本书针对具有多维信号处理中产生的信息几何与几何不变量问题，探索一种新的多维信号处理方法。从信息学角度出发，给出几何不变量，并研究其几何不变量的性质，为实现具有多维信号处理问题提供有效的解决方案。本书适合从事智能信息处理、人工智能、计算机视觉等领域工作的学者和研究人员阅读人参考，同时也可以作为理工科大学相关专业研究生的教学参考书。

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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 Representation Model for Multi-channel Image Based on Reduced Geometric Algebra 158
8.1 Introduction 158
8.2 Related Work 160
8.2.1 Sparse Representation Models for Color Image 160
8.2.2 The Basic of Geometric Algebra 161
8.3 Reduced Geometric Algebra (RGA) 163
8.3.1 Defnition 163
8.3.2 The Properties of Reduced Geometric Algebra 163
8.3.3 Singular Value Decomposition (SVD) of RGA 165
8.4 The RGA-sparse Representation Model for Multi-channel Image 166
8.5 RGA-based Dictionary Training 168
8.5.1 RGA Dictionary Training 168
8.5.2 Further Analysis 170
8.6 Experiments and Analysis 172
8.6.1 Data Sets 172
8.6.2 Color Image Reconstruction 173
8.6.3 Color Image Denosing 175
8.7 Conclusions 178
References 179
Chapter 9 An Extended Multilayer Perceptron Model Using Reduced Geometric Algebra 182
9.1 Introduction 182
9.2 Reduced Geometric Algebra (RGA) 184
9.3 Multilayer Perceptron Model Based on RGA for Color Images 186
9.3.1 RGA Neuron Model for Color Images 186
9.3.2 The Structure of RGA-MLP Model 188
9.3.3 Learning Algorithm 189？
9.4 Experiments and Analysis on Color Images Classifcation 190
9.4.1 Experimental Setup 190
9.4.2 Color Image Classifcation 191
9.5 Conclusions 193
References 194
Chapter 10 RGA-CNNs: Convolutional Neural Networks Based on Reduced Geometric Algebra 196
10.1 Introduction 196
10.2 Related Work 198
10.2.1 Convolutional Neural Network (CNN) 198
10.2.2 The Basics of Geometric Algebra 200
10.3 Reduced Geometric Algebra (RGA) 201
10.3.1 The Basics of RGA 201
10.3.2 Convolution in RGA 203
10.4 Convolutional Neural Networks Based on Reduced Geometric Algebra 203
10.4.1 The Structure of RGA-CNNs 203
10.4.2 RGA Neuron Model 204
10.4.3 RGA Multilayer Perceptron (RGA-MLP) and Its Learning Algorithm 206
10.5 Classifcation Experiments and Analysis 208
10.5.1 Data Sets 208
10.5.2 Experimental Setup 210
10.5.3 3D Geometrical Shapes Classiˉcation 210
10.5.4 Color Image Classiˉcation 213
10.5.5 Complexity Analysis 216
10.6 Conclusions 217
References 218
Color Figures