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