Kooperativer Bibliotheksverbund

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Format: VII, 181 p. 47 illus., 45 illus. in color. , online resource.

Edition: 1st ed. 2024.

ISBN: 9783031617126

Content: This book introduces in a constructive manner a general framework for regression and fitting methods for many applications and tasks involving data on manifolds. The methodology has important and varied applications in machine learning, medicine, robotics, biology, computer vision, human biometrics, nanomanufacturing, signal processing, and image analysis, etc. The first chapter gives motivation examples, a wide range of applications, raised challenges, raised challenges, and some concerns. The second chapter gives a comprehensive exploration and step-by-step illustrations for Euclidean cases. Another dedicated chapter covers the geometric tools needed for each manifold and provides expressions and key notions for any application for manifold-valued data. All loss functions and optimization methods are given as algorithms and can be easily implemented. In particular, many popular manifolds are considered with derived and specific formulations. The same philosophy is used in all chapters and all novelties are illustrated with intuitive examples. Additionally, each chapter includes simulations and experiments on real-world problems for understanding and potential extensions for a wide range of applications.

Note: Introduction -- Spline Interpolation and Fitting in R𝒏 -- Spline Interpolation on the Sphere S𝒏 -- Spline Interpolation on the Special Orthogonal Group 𝑺𝑶(𝒏) -- Spline Interpolation on Stiefel and Grassmann manifolds -- Spline Interpolation on the Manifold of Probability Measures -- Spline Interpolation on the Manifold of Probability Density Functions -- Spline Interpolation on Shape Space -- Spline Interpolation on Other Riemannian Manifolds.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783031617119

Additional Edition: Printed edition: ISBN 9783031617133

Additional Edition: Printed edition: ISBN 9783031617140

Language: English

DOI: 10.1007/978-3-031-61712-6

URL: https://doi.org/10.1007/978-3-031-61712-6

URL: Volltext  (URL des Erstveröffentlichers)

URL: lizenzpflichtig

UID:

Format: 1 online resource (0 pages)

Edition: 1st ed.

