Proximity graphs: reconstruction of curves and surfaces
description
Transcript of Proximity graphs: reconstruction of curves and surfaces
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