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![Page 1: Proximity graphs: reconstruction of curves and surfaces Duality between the Voronoi diagram and the Delaunay triangulation. Power diagram. Alpha.](https://reader035.fdocuments.net/reader035/viewer/2022062322/56649cb85503460f9497e1ef/html5/thumbnails/1.jpg)
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
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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)
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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
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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)
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
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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
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Power diagram and regular triangulation (3)
1 21 2
1 21 2
R1 R2R1 R2
R1 R2R1 R2
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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 point
ijx),px),p(x;)R(p jii (
ip is defined by:
.)R(pi
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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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T
.Tp
R(p)
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)
.et TSbSb αα
of 3,0 simplex,- a is kkT
b ball a exists there iff S of shapeα :that such radius of 0
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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 = 10 alpha = 20
alpha = 40 alpha = 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)
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)
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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
p3v1
v2
minα
p1v2p1v1,maxmaxα
2-simplex1-simplex
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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.
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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)
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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)
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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
If points. of set finite a is and
boundary, withoutmanifold1- compact a beLet
CS
RC 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.
Cb
bC
Rb
ρ
2
S, of point one least at contains
, on centered radius of ball open An2. C
qpC ,S
CqSpα
α
minmax )D( and C tophic
-homeomor is , S, of ,S shape, the then 2
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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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 for
and T,p all for
0
0x)(p,
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p’
x’
p"
0x)(p,
p"
Weighted alpha shape (2)
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Weighted alpha shape (3)
0x),(p1,2
0x),(p5
),(x'x
shape-Euclidean ]p[p 21
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Weighted alpha shape (4)
0x),(p1,2
0x),(p5
),(x'x
shape-Euclidean ]p[p 21
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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.
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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)
.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
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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)
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.
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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)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.
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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 = 1.1 beta = 1.4
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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(p
vv
ppcosv2pt
2111
21
21111,2
2
12
.1 1,221 t0 iff S of skeleton- of edge an is pp
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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 discis an approximationof 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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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)ps
SsCp
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 <0.297. The -skeleton of S contains exactly the edges between adjacent verticeson the curve, for = 1.70.
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A-shape and Crust (1)
of 2,0 simplex,- a is kkTb 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 point
a least at to neighbors are T of points the iff
S of shape- Aof 2,0 simplex,- a is
kkT
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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)
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'
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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)
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
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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.
.
1/3,
-
E e ifonly and if e edge
an outputs Crust-NN algorithm the
withcurve closed a for S samplean Given
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3D reconstruction: an example