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)

Crystalline Flow

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

Crystalline Flow

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.

Facet Extraction

Facet Extraction

Facet Extraction

Conclusions

•Crystalline for Contour Shape Analysis•Wulff Shape Selection!