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Contents Preface Chapter 1 L1-norm Minimization for Multi-dimensional Signals Based on Geometric Algebra 1 1.1 Introduction 1 1.2 Related Work 3 1.2.1 Preliminaries of Geometric Algebra 3 1.2.2 L1-norm Minimization 4 1.3 The Proposed Algorithm 5 1.3.1 Noiseless Case 5 1.3.2 Noise Case 9 1.4 Multi-dimensional Signal Processing in G2£¬ G3 Space 10 1.4.1 Multi-dimensional Signal Processing in G2 Space 10 1.4.2 Multi-dimensional Signal Processing in G3 Space 11 1.5 Experiments Results and Analysis 13 1.5.1 4-dimensional Signal Reconstruction in G2 Space 13 1.5.2 8-dimensional Signal Reconstruction in G3 Space 16 1.6 Conclusions 21 References 21 Chapter 2 GA-SVD: A Novel Singular Value Decomposition Algorithm for Multispectral Image Based on Geometric Algebra 24 2.1 Introduction 24 2.2 Related Work 26 2.2.1 The Basics of Geometric Algebra 26 2.2.2 Singular Value Decomposition (SVD) 27 2.3 The GA-SVD Algorithm for Multispectral Image 27 2.3.1 Representation of Multispectral Image in GA 28 2.3.2 The Implementation of GA-SVD Algorithm 29 2.3.3 The Reconstruction of Multispectral Image Based on GA-SVD 31 2.4 The SVD Algorithm in G2£¬ G3 Space 32 2.4.1 The SVD Algorithm in G2 Space 32 2.4.2 The SVD Algorithm in G3 Space 33 2.5 Experimental Analysis 34 2.5.1 Data Sets 34 2.5.2 Multispectral Image Compression 36 2.5.3 Multispectral Image Denoising 38 2.6 Conclusions 40 References 41 Chapter 3 Multivector Sparse Representation for Multispectral Images Using Geometric Algebra 44 3.1 Introduction 44 3.2 Related Work 46 3.2.1 Review of Current Sparse Representation Models 46 3.2.2 Representation Models for Multispectral Images 48 3.2.3 The Basics of Geometric Algebra 48 3.3 The Multivector Sparse Represention Model for Multispectral Images 50 3.3.1 Representation of Multispectral Images Using GA 50 3.3.2 GA-Multivector Sparse Representation Model for Multispectral Images 51 3.4 GA-based Dictionary Training 53 3.4.1 GA Dictionary Training Analysis 54 3.4.2 Further Analysis 56 3.5 Experimental Analysis 58 3.5.1 Data Sets 58 3.5.2 Multispectral Images Reconstruction 60 3.5.3 Multispectral Image Denoising 62 3.6 Conclusions 66 Appendix A 66 References 68 Chapter 4 GA-SURF: A New Speeded-up Robust Feature Extraction Algorithm for Multispectral Images Based on Geometric Algebra 72 4.1 Introduction 72 4.2 Related Work 73 4.2.1 SURF Algorithm 73 4.2.2 The Basics of Geometric Algebra 75 4.2.3 GA-SIFT Algorithm 75 4.3 The Proposed GA-SURF Algorithm 76 4.3.1 The Construction of the Hessian Matrix 76 4.3.2 Detection and Descriptor of Interest Points in a Multispectral Image 78 4.3.3 The Implementation of GA-SURF 79 4.4 The Proposed GA-SURF Algorithm 80 4.4.1 Data Set 80 4.4.2 Evaluation Metrics 81 4.4.3 Experimental Results 82 4.5 Conclusions 85 References 85 Chapter 5 Multi-modal Medical Image Registration Based on Feature Spheres in Geometric Algebra 88 5.1 Introduction 88 5.2 Method 90 5.2.1 SURF Algorithm 90 5.2.2 The Basics of Geometric Algebra 91 5.2.3 The GA-SURF Algorithm 92 5.2.4 Applying GA-SURF to the Medical Images 94 5.2.5 Construct Feature Spheres 96 5.2.6 Conformal Geometric Algebra 99 5.3 Results 101 5.4 Conclusions 106 References 106 Chapter 6 GA-STIP: Action Recognition in Multi-channel Videos with Geometric Algebra Based Spatio-temporal Interest Points 109 6.1 Introduction 109 6.2 Related Work 111 6.2.1 Feature Extraction Algorithms Based on Hessian Matrix 111 6.2.2 Geometric Algebra (GA) 112 6.3 The GA-STIP Algorithm for Multi-channel Video 113 6.3.1 Representation of Multi-channel Video in GA 114 6.3.2 Spatio-temporal Interest Points of Multi-channel Video 115 6.3.3 Spatio-temporal Descriptors of Feature Points 121 6.3.4 Action Recognition of the Multi-channel Video 122 6.3.5 The Implementation of GA-STIP 124 6.4 Experimental Analysis 125 6.4.1 Data Sets 125 6.4.2 Experimental Analysis 125 6.4.3 Experimental Results 130 6.5 Conclusions 133 References 134 Chapter 7 GA-CNNs: Convolutional Neural Networks Based on Geometric Algebra 138 7.1 Introduction 138 7.2 Related Work 139 7.2.1 Basics of Geometric Algebra 139 7.2.2 Neural Networks Based on Geometric Algebra 141 7.3 Convolutional Neural Networks Based on Geometric Algebra (GA-CNNs) 142 7.3.1 Convolutional Layer 143 7.3.2 Pooling Layer 144 7.3.3 Fully-connected Layer 144 7.3.4 Backpropagation Algorithm 145 7.4 Experiments and Analysis 147 7.4.1 Experiment on Synthetic Data 147 7.4.2 Experiment on Color Images 149 7.4.3 Experiment on Hyperspectral Images 151 7.5 Conclusions 155 References 155 Chapter 8 Joint Sparse