. Conjugate of a quaternion
			D8. Modulus of a quaternion
			T2. (Properties of quaternion conjugate and modulus)
		2.3 Polar Representation
		2.4 Unit-Modulus Quaternions as 3-D Rotation Operators
Ch4 Support Vector Machines and Reproducing Kernel Hilbert Spaces
	§1 Human Learning and Machine Learning
	§2 Unsupervised Learning and Supervised Learning
	§3 Linear Support Vector Machines
	§4 Nonlinear Support Vector Machines and Reproducing Kernel Hilbert Spaces
Ch5 Manifold Learning in Machine Learning
	§1 The Need for Dimensionality Reduction
	§2 Locally Linear Embedding
	§3 Isomap
	Appendix
		A1. Principal Component Analysis
Bibliography
Index