Application of Persistent Homology in Signal and Image Denoising

by Yi Zheng
Doctoral thesis
Date of Examination:2015-06-12
Date of issue:2015-07-29
Advisor:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Prof. Dr. Armin Iske
Persistent Address: http://dx.doi.org/10.53846/goediss-5200

 

 

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Abstract

English

Motivated by recent developments in topological persistence for assessment of the importance of features in data sets, we study the ideas of persistence homology for one-dimensional digital signals and its application in signal and image denoising. The notions of persistence homology and persistence pairs were introduced in for measuring the topological complexity of point sets in R3. Persistence pairs and corresponding persistence diagrams are well suited to quantify the topological significance of data structures and to develop a formalism for topological simplification. In case of one-dimensional digital signals the idea of topological persistence boils down to the problem of pairing suitable local minima and maxima of the signal. Considering the persistence pairs and the corresponding persistences not only for the signal f but also for -f, we propose the new notion of persistence distance of f. Transferring from f to -f switches the roles of the sets of local minima and local maxima of f. A comparison of the persistence pairs obtained for f and for -f already provides us with an important categorization tool. Persistence pairs occurring for both, f and -f, are less significant than those occurring only once, for f or for -f. We show that the persistence distance has a lot of favorable properties. Particularly, we show that the persistence distance is very closely related to the discrete total variation of f. This relation motivates us to employ the new notion of persistence distance for signal and image denoising. The new signal denosing method is based on the basic scheme of the celebrated ROF-model and makes use of the additional information given by persistence. It replaces the regularization term of ROF-model with persistence-weighted regularization term to obtain the persistence weighted ROF-model. In particular, the weights are taken in a way such that the denoised signal obtained by minimization of weighted ROF-functional preserves the essential peaks (discontinuities) of f well and yields good denoising performance at smooth subregions in f. Furthermore, a two dimensional persistence-weighted ROF model can be established where the weights are determined according to the one dimensional persistence information.
Keywords: discrete total variation; persistence homology; persistence pairs; persistence distance; image denoising
 
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