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Proximity graphs: reconstruction of curves and surfaces
Framework 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. M. Melkemi

Duality: Voronoi diagram and Delaunay triangulation (1)
A Voronoi region of a point is defined by: The Voronoi diagram of the set S, DV(S), is the set of the regions 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 (2)
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 (3)
Examples 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.

Duality: Voronoi diagram and
Delaunay triangulation (4)

Duality: Voronoi diagram and
Delaunay triangulation (5)

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

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

Power diagram and regular triangulation (3)
1 2 1 2 R1 R2 R1 R2 1 2 1 2 R1 R2 R1 R2

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

Power diagram and regular
triangulation (5)

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.

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

Alpha-shape of a set of points (1)

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

Alpha-shape of a set of points: example(3)

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

Alpha-shape of a set of points: properties(5)
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 (6)
Theorem (2D case)

Alpha-shape of a set of points (7)

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

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

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

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

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

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

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.

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

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

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.

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

Weighted alpha shape (2)
x’

Weighted alpha shape (3)

Weighted alpha shape (4)

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

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

Gabriel Graph: definition (1)

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

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

Gabriel Graph: example (4)

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

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

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

Beta skeleton (3) A beta-skeleton edge

Beta skeleton (4) beta = 1.1 beta = 1.4

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

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.

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

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.

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

Reconstruction : e-sampling condition(2)

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

A-shape and Crust (1)

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

A-shape and Crust (3)

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.

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

Crust : algorithm (6) 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 : reconstruction (7)
The crust of S (S being an e-sampling of C) reconstructs the curve C if e <1/5.

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.

Local Crust : definition and properties (2)

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

Voronoi Gabriel Graph (VGG)
Local Crust and Gabriel Graph (4) Voronoi Gabriel Graph (VGG) [v v’] is an edge of the VGG of S iff [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.

Local Crust and Gabriel Graph (5)

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

Local Crust and Gabriel Graph (7)
Voronoi Gabriel Graph

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

3D reconstruction: an example

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