Title Topological signal processing / Michael Robinson

Published Berlin : Springer, [2014] ©2014

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

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Description 1 online resource : illustrations Series Mathematical engineering Mathematical engineering. Contents 1. Introduction and informal discussion -- 2. Parametrization -- 3. Signals -- 4. Detection -- 5. Transforms -- 6. Noise Summary Signal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems Bibliography Includes bibliographical references and index Notes Print version record Subject Signal processing. TECHNOLOGY & ENGINEERING -- Mechanical. Ingénierie. Signal processing Form Electronic book ISBN 9783642361043 3642361048

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