Contents Preface Chapter1 Clifford Fuzzy Support Vector Machines for Classification 1 1.1 Introduction 1 1.2 Related Works 2 1.2.1 Multiple Classification Methods 2 1.2.2 Fuzzy SVMs 4 1.3 Clifford SVM 5 1.3.1 Geometric Algebra 5 1.3.2 Clifford SVM 6 1.4 Clifford Fuzzy SVM 8 1.4.1 Fuzzy Property of Multiple Input 8 1.4.2 Reformulation Linear CFSVM for Multiple Classification 9 1.4.3 Reformulation Nonlinear CFSVM for Multiple Classification 11 1.4.4 Fuzzy Membership Discussion 14 1.4.5 Generating the Fuzzy Memberships 14 1.5 Experimental Analysis 15 1.5.1 Linear Classification of Valency-State Classes Using CFSVMs 15 1.5.2 Nonlinear Classification Using CFSVMs: 3D Spiral Data with Noise 21 1.6 Conclusion 23 References 24 Chapter2 3D Point Cloud Model and Its Application 27 2.1 Introduction 27 2.2 The Geometric Representation of the Point Cloud Data in Space G3 28 2.3 Clifford Fourier Transform in Space G3 30 2.4 Watermark Embedding and Extracting 31 2.4.1 Watermark Embedding 32 2.4.2 Watermark Extracting 33 2.5 Experiments Analysis 34 2.6 Conformal Geometric Algebra (CGA) 37 2.6.1 Geometric Product in CGA 37 2.6.2 Representation of 3D Point Cloud Based on CGA 38 2.7 Measure of Point Clouds in Space G3 42 2.8 Feature Extraction Method of 3D Point Cloud 45 2.8.1 Delaunay Triangulation Based on CGA 46 2.8.2 Voronoi Diagram Based on CGA 46 2.8.3 Analysis of Normal Vector 48 2.9 Conclusion 53 References 54 Chapter3 Sparse Fast Clifford Fourier Transform 55 3.1 Introduction 55 3.2 Related Work 56 3.2.1 Clifford Algebra 56 3.2.2 Clifford Convolution 57 3.2.3 Clifford Fourier Transform 58 3.3 Sparse Fast Clifford Fourier Transform 60 3.3.1 Sparse Fast Fourier Transform 60 3.3.2 Sparse Fast Clifford Fourier Transform 61 3.3.3 The Proposed Algorithm 65 3.4 Experiment Analysis 66 3.4.1 Performance in Scalar Field 67 3.4.2 Performance in 2D-Grayscale Image 69 3.4.3 Performance in Color Image 70 3.5 Conclusion 72 References 72 Chapter4 Coverage Analysis for Sensor Networks Based on Clifford Algebra 75 4.1 Introduction 75 4.2 Clifford Geometric Algebra 78 4.2.1 Clifford Product 78 4.2.2 Projections in Space Gn 80 4.2.3 Rotations in Space G3 80 4.3 Coverage Analysis Based on Clifford Algebra 82 4.4 Distance Measure Based on Clifford Algebra in Space G3 86 4.4.1 Distance between Two Points in Space G3 86 4.4.2 Distance between a Point in Space G3 and a Line in 1-Dimensional Subspace 86 4.4.3 Distance between a Point in Space G3 and a Plane in 2-Dimensional Subspace 87 4.5 Coverage Model and Analysis for Sensor Networks with Hybrid Types of Targets 88 4.5.1 Coverage Model for Sensor Networks with Hybrid Types of Targets 89 4.5.2 Maximal Support Path, Maximal Breach Path and Boundary Analysis of Sensor Network Coverage 90 4.6 Conclusion and Prospect 92 Appendix A The Meaning of the Outer Product in Space G3 92 Appendix B Properties of Geometric Products 95 Appendix C Proof of Theorem 1 95 Appendix D Proof of Theorem 2 95 References 96 Chapter5 Multispectral Image Edge Detection via Clifford Gradient 98 5.1 Introduction 98 5.2 The Definition and Properties on Clifford Algebra of Multispectral Image 99 5.2.1 The Clifford Algebra Space of Multispectral Image 99 5.2.2 Clifford Algebra Expression of Multispectral Image 100 5.2.3 Clifford Differentiation Operator and Clifford Gradient Operator of Multispectral Image 103 5.3 The New Edge Detection Algorithm Based on Clifford Algebra 104 5.4 Experiments and Analysis 106 5.4.1 Experimental Data and the Corresponding Environment 106 5.4.2 Experiments and Result Analysis 106 5.5 Conclusion 111 References 112 Chapter6 Information Geometry Theory of High-Dimension Space and Application for Speaker Independent Continuous Digit Speech Recognition 114 6.1 Introduction 115 6.2 Basic Concept in HDS 116 6.2.1 HDS and the Point in Space 116 6.2.2 Basic Graph in HDS 117 6.2.3 Graph Transformation in Space 119 6.2.4 Measure in HDS 124 6.3 Point Cover 127 6.3.1 Cover 127 6.3.2 Cover Ratio 127 6.3.3 Local Vertex Cover 130 6.3.4 Covering Product 131 6.4 PCA (Principal Component Analysis) and HDS Geometric Meaning 131 6.4.1 Brief Introduction of PCA 132 6.4.2 HDS Geometry Meaning of PCA 133 6.5 Speech Morphological Analysis in HDS 134 6.5.1 Speech Distribution in HDS 135 6.5.2 Covering Method for Different Classes of Speech 139 6.5.3 Advantages of Point Covering Method 140 6.6 Speaker-Independent Continuous Digit Speech Recognition Based on HDS Covering Dynamic Searching 141 6.6.1 Characteristics of Continuous Speech 142 6.6.2 Feature Extraction Method for Speaker Independent Continuous Speech Recognition and Neural Network Structure in HDS Classification Covering Area 143 6.6.3 New Algorithm and Its Implement of Speaker Independent Continuous Speech Recognition Based on the Dynamic Search Theory of HDS Point Cover 147 6.6.4 Experimental Result and Discussion 149 6.6.5 Comparing Result and Discussion with HMM Model 151 6.7 Mandarin Speech Emotion Recognition 157 6.7.1 The Algorithm of High Dimensional Geometry Theory 157 6.7.2 Algorithm 157 6.7.3 Experiment and Analysis 159 6.7.4 Comparison Using GSVM and HDGT 161 6.8 Conclusion 162 References 163 Chapter7 High Dimensional Imagery Geometry 165 7.1 Introduction 165 7.1.1 The Relations between Information Science and Points Distribution in High-Dimensional Space 165 7.1.2 The Several Main Stages of High Dimensional Imagery Geometry 166 7.2 The Method of HDIG about the Analysis of Point Distribution in High-Dimensional Space 167 7.2.1 The Main Feature of the High Dimensional Imagery Geometry Methods 167 7.2.2 The General Concepts of High Dimensional Imagery Geometry 169 7.2.3 The Instances of the High Dimensional Imagery Geometry Methods 171 7.3 The Software Implementation of Computational Information Geometry 178 7.3.1 The Algorithms of Calculating the Distance from a Point to an Infinite Subspace in High-Dimension-Space 178 7.3.2 The Algorithm to Identifying whether a Point is Inside a Simplex or Not 181 7.4 Viruses' Complete Genomes Classification 182 7.4.1 Material and Method 182 7.4.2 Feature Extraction of Genome Sequence 182 7.4.3 Result and Discussion 183 References 184 Color Figures
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