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We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental… (More)

- Herbert Edelsbrunner
- EATCS Monographs on Theoretical Computer Science
- 1987

- David Cohen-Steiner, Herbert Edelsbrunner, John Harer
- Discrete & Computational Geometry
- 2005

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply only small changes in the diagram. We apply this result to estimating the homology of sets in a metric space and… (More)

- Herbert Edelsbrunner, Ernst P. Mücke
- ACM Trans. Graph.
- 1992

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of α-shapes of a finite point set in R<supscrpt>3</supscrpt>. Each shape is a… (More)

- J Liang, H Edelsbrunner, C Woodward
- Protein science : a publication of the Protein…
- 1998

Identification and size characterization of surface pockets and occluded cavities are initial steps in protein structure-based ligand design. A new program, CAST, for automatically locating and measuring protein pockets and cavities, is based on precise computational geometry methods, including alpha shape and discrete flow theory. CAST identifies and… (More)

- Herbert Edelsbrunner, David G. Kirkpatrick, Raimund Seidel
- IEEE Trans. Information Theory
- 1983

ACKNOWLEDGMENT The authors express their appreciation for numerous constructive suggestions, which led to improvements on various phases of the manuscript, to Dr. A high performance multiple data rate burst modem for satellite packet communication, " EASCON Conference Record, Nov. 198 1. A microprocessor-based PSK modem for packet transmission over… (More)

- Herbert Edelsbrunner
- Discrete & Computational Geometry
- 1993

Efficient algorithms are described for computing topological,combinatorial, and metric properties of the union of finitely many ballsin <inline-equation><f><blkbd>R</blkbd><sup>d</sup></f></inline-equation>. These algorithms are based on a simplicial complexdual to a certain decomposition of the union of balls, and on shortinclusion-exclusion formulas… (More)

- J Liang, H Edelsbrunner, P Fu, P V Sudhakar, S Subramaniam
- Proteins
- 1998

The size and shape of macromolecules such as proteins and nucleic acids play an important role in their functions. Prior efforts to quantify these properties have been based on various discretization or tessellation procedures involving analytical or numerical computations. In this article, we present an analytically exact method for computing the metric… (More)

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale 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… (More)