Computer Vision November 2002 L1.1© 2002 by Davi Geiger

Symmetry Axis

Stretching

Occlusion Articulation

Stretching

Symmetry Axis

Computer Vision November 2002 L1.2© 2002 by Davi Geiger

Shape Axis (SA)

SA-Tree

Shape

Computer Vision November 2002 L1.3© 2002 by Davi Geiger

A variational shape representation model based

on self-similarity of shapes.

For each shape contour, first compute its shape axis then derive a unique shape-axis-tree (SA-tree) or shape-axis-forest (SA-forest) representation.

Shape Axis (SA) SA-TreeShape Contour

Shape Representation via Self-Similaritywork of Liu, Kohn and Geiger

Computer Vision November 2002 L1.4© 2002 by Davi Geiger

Use Use twotwo different parameterizations to compute the different parameterizations to compute the represention of a shape.represention of a shape.

Construct a cost functional to measure the Construct a cost functional to measure the goodnessgoodness of a match between the two parameterizations.of a match between the two parameterizations.

The cost functional is decided by the self-similarity The cost functional is decided by the self-similarity criteria.criteria.

Useful self-similarity criteria include Useful self-similarity criteria include symmetrysymmetry, , parallelismparallelism (translation), (translation), convexityconvexity and and distancedistance..

The Insight

1),(|2

)(~)(ts

txsx counterclockwise clockwise

Computer