Contour Shape Analysis Using Crystalline Flow

1
Contour Shape Analysis Using Crystalline Flow

• 2001.Nov.7

2
Crystalline Flow
Evolution of a polygon
3
Crystalline Flow
Outward Normal Velocity depends on Nonlocal
Weighted Curvature
4
Curvature
Inscribing Circle
Wulff Shape
5
Wulff Shape
A Convex m-Polygon
6
Wulff Shape
A Set of Unit Vectors
7
Admissible Crystal
Wulff Shape
8
Crystalline Flow
9
Crystalline Flow
Facet Disappearing at t T
Case A

• The polygon becomes convex near T
• and all facet disappear at T.

All facets disappear at t T.
10
Crystalline Flow
Facet Disappearing at t T
Case B

• Two parallel facets meet together.

11
Crystalline Flow
Case B does not occur if
is locally Lipschitz on
is nondecreasing on
for all
12
Crystalline Flow
Facet Disappearing at t T
Case C

• At most two consecutive facets disappear.

13
Crystalline Flow
2
10
2
2
14
Crystalline Flow
15
Chain Coded Contour
6
0
0
0
0
7
5
1
4
0
2
1
3
2
2
4
4
4
4
4
2
0
0
0
2
7
16
Crystalline Flow
17
Scale Space Analysis
White Convex Black Concave
Time
Facet Number in Original
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Facet Extraction
Trace concave(convex) facets back to t0.
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Facet Extraction
As the extraction scale increases, more important
facets are extracted.
20
Facet Extraction
21
Facet Extraction
22
Facet Extraction
23
Conclusions

• Crystalline for Contour Shape Analysis
• Wulff Shape Selection!