Abstract
Signals are a collection of related measurements. Of course, all measurements contain errors, so error tolerance is a desirable feature of any signal processing system. At first sight, topological methods appear to be both very tolerant and very intolerant to errors. This tension is characterized by the fact that unknown deformations in the configuration of sensors can produce little or no effect on the output of a topological filter or detector, yet deformations in the values returned by each sensor can dramatically change the output of both topological filters and detectors. We can improve the error tolerance of topological detectors by changing how the measurement values are interpreted. Rather than taking a statistical approach, which assumes a model of randomness associated to each measurement and its relation to those nearby, we assume that whatever values are taken must be self-consistent.
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Notes
- 1.
A multiset is a set that can contain duplicate elements.
- 2.
Two maps belong to the same homotopy class if there is a homotopy between them.
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Robinson, M. (2014). Noise. In: Topological Signal Processing. Mathematical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36104-3_6
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DOI: https://doi.org/10.1007/978-3-642-36104-3_6
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