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Proximity graphs: reconstruction of curves and surfaces Duality between the Voronoi diagram and the Delaunay triangulation. Power diagram. Alpha shape and weighted alpha shape. The Gabriel Graph. The beta-skeleton Graph. A-shape and Crust. Local Crust and Voronoi Gabriel Graph. NN-crust. Framework M. Melkemi

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The Voronoi diagram of the set S, DV(S), is the set of the regions A Voronoi region of a pointis defined by: A 3-cell is a Voronoi polyhedron, a 2-cell is a face,a 1-cell is an edge of DV(S). Duality: Voronoi diagram and Delaunay triangulation (1)

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is a k-simplex of the Delaunay triangulation D(S) iff there exists an open ball b such that: Duality: Voronoi diagram and Delaunay triangulation (2)

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Duality: Voronoi diagram and Delaunay triangulation (3) A Delaunay triangle corresponds to a Voronoi vertex. An edge of D(S) corresponds to a Voronoi edge. A Delaunay vertex corresponds to a Voronoi region. Examples

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Duality: Voronoi diagram and Delaunay triangulation (4)

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Duality: Voronoi diagram and Delaunay triangulation (5)

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Power diagram and regular triangulation (1) A weighted point is denoted as p=(p’,p’’), with its location andits weight.For a weighted points, p=(p’,p’’), the power distance of a point x to p is defined as follows: (p,x) x p’

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Power diagram and regular triangulation (2) The locus of the points equidistant from two weighted points is a straight line.

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Power diagram and regular triangulation (3) R1R2 R1 R2 R1R2 R1R2

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Power diagram and regular triangulation (4) The power diagram of the set S, P(S), is the set of the regions A power region of a pointis defined by:

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Power diagram and regular triangulation (5)

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Power diagram and regular triangulation (6) A power region may be empty. A power region of p may be does not contain the point p. A point on the convex hull of S has an unbounded or an empty region.

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Power diagram and regular triangulation (7) is a k- simplex of the regular triangulation of S iff

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Alpha-shape of a set of points (1)

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Alpha-shape of a set of points: example (2)

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Alpha-shape of a set of points: example(3) alpha = 10alpha = 20 alpha = 40alpha = 60

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Alpha-shape of a set of points: example(4)

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The alpha shape is a sub-graph of the Delaunay triangulation. The convex hull is an element of the alpha shape family. Alpha-shape of a set of points: properties(5)

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Theorem (2D case) Alpha-shape of a set of points (6)

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Alpha-shape of a set of points (7)

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Input: the point set S, output: -shape of S Compute the Voronoi diagram of S. For each edge e compute the values min (e) and max (e). For each edge e If ( min (e)<= <= max (e)) then e is in the - shape of S. Alpha-shape of a set of points: algorithm(8)

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Alpha-shape of a set of points : 3D case(9) p1 p2 p3 v1 v2 2-simplex 1-simplex

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Simplicial Complex Alpha-shape of a set of points (10) A simplicial complex K is a finite collection of simplices with the following two properties: A Delaunay triangulation is a simplicial complex.

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Alpha Complex Alpha-shape of a set of points (11)

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Alpha Complex Alpha-shape of a set of points (12)

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Alpha-shape of a set of points (13) Alpha Complex : example

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Alpha-shape of a set of points (14) Curve reconstruction: definition The problem of curve reconstruction takes a set, S, of sample points on a smooth closed curve C, and requires to produce a geometric graph having exactly those edges that connect sample points adjacent in C.

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A set of points S The reconstructed surface Alpha-shape of a set of points (15) Surface reconstruction

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Curve reconstruction : theorem Alpha-shape of a set of points (16)

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Alpha-shape of a set of points (17) The sampling density must be such that the center of the “disk probe” is not allowed to cross C without touching a sample point. Examples of non admissible cases of probe-manifold intersection.

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For two weighted points, (p’, p ’’) and x=(x’,x’’), we define Weighted alpha shape (1)

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p’ x’ Weighted alpha shape (2)

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Weighted alpha shape (3)

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Weighted alpha shape (4)

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Weighted alpha shape (5) The weighted alpha shape is a sub-graph of the regular triangulation.

