Alpha Shapes — a Survey

Alpha shapes have been conceived in 1981 as an attempt to define the shape of a finite set of point in the plane. Since then, connections to diverse areas in the sciences and engineering have developed, including to pattern recognition, digital shape sampling and processing, and structural molecular biology. This survey begins with a historical account and discusses geometric, algorithmic, topological, and combinatorial aspects of alpha shapes in this sequence.

[1]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. , 1908 .

[2]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[3]  B. Lee,et al.  The interpretation of protein structures: estimation of static accessibility. , 1971, Journal of molecular biology.

[4]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[5]  Karl W. Kratky,et al.  The area of intersection of n equal circular disks , 1978 .

[6]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[7]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[8]  David G. Kirkpatrick,et al.  On the shape of a set of points in the plane , 1983, IEEE Trans. Inf. Theory.

[9]  M. L. Connolly.  Analytical molecular surface calculation , 1983 .

[10]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[11]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[12]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[13]  Rex A. Dwyer,et al.  Average-case analysis of algorithms for convex hulls and Voronoi diagrams , 1988 .

[14]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[15]  B. Joe.  Three-dimensional triangulations from local transformations , 1989 .

[16]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[17]  Donald E. Knuth,et al.  Axioms and Hulls , 1992, Lecture Notes in Computer Science.

[18]  Herbert Edelsbrunner,et al.  Three-dimensional alpha shapes , 1992, VVS.

[19]  Herbert Edelsbrunner,et al.  Weighted alpha shapes , 1992 .

[20]  D. Naiman,et al.  INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS , 1992 .

[21]  Herbert Edelsbrunner,et al.  The union of balls and its dual shape , 1993, SCG '93.

[22]  R. Seidel.  Backwards Analysis of Randomized Geometric Algorithms , 1993 .

[23]  Achille C. Varzi,et al.  Holes and Other Superficialities , 1994 .

[24]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[25]  G. Ziegler.  Lectures on Polytopes , 1994 .

[26]  Herbert Edelsbrunner,et al.  An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere , 1995, Comput. Aided Geom. Des..

[27]  Herbert Edelsbrunner,et al.  On the Definition and the Construction of Pockets in Macromolecules , 1998, Discret. Appl. Math..

[28]  Ho-Lun Cheng,et al.  Shape space from deformation , 1998, Proceedings Pacific Graphics '98. Sixth Pacific Conference on Computer Graphics and Applications (Cat. No.98EX208).

[29]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[30]  Herbert Edelsbrunner,et al.  Deformable Smooth Surface Design , 1999, Discret. Comput. Geom..

[31]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[32]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[33]  Herbert Edelsbrunner,et al.  The weighted-volume derivative of a space-filling diagram , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Jean-Daniel Boissonnat,et al.  Complexity of the delaunay triangulation of points on surfaces the smooth case , 2003, SCG '03.

[35]  Joachim Giesen,et al.  The flow complex: a data structure for geometric modeling , 2003, SODA '03.

[36]  H. Edelsbrunner.  Surface Reconstruction by Wrapping Finite Sets in Space , 2003 .

[37]  Günter Rote,et al.  Incremental constructions con BRIO , 2003, SCG '03.

[38]  Herbert Edelsbrunner,et al.  The Area Derivative of a Space-Filling Diagram , 2004, Discret. Comput. Geom..

[39]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type as its medial axis , 2004, Comput. Aided Des..

[40]  Gunnar E. Carlsson,et al.  Topological estimation using witness complexes , 2004, PBG.

[41]  Herbert Edelsbrunner.  Biological applications of computational topology , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[42]  William L. Seaver,et al.  Random Graphs for Statistical Pattern Recognition , 2005, Technometrics.

[43]  Herbert Edelsbrunner,et al.  Incremental topological flipping works for regular triangulations , 1992, SCG '92.

[44]  Herbert Edelsbrunner,et al.  Inclusion-exclusion formulas from independent complexes , 2005, Symposium on Computational Geometry.

[45]  Herbert Edelsbrunner,et al.  The geometry of biomolecular solvation , 2005 .

[46]  D. Cohen-Steiner,et al.  Geometric Inference , 2007 .

[47]  R. Pollack,et al.  Surveys on discrete and computational geometry : twenty years later : AMS-IMS-SIAM Joint Summer Research Conference, June 18-22, 2006, Snowbird, Utah , 2008 .

[48]  H. Edelsbrunner,et al.  Persistent Homology — a Survey , 2022 .