Table of Contents |
1. | Introduction and Informal Discussion | 1 |
1.1. | Meet the Case Studies | 2 |
1.2. | Discrete Approximations of Spaces | 4 |
1.3. | Local Data: Filtering | 5 |
1.4. | The Interplay Between Local Data and Global Inference: Detection | 6 |
1.5. | Coda: An Invitation | 7 |
2. | Parametrization | 9 |
2.1. | Abstract Spaces | 10 |
2.1.1. | CW Complexes | 10 |
2.1.2. | Cellular Maps and Homotopy | 16 |
2.2. | Representation of Spaces | 17 |
2.2.1. | Abstract Simplicial Complexes | 17 |
2.2.2. | Manifolds and Embeddings | 20 |
2.3. | Case Study: Signal Manifolds for Localization, Tracking, and Navigation | 28 |
2.3.1. | Signal Manifold Fingerprinting | 32 |
2.3.2. | Multiple Target Detection and Localization | 34 |
2.4. | Open Questions | 36 |
| References | 37 |
3. | Signals | 39 |
3.1. | Locality: Principles and Axioms | 39 |
3.1.1. | Sheaf Morphisms | 46 |
3.2. | Global Sections | 48 |
3.3. | Operations on Sheaves | 52 |
3.3.1. | Pushforwards and Pullbacks | 53 |
3.3.2. | Algebraic Operations | 59 |
3.4. | Case Study: Topological Filters | 61 |
3.4.1. | Linear Shift-Invariant Systems | 61 |
3.4.2. | Linear Filtering on Nontrivial Base Spaces | 65 |
3.4.3. | Thresholding Filters | 67 |
3.4.4. | Angle-Valued Filters | 70 |
3.5. | Case Study: Indoor Wave Propagation | 74 |
3.5.1. | Transmission Line Sheaves | 75 |
3.5.2. | Sheaf Pushforwards and Edge Collapse | 78 |
3.6. | Open Questions | 82 |
| References | 83 |
4. | Detection | 85 |
4.1. | Categories and Functors | 85 |
4.1.1. | Detectors are Functors | 88 |
4.2. | Exact Sequences | 89 |
4.3. | Sheaf Cohomology | 95 |
4.3.1. | Orientation | 95 |
4.3.2. | Definition of Sheaf Cohomology | 97 |
4.3.3. | Interpretation and Examples | 104 |
4.4. | Long Exact Sequences for Cohomology | 107 |
4.4.1. | Mayer-Vietoris Sequences for Sheaves | 107 |
4.5. | General Sampling Theorem for Signal Sheaves | 109 |
4.5.1. | The Shannon-Nyquist Theorem | 111 |
4.5.2. | Sampling of Heterogeneous, Non-bandlimited Signals | 113 |
4.5.3. | Sampling in Topological Filters | 115 |
4.6. | Case Study: Tracking Water Pollution | 117 |
4.6.1. | A Sheaf of Concentrations | 117 |
4.6.2. | Elementary Water Flow Networks | 118 |
4.6.3. | Measurement of Larger Networks | 122 |
4.7. | Case Study: Extracting Topology from Intersections in Coverage | 123 |
4.7.1. | The Nerve Model of a Space | 123 |
4.8. | Open Questions | 131 |
| References | 131 |
5. | Transforms | 133 |
5.1. | The Euler Characteristic | 134 |
5.1.1. | Valuations | 137 |
5.1.2. | The Euler Integral | 138 |
5.2. | Case Study: Target Enumeration | 143 |
5.3. | Euler Integral Transforms | 146 |
5.3.1. | The Euler--Fourier Transform | 149 |
5.3.2. | Euler--Bessel Transform | 151 |
5.3.3. | Sidelobe Cancellation | 156 |
5.4. | Case Study: Shape Recognition in Computer Vision | 159 |
5.5. | Open Questions | 159 |
| References | 161 |
6. | Noise | 163 |
6.1. | Persistence | 165 |
6.1.1. | Persistence Sheaves | 165 |
6.1.2. | Interpretation of Persistent Cohomology | 168 |
6.2. | Case Study: Experimental Validation of Topology Extraction | 169 |
6.3. | Persistent Cohomology is a Robust Detector | 173 |
6.3.1. | Historical Context | 176 |
6.4. | Case Study: Quasi-Periodic Signals | 176 |
6.4.1. | Experimental Setup | 178 |
6.4.2. | Results of Persistent Cohomology | 180 |
6.5. | Recovering a Space from a Point Cloud | 181 |
6.6. | Case Study: Recovery of a Space from Measurements of Waves | 187 |
6.7. | Open Questions | 191 |
| References | 192 |
Appendix A | Topological Spaces and Continuity | 195 |
Appendix B | Topological Groups | 199 |
| Index | 203 |