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Input: the points set S, output: weighted -shape of S. Compute the power diagram of S. For each edge e of the regular triangulation of S compute the values min (e) and max (e). For each edge e If ( min (e)<= <= max (e)) then e is in the weighted - shape of S. Weighted alpha-shape (6)

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Gabriel Graph: definition (1)

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Gabriel Graph: example (2) An edge of Gabriel This edge is not in the GG

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Gabriel Graph: properties (3) 1) The Gabriel graph of S is a sub graph of the Delaunay triangulation of S.

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Gabriel Graph: example (4)

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Compute the Voronoi diagram of S. A Delaunay edge e belongs to the Gabriel Graph of S iff e cuts its dual Voronoi-edge. Gabriel Graph: algorithm (5)

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Beta skeleton (1) -neighborhood, neighborhood, The Gabriel graph is an element of the -skeleton family ( = 1). The -skeleton is a sub-graph of the Delaunay triangulation.

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Beta skeleton (2) Examples of -neighborhood : Forbidden regions

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A beta-skeleton edge (3)Beta skeleton

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Beta skeleton (4) beta = 1.1beta = 1.4

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Beta skeleton : algorithm (5) The coordinates of these centers are:

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Medial axis (1) The medial axis of a region, defined by a closed curves C, is the set of points p which have a same distance to at least two points of C.

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Medial axis and Voronoi diagram(2) A Delaunay disc is an approximation of a maximal ball

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Medial axis and Voronoi diagram (3) Let S be a regular sampling of C. Compute the Voronoi diagram of S. A Voronoi edge vv’ is in an approximation of the medial axis of C if it separates two non adjacent samples on C.

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S is an -sampling ( <1) of a curve C iff Reconstruction : -sampling condition(1)

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Reconstruction : -sampling condition(2)

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Reconstruction : -skeleton (3) Let S -sample a smooth curve, with < The -skeleton of S contains exactly the edges between adjacent vertices on the curve, for = 1.70.

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A-shape and Crust (1)

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A-shape and Crust (2) An edge of A-shape

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A-shape and Crust (3)

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A-shape et Crust (4) Crust of S is an A-shape of S when A is the set of the vertices of the Voronoi diagram of S.

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A-shape et Crust (5) Voronoi vertex crust Voronoi crust

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Compute the Voronoi diagram of S, DV(S). Compute the Voronoi diagram of SUV, DV(SUV), V being the set of the Voronoi vertices of DV(S). A k-simplex, conv(T), of the Delaunay triangulation of SUV, belongs to the crust of S iff the points of T have a same neighbor belonging to V. Crust : algorithm (6)

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The crust of S (S being an -sampling of C) reconstructs the curve C if <1/5. Crust : reconstruction (7)

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Local Crust : definition and properties (1) v v’ is the dual Voronoi edge of pp’, b(p p’ v) is the ball which circumscribes the points p, p’,v.

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Local Crust : definition and properties (2)

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Local Crust and Gabriel Graph (3) Local crust of S is a sub graph of the Gabriel Graph of S.

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Voronoi Gabriel Graph (VGG) Local Crust and Gabriel Graph (4) [v v’] is the dual Voronoi edge of the Delaunay edge [pp’]. b(v v’) is the ball of diameter v v’. An edge pp’ belongs to the Local crust of S iff vv’ belongs to the VGG of S. [v v’] is an edge of the VGG of S iff

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Local Crust and Gabriel Graph (5)

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The Local crust of S (S being an -sampling of C) reconstructs the curve C, if <0.42. Local Crust : reconstruction (6)

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Local Crust and Gabriel Graph (7) Local crust Voronoi Gabriel Graph

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NN-Crust: curve reconstruction 1.Compute the Delaunay triangulation of S. E is empty. 2.For each p in S do 1.Compute the shortest edge pq in D(S). 2.Compute the shortest edge ps so that the angle (pqs) more than . E= E U {pq, ps}. 3.E is the NN-crust of S.

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3D reconstruction: an example

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