Proximity graphs: reconstruction of curves and

surfacesDuality between the Voronoi diagram andthe 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

The Voronoi diagram of the set S, DV(S), is the set of the regions

A Voronoi region of a point ijppppp; )R(p jii

ip is defined by:

.)R(pi

isR cell-k a3k0k,4 T, TS,T

Tp

R(p)

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)

conv(T)3,k01,kTS,T T

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

TSbSb et

Duality: Voronoi diagram and Delaunay triangulation (2)

D(S)-kT ofsimplexais

Tp

DV(S)k3R(p)

ofcell)(ais

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

Duality: Voronoi diagram and Delaunay triangulation (4)

Duality: Voronoi diagram and Delaunay triangulation (5)

Power diagram and regular triangulation (1)

points. weightedof set finite a beLet RRS d

A weighted point is denoted as p=(p’,p’’), with dRp'

Rp"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"xp'x)(p, 2

(p,x)

xp’

"p

Power diagram and regular triangulation (2)

The locus of the points equidistant from two weighted points is a straight line.

x),(px),(p ji

)/2pyxpy(x)yy(y)xx(x "j

2j

2j

"i

2i

2iijij

Power diagram and regular triangulation (3)

1 21 2

1 2 1 2

R1 R2R1 R2

R1 R2R1 R2

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 point

ijx),px),p(x;)R(p jii (

ip is defined by:

.)R(pi

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.

T

.Tp

R(p)

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)

.et TSbSb αα

of 3,0 simplex,- a is kkT

b ball a exists there iff S of shapeα :that such radius of 0

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

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

alpha = 10 alpha = 20

alpha = 40 alpha = 60

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

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)

Theorem (2D case)

there ]p[peedgeDelaunayeachFor ji

that suchandexists 0(e)α0(e)α maxmin

.αααα maxmin iff S of shape]p[pe ji

Alpha-shape of a set of points (6)

Alpha-shape of a set of points (7)

Input: the point set S, output: -shape of S Compute the Voronoi diagram of S. For each edge ecompute the values min(e) and max(e). For each edge eIf (min(e)<=<=max(e)) then e is in the -

shape of S.

Alpha-shape of a set of points: algorithm(8)

Alpha-shape of a set of points : 3D case(9)

p1

p2

p3v1

v2

minα

p1v2p1v1,maxmaxα

2-simplex 1-simplex

TUK,σKσ UT thenIf

VUVUV K thenIf ,,U

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.

Alpha Complex

D(S),T each For

ball. this ofcenter the is boundary its on are T of points the that

such radius smalest the has b ballThe

T

TTT

y

),(y.

conflict. has else iff free conflict is

T ,s),b(y TTT

Alpha-shape of a set of points (11)

Alpha Complex

:that such σ simplices allby formed ofcomplex -sub a is S ofcomplex -alpha The

T D(S)

S. ofcomplex - and of face ais(b)or free, conflit is and (a)

UU

T

TTT ),b(y

Alpha-shape of a set of points (12)

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

Alpha-shape of a set of points (14)Curve reconstruction: definitionThe 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.

A set of points S The reconstructed surface

Alpha-shape of a set of points (15)Surface reconstruction

Curve reconstruction : theorem

If points. of set finite a is andboundary, withoutmanifold1- compact a beLet

CSRC 2

; int(I) )int( that such I, ball1- closed a tomorphic

-homeo (c) p; point single a (b) empty; (a) : either is

, radius of disk closedany For 1.

CbbC

Rb

ρ

2

S, of point one least at contains , on centered radius of ball open An2. C

qpC ,S

CqSpαα

minmax )D( and C tophic

-homeomor is , S, of ,S shape, the then 2

Alpha-shape of a set of points (16)

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.

points. weightedof set finite aLet RRS d

p"-x"x'p'x)(p, 2

For two weighted points, (p’, p ’’) and x=(x’,x’’), we define

Weighted alpha shape (1)

S of shape- weightedtheofsimplexais -kT

that so ),(x' xpoint weighteda exists there iff

T-Sp all forand T,p all for

00

x)(p,

p’x’

p"

0x)(p,

p"

Weighted alpha shape (2)

Weighted alpha shape (3)

0x),(p1,2

0x),(p5

),(x'x

shape-Euclidean ]p[p 21

Weighted alpha shape (4)

0x),(p1,2

0x),(p5

),(x'x

shape-Euclidean ]p[p 21

Weighted alpha shape (5)

][,0),(max 211 vvxxp max

][,0),(min 211 vvxxp min

The weighted alpha shape is a sub-graph of the regular triangulation.

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 eIf (min(e)<=<=max(e)) then e is in the weighted -

shape of S.

Weighted alpha-shape (6)

Gabriel Graph: definition (1)

.et jijiji ppSpb(pSppb ,))(

Gabriel the ofsimplex 1- a is ][ edge An ji pp

iff S of graph

.)( jiji pp ppb diameter of ball a being

Gabriel Graph: example (2)

An edge of Gabriel

This edge is not in the GG

Gabriel Graph: properties (3)

222

]

kjkiji

k

ji

pppppp

:p all for iff S of G G

the to belongs p[p edgeDelaunay A 2)

1) The Gabriel graph of S is a sub graph of the Delaunay triangulation of S.

Gabriel Graph: example (4)

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)

Beta skeleton (1)two of union the is andof 1, ji p p

jiji pppp2

radii of and and through passing balls

iff S of skeletonthe of edge an is ][ - pp ji

contain not does ,p and p of ji

-neighborhood,

neighborhood,

S. of pointany

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

Beta skeleton (2)

Examples of -neighborhood :Forbidden regions

A beta-skeleton edge

(3)Beta skeleton

Beta skeleton (4)

beta = 1.1 beta = 1.4

Beta skeleton : algorithm (5)

.2121 pp to dual edge Voronoithe be vv Let

ball the of center a bev tt)v(1- c(t) Let 21

.2

, 2121 pp radius of andpp points the through passing

The coordinates of these centers are:

)vv,v(pvv

ppcosv2pt

2111

21

21111,2

2

12

.1 1,221 t0 iff S of skeleton- of edge an is pp

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 discis an approximationof 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.

C. of axis medial the of point nearest the to p of distance the , , call We Cpf(p)

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

. that such point a exists there ,

f(p)psSsCp

Reconstruction : -sampling condition(1)

Reconstruction : -sampling condition(2)

Reconstruction : -skeleton (3)

Let S -sample a smooth curve, with <0.297. The -skeleton of S contains exactly the edges between adjacent verticeson the curve, for = 1.70.

A-shape and Crust (1)

of 2,0 simplex,- a is kkT

b ball a exists there iff S of shape-A

points. of set finite a beingA and A,of

point a and T of points the through passing.Sb

A).DV(S in A of pointa least at to neighbors are T of points the iffS of shape- Aof 2,0 simplex,- a is

kkT

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

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)

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

Crust : reconstruction (7)

Local Crust : definition and properties (1)

iff S of crust Local of edge an is ][pp'

)v'p'b(pv et v)p'b(pv'

v v’ is the dual Voronoi edge of pp’, b(p p’ v) is the ball which circumscribes the points p, p’,v.

).v' v,D(S of edge an is ][pp'iff S of crust Local of edge an is ][pp'

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)

S)v'b(v

[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

Local Crust and Gabriel Graph (5)

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

Local Crust : reconstruction (6)

Local Crust and Gabriel Graph (7)

Local crust

Voronoi Gabriel Graph

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.

. 1/3,

-

E e ifonly and if e edge an outputs Crust-NN algorithm the

withcurve closed a for S samplean Given

3D reconstruction: an example