Relation between total variation and persistence distance and its application in signal processing
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In this paper we establish the new notion of persistence distance for discrete signals and study its main properties. The idea of persistence distance is based on recent developments in topological persistence for assessment and simplification of topological features of data sets. Particularly, we establish a close relationship between persistence distance and discrete total variation for finite signals. This relationship allows us to propose a new adaptive denoising method based on persistence that can also be regarded as a nonlinear weighted ROF model. Numerical experiments illustrate the ability of the new persistence based denoising method to preserve significant extrema of the original signal.
KeywordsDiscrete total variation Persistence homology Persistence pairs Persistence distance
Mathematics Subject Classifications (2010)41A15 49M25 57M50 65D10 65D18 94A12
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