Results 1  10
of
331
Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
Abstract

Cited by 152 (21 self)
 Add to MetaCart
(Show Context)
We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
MorseSmale Complexes for Piecewise Linear 3Manifolds
, 2003
"... We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatori ..."
Abstract

Cited by 133 (28 self)
 Add to MetaCart
We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
Abstract

Cited by 117 (4 self)
 Add to MetaCart
(Show Context)
An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
Barcodes: The persistent topology of data
, 2007
"... Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persis ..."
Abstract

Cited by 107 (3 self)
 Add to MetaCart
(Show Context)
Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persistent homology—and a novel representation of this algebraic characterization— barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a fourdimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object? ” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer threedimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations
Removing excess topology from isosurfaces
 ACM Trans. Graph
, 2004
"... Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studie ..."
Abstract

Cited by 85 (1 self)
 Add to MetaCart
Many highresolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this paper we present an efficient method for removing handles in an isosurface. Our algorithm makes an axisaligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed for outofcore execution. It finds the handles by incrementally constructing and analyzing a surface Reeb graph. The size of a handle is measured by a short surface loop that breaks it. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefit for subsequent surface processing.
Proximity of persistence modules and their diagrams
, 2008
"... Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case o ..."
Abstract

Cited by 82 (15 self)
 Add to MetaCart
(Show Context)
Topological persistence has proven to be a key concept for the study of realvalued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications.
ShapeGoogle: geometric words and expressions for invariant shape retrieval
, 2010
"... The computer vision and pattern recognition communities have recently witnessed a surge of featurebased methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words ” and treat them using text search approaches following the ..."
Abstract

Cited by 82 (13 self)
 Add to MetaCart
The computer vision and pattern recognition communities have recently witnessed a surge of featurebased methods in object recognition and image retrieval applications. These methods allow representing images as collections of “visual words ” and treat them using text search approaches following the “bag of features ” paradigm. In this paper, we explore analogous approaches in the 3D world applied to the problem of nonrigid shape retrieval in large databases. Using multiscale diffusion heat kernels as “geometric words”, we construct compact and informative shape descriptors by means of the “bag of features ” approach. We also show that considering pairs of “geometric words ” (“geometric expressions”) allows creating spatiallysensitive bags of features with better discriminativity. Finally, adopting metric learning approaches, we show that shapes can be efficiently represented as binary codes. Our approach achieves stateoftheart results on the SHREC 2010 largescale shape retrieval benchmark.
Vector field design on surfaces
, 2006
"... Vector field design on surfaces is necessary for many graphics applications: examplebased texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vecto ..."
Abstract

Cited by 72 (20 self)
 Add to MetaCart
Vector field design on surfaces is necessary for many graphics applications: examplebased texture synthesis, nonphotorealistic rendering, and fluid simulation. For these applications, singularities contained in the input vector field often cause visual artifacts. In this article, we present a vector field design system that allows the user to create a wide variety of vector fields with control over vector field topology, such as the number and location of singularities. Our system combines basis vector fields to make an initial vector field that meets user specifications. The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated due to the PoincaréHopf index theorem. To reduce the visual artifacts caused by these singularities, our system allows the user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations offer topological guarantees for the vector field in that they only affect userspecified singularities. We develop efficient implementations of these operations based on Conley index theory. Our system also provides other editing operations so that the user may change the topological and geometric characteristics of the vector field. To create continuous vector fields on curved surfaces represented as meshes, we make use of the ideas of geodesic polar maps and parallel transport to interpolate vector values defined at the vertices of the mesh. We also use geodesic polar maps and parallel transport to create basis vector fields on surfaces that meet the user specifications. These techniques enable our vector field design system to work for both planar domains and curved surfaces.
Persistent Homology  a Survey
 CONTEMPORARY MATHEMATICS
"... Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. ..."
Abstract

Cited by 66 (3 self)
 Add to MetaCart
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.