Contour Shape Analysis Using Crystalline Flow
2001.Nov.7
Crystalline Flow
Evolution of a polygon
Crystalline Flow
Vi
Outward Normal Velocitydepends onNonlocal Weighted Curvature
Vi+1Vià 1
Curvature
ô = 1=r
Inscribing Circle
Wulff Shape
Wulff Shape
A Convex m-Polygon
Wulff Shape
n1n2
n3
nm
ni
A Set of Unit Vectors:
N = f nig
Admissible Crystal
•A simple Polygon•All outward normals belong to•The normal of adjacent facet is parallel to the normal adjacent in the Wulff Shape
N:
n1n2
n3
nm
ni
n1n2
n3
n2
Admissible
WulffShape
NotAdmissible n1
n3
Jump!
Crystalline Flow
Vi = g(ni;ÿiÉ (ni)=L i(t))
g(á;á) : Nondecreasing in 2nd variableÿi : Transition Number
-1 0 +1
É (ni) : Length of facet of Wulff Shape
É (n1)
L i(t)L i(t) : Length of i-th facet
É (n1)L i(t)
Nonlocal Curvature
Crystalline Flow
Facet Disappearing at t = T*
Case A:ÿj6=0•The polygon becomes convex near T* and all facet disappear at T*.
All facets disappear at t = T*.
Crystalline Flow
Facet Disappearing at t = T*
Case B:ÿj6=0•Two parallel facets meet together.
Crystalline Flow
õ7! g(ni;õ)
limõ! æ1 g(ni;õ) =æ1 ni 2 N
g2 Dg
RIæg(ni;õ)õ
à2dõ =æ1
is locally Lipschitz on R nf 0g
õ7! g(ni;õ) is nondecreasing on R
for all
Case B does not occur if
I+ = (1;1 ); I à = (à 1 ;0)
Crystalline Flow
Facet Disappearing at t = T*
Case C: ÿj = 0•At most two consecutive facets disappear.
Crystalline Flow
2
2
2
10
Vi = ÿiÉ (ni)=L i(t)
Chain Coded Contour
0
12
3
4
5
67
0 0 0 0
1
2
2
444442
2 70 0 0
33
1 0Make given polygon Admissible
Scale Space Analysis
Facet Number in Original
Time
White: ConvexBlack: Concave
Facet Extraction
Trace concave(convex) facets back to t=0.
Extraction Scale
Extracted Facets
Facet Extraction
As the extraction scale increases, more important facets are extracted.
Conclusions
•Crystalline for Contour Shape Analysis•Wulff Shape Selection!