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دسته بندی: سازمان و پردازش داده ها ویرایش: نویسندگان: Hongyu Guo سری: ISBN (شابک) : 9814449326, 9789814449328 ناشر: World Scientific Publishing سال نشر: 2014 تعداد صفحات: 509 زبان: English فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود) حجم فایل: 9 مگابایت
کلمات کلیدی مربوط به کتاب ریاضیات مدرن و برنامه های کاربردی در گرافیک کامپیوتر و بینایی: علوم و مهندسی کامپیوتر، پردازش داده های رسانه ای، پردازش تصویر
در صورت تبدیل فایل کتاب Modern Mathematics and Applications in Computer Graphics and Vision به فرمت های PDF، EPUB، AZW3، MOBI و یا DJVU می توانید به پشتیبان اطلاع دهید تا فایل مورد نظر را تبدیل نمایند.
توجه داشته باشید کتاب ریاضیات مدرن و برنامه های کاربردی در گرافیک کامپیوتر و بینایی نسخه زبان اصلی می باشد و کتاب ترجمه شده به فارسی نمی باشد. وبسایت اینترنشنال لایبرری ارائه دهنده کتاب های زبان اصلی می باشد و هیچ گونه کتاب ترجمه شده یا نوشته شده به فارسی را ارائه نمی دهد.
ساختارهای ریاضی; جبر: جبر خطی; جبر تانسور; جبر بیرونی; جبر هندسی; هندسه: هندسه تصویری; هندسه دیفرانسیل; هندسه غیر اقلیدسی; توپولوژی و بیشتر: توپولوژی عمومی; منیفولدها؛ فضاهای هیلبرت؛ اندازه گیری فضاها و فضاهای احتمال. برنامه های کاربردی: فضاهای رنگی. تحلیل چشم انداز تصاویر; کواترنیون ها و چرخش های سه بعدی. پشتیبانی از ماشینهای برداری و بازتولید فضاهای هیلبرت هسته. یادگیری چندگانه در یادگیری ماشینی؛
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;
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