Given a real-valued function f defined over a manifold M embedded in R^d, we are interested in recovering structural information about f from the sole information of its values on a finite sample P. Existing methods provide approximation to the persistence diagram of f when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice. We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.
@InProceedings{buchet_et_al:LIPIcs.SOCG.2015.827, author = {Buchet, Micka\"{e}l and Chazal, Fr\'{e}d\'{e}ric and Dey, Tamal K. and Fan, Fengtao and Oudot, Steve Y. and Wang, Yusu}, title = {{Topological Analysis of Scalar Fields with Outliers}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {827--841}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.827}, URN = {urn:nbn:de:0030-drops-51052}, doi = {10.4230/LIPIcs.SOCG.2015.827}, annote = {Keywords: Persistent Homology, Topological Data Analysis, Scalar Field Analysis, Nested Rips Filtration, Distance to a Measure} }
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