contour shape analysis using crystalline flow 2001.nov.7
TRANSCRIPT
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
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.