Herbert Edelsbrunner

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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 &#8220;shape&#8221; of the set. For that purpose, this article introduces the formal notion of the family of &#945;-shapes of a finite point set in R<supscrpt>3</supscrpt>. Each shape is a(More)
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)
A generalization of the convex hull of a finite set of points in Akl and Toussaint [ 11, for instance, discuss the relevance the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, “o-shapes,” which seem to capture the intuitive of the convex hull problem to pattern recognition. By notions of “fine shape” and(More)
Given a subspace<inline-equation><f><blkbd>X&#8838;R</blkbd><sup>d</sup></f></inline-equation> and a finite set <inline-equation><f>S&#8838;<blkbd>R</blkbd><sup>d</sup></f></inline-equation>, we introduce the Delaunay simplicial complex, <inline-equation><f><sc>D</sc><inf><blkbd>X</blkbd></inf></f></inline-equation>, restricted by(More)
A set ofn weighted points in general position in ℝ d defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next(More)
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)
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)