1 - 6 of 6 Chapters
[Signal processing is the discipline of extracting information from related collections of measurements. To be effective, measurements must be organized and then filtered, detected, or transformed in some way to expose the desired information.]
[This chapter develops several models of topological spaces associated to collections of measurements.]
[The theory of sheaves as local data over a topological space is a compelling framework for the study of measurements. Although the theory of sheaves is generally acclaimed to be technically challenging, a variant developed for sheaves on cell complexes is much more manageable and well-suited to...
[Detectors are tools which extract and emphasize important features of a signal. In order to be useful, a detector should be functorial, in that it preserves the features of the signal. In most practical settings, it is desirable for detectors to be lossy. Specifically, one usually wants to...
[Many filters arise from operating locally in a transformed space of functions. In contrast to filters, the transforms themselves are non-local and have global symmetries. For instance, the uncertainty principle in the Fourier transform ensures that the value of a function at a particular point...
[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...
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