ISBN: 9783031617126

Note: Intro -- Contents -- 1 Introduction -- 1.1 Motivation and Applications -- 1.2 Challenges and Concerns -- 1.3 Organization of This Book -- References -- 2 Spline Interpolation and Fitting in double struck upper R Superscript nmathbbRn -- 2.1 The Problem in a Continuous Setting -- 2.2 Bézier Splines -- 2.2.1 Bernstein Polynomials and Bézier Curves -- 2.2.2 The de Casteljau Algorithm in double struck upper R Superscript nmathbbRn -- 2.2.3 Bézier Splines and Continuity Conditions -- 2.3 Solving Interpolation Problem in double struck upper R Superscript nmathbbRn -- 2.3.1 upper C Superscript 1C1 Bézier Spline Interpolation in double struck upper R Superscript nmathbbRn -- 2.3.2 upper C squaredC2 Bézier Spline Interpolation in double struck upper R Superscript nmathbbRn -- References -- 3 Spline Interpolation on the Sphere double struck upper S Superscript nmathbbSn -- 3.1 Problem Statement -- 3.2 Geometry of the Sphere double struck upper S Superscript nmathbbSn -- 3.3 Spherical Bézier Splines -- 3.3.1 Spherical Bézier Curves -- 3.3.2 Spherical Bézier Splines and Continuity Conditions -- 3.4 Solving Interpolation Problem on the Sphere double struck upper S Superscript nmathbbSn -- 3.4.1 upper C Superscript 1C1 Spherical Bézier Spline -- 3.4.2 upper C squaredC2 Spherical Bézier Spline -- References -- 4 Spline Interpolation on the Special Orthogonal Group upper S upper O left parenthesis n right parenthesisSO(n) -- 4.1 Problem Formulation -- 4.2 Geometry of upper S upper O left parenthesis n right parenthesisSO(n) -- 4.3 Bézier Splines on upper S upper O left parenthesis n right parenthesisSO(n) -- 4.3.1 Bézier Curves on upper S upper O left parenthesis n right parenthesisSO(n) -- 4.3.2 Bézier Splines and Continuity Conditions -- 4.4 Solving Interpolation Problem on upper S upper O left parenthesis n right parenthesisSO(n). , 4.4.1 Proposed Interpolating Bézier Spline on upper S upper O left parenthesis n right parenthesisSO(n) -- 4.4.2 Experiments -- References -- 5 Spline Interpolation on Stiefel and Grassmann Manifolds -- 5.1 Problem Formulation -- 5.2 Geometry of the Stiefel Manifold -- 5.3 Geometry of the Grassmann Manifold -- 5.4 Interpolation Problem on the Stiefel Manifold -- 5.4.1 Bézier spline on the Stiefel Manifold -- 5.4.2 upper C Superscript 1C1 Bézier Spline on the Stiefel Manifold -- 5.4.3 Experiments -- 5.5 Interpolation Problem on the Grassmann Manifold -- References -- 6 Spline Interpolation on the Manifold of Probability Measures -- 6.1 Problem Formulation -- 6.2 Geometry of the Space of Probability Measures -- 6.2.1 Manifold Structure -- 6.2.2 Fisher-Rao Metric on script upper P Subscript plusmathcalP+ -- 6.2.3 Geodesics on script upper P Subscript plusmathcalP+ -- 6.2.4 Levi-Civita Parallel Transport on script upper P Subscript plusmathcalP+ -- 6.3 Interpolation Problem on Space of Probability Measures -- 6.3.1 Measure Interpolation Spline Using De Casteljau Algorithm -- 6.3.2 upper C squaredC2 Bézier Splines on script upper P Subscript plusmathcalP+ -- 6.4 Experiments -- 6.4.1 Numerical Examples -- 6.4.2 Medical Examples -- References -- 7 Spline Interpolation on the Manifold of Probability Density Functions -- 7.1 Problem Formulation -- 7.2 Geometry of the Space of Probability Density Functions -- 7.2.1 Fisher-Rao Geometry -- 7.2.2 Riemannian Isometry -- 7.2.3 Riemannian Structure of script upper M mathcalM -- 7.2.4 Geodesic Curves on script upper PmathcalP -- 7.3 Interpolation Problem on the Space of Probability Density Functions -- 7.3.1 upper C Superscript 1C1 Spline on script upper MmathcalM -- 7.3.2 upper C squaredC2 Bézier Splines on script upper MmathcalM -- 7.3.3 Properties of Splines on script upper PmathcalP -- 7.4 Experiments. , References -- 8 Spline Interpolation on Shape Space -- 8.1 Problem Formulation -- 8.2 Geometry of Shape Space -- 8.2.1 Curve Representation and Shape Space -- 8.2.2 Geodesics in Shape Space, Exponential Map and Parallel Transport -- 8.3 Interpolation Problem on the Shape Space -- 8.4 Experiments -- 8.4.1 Example 1 -- 8.4.2 Example 2 -- References -- 9 Spline Interpolation on Other Riemannian Manifolds -- 9.1 Spline Interpolation on Symmetric Positive Definite Matrices script upper P Subscript n Superscript plusmathcalPn+ -- 9.1.1 Geometry of script upper P Subscript n Superscript plusmathcalPn+ -- 9.1.2 upper C squaredC2 Bézier Spline on script upper P Subscript n Superscript plusmathcalPn+ -- 9.2 Spline Interpolation on Hyperbolic Space script upper H Subscript nmathcalHn -- References -- Appendix A Background Material -- A.1 Basic Elements of Riemannian Geometry -- A.1.1 Charts and Manifolds -- A.1.2 Tangent Spaces and Tangent Vectors -- A.1.3 Riemannian Manifolds, Metrics and Isometries -- A.1.4 Connections and Covariant Derivative -- A.1.5 Distances, Geodesic Curves and Parallel Translation -- A.1.6 Exponential and Logarithmic Maps -- A.1.7 Riemannian Homogeneous Manifolds -- A.1.8 Riemannian Symmetric Spaces -- A.2 Infinite-Dimensional Riemannian Manifolds -- References -- Index.

Additional Edition: ISBN 9783031617119

Language: English

UID:

Format: 1 online resource (0 pages)

Edition: 1st ed. 2024.

ISBN: 9783031617126

Content: This book introduces in a constructive manner a general framework for regression and fitting methods for many applications and tasks involving data on manifolds. The methodology has important and varied applications in machine learning, medicine, robotics, biology, computer vision, human biometrics, nanomanufacturing, signal processing, and image analysis, etc. The first chapter gives motivation examples, a wide range of applications, raised challenges, raised challenges, and some concerns. The second chapter gives a comprehensive exploration and step-by-step illustrations for Euclidean cases. Another dedicated chapter covers the geometric tools needed for each manifold and provides expressions and key notions for any application for manifold-valued data. All loss functions and optimization methods are given as algorithms and can be easily implemented. In particular, many popular manifolds are considered with derived and specific formulations. The same philosophy is used in all chapters and all novelties are illustrated with intuitive examples. Additionally, each chapter includes simulations and experiments on real-world problems for understanding and potential extensions for a wide range of applications.

Note: Introduction -- Spline Interpolation and Fitting in R𝒏 -- Spline Interpolation on the Sphere S𝒏 -- Spline Interpolation on the Special Orthogonal Group 𝑺𝑶(𝒏) -- Spline Interpolation on Stiefel and Grassmann manifolds -- Spline Interpolation on the Manifold of Probability Measures -- Spline Interpolation on the Manifold of Probability Density Functions -- Spline Interpolation on Shape Space -- Spline Interpolation on Other Riemannian Manifolds.

Additional Edition: ISBN 9783031617119

Language: English

DOI: 10.1007/978-3-031-61712-6