Persistence stability for geometric complexes

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact spaces. Using recent developments in the theory of topological persistence we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Cech complexes built on top of compact spaces.Comment: We include a discussion of ambient Cech complexes and a new class of examples called Dowker complexe

The observable structure of persistence modules

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of this paper is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persistence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition

The structure and stability of persistence modules

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for persistence module

OrthoMap: Homeomorphism-guaranteeing normal-projection map between surfaces

Consider two (n—1)-dimensional manifolds, S and Sʹ in Rn. We say that they are projection-homeomorphic when the closest projection of each one onto the other is a homeomorphism. We give tight conditions under which S and Sʹ are projection-homeomorphic. These conditions involve the local feature size for S and for Sʹ and the Hausdorff distance between them. Our results hold for arbitrary n

Persistence Stability for Geometric complexes

Non UBCUnreviewedAuthor affiliation: INRIA Saclay Ile-de-FranceFacult

Persistence-based Structural Recognition

International audienceThis paper presents a framework for object recognition using topological persistence. In particular, we show that the so-called persistence diagrams built from functions defined on the objects can serve as compact and informative descriptors for images and shapes. Complementary to the bag-of-features representation, which captures the distribution of values of a given function, persistence diagrams can be used to characterize its structural properties, reflecting spatial information in an invariant way. In practice, the choice of function is simple: each dimension of the feature vector can be viewed as a function. The proposed method is general: it can work on various multimedia data, including 2D shapes, textures and triangle meshes. Extensive experiments on 3D shape retrieval, hand gesture recognition and texture classification demonstrate the performance of the proposed method in comparison with state-of-the-art methods. Additionally, our approach yields higher recognition accuracy when used in conjunction with the bag-of-features

B-morph

Key b-morph trajectories (black) between the b-roundings (solid green and blue) of two non-compatible shapes. Incompatible features (dotted) were removed by the b-rounding. Note the trajectory fans reaching the circular caps that replace these incompatible features. Extended trajectories are faded. Right: Composite b-morph (yellow) and trajectories (faded) between incompatible shapes obtained by recursively b-rounding and morphing the original shapes with their b-roundings. We investigate tools for comparing and morphing between pairs of registered, nearly similar shapes, P and Q. The traditional (closest point) c-morph moves a point p of P at constant velocity towards a closest point q on Q. Our new (tangent ball) b-morph moves each point along a circular arc incident orthogonally onto both P and Q. The b-morph has several advantages over the c-morph: it is symmetric, orientation sensitive, and yields smaller travel distance and distortion. It may help to measure and exaggerate the discrepancy between shapes, morph shapes and their textures, represent one shape as the offset of another, and register shapes. We provide an algorithm for computing bmorphs of b-compatible 2D shapes and extend it to incompatible shape-pairs through recursive b-rounding

Efficient and Robust Topological Data Analysis on Metric Spaces

We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to approximate the sub-level sets of this function by a union of metric balls and extend previous results on sparse Rips filtrations to this setting. This robust and efficient approach to topological data analysis is illustrated with several examples from an implementation

Data-Driven Trajectory Smoothing

Motivated by the increasing availability of large collections of noisy GPS traces, we present a new data-driven framework for smoothing trajectory data. The framework, which can be viewed of as a generalization of the classical moving average technique, naturally leads to efficient algorithms for various smoothing objectives. We analyze an algorithm based on this framework and provide connections to previous smoothing techniques. We implement a variation of the algorithm to smooth an entire collection of trajectories and show that it perform well on both synthetic data and massive collections of GPS traces