Persistent homology transform for modeling shapes and surfaces

Abstract

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in R 3 and shapes in R 2 . This statistic is a collection of persistence diagrams -multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in R 3 into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for probability densities on the space of piecewise linear shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data. persistence homology, surfaces, shape spaces, sufficient shape statistics Insert classification here 1 arXiv:1310.1030v2 [math.ST]

Persistence Paths and Signature Features in Topological Data Analysis

“…, n}, and forms an n-dimensional path by running in the i-th coordinate with unit speed if the i-th bar is active (otherwise remaining constant). Euler embedding ι χ : reduces a barcode to a single Euler characteristic curve (see [33,Sec. 3.2]).…”

Fréchet-Stable Signatures Using Persistence Homology

“…It simply sees past the parameterization. Our approach is very similar to both the persistent homology transform of Turner et al [26] and the persistence distortion distance of Dey et al [14]. However, we extend these ideas to arbitrary metrics and give the first connections between these approaches and the Fréchet distance (see Section 4).…”

“…In that case, the signature could be computed using the Vineyard algorithm of Cohen-Steiner et al [12]. This would be the first step towards a coordinate-free version of the persistent homology transform of Turner et al [26].…”

Topological Data Analysis

“…The Euler characteristic has also been used for classification of shapes (Richardson & Werman, 2014). See also Turner et al (2014). Bendich et al (2010) use topological methods to study the interactions between root systems of plants.…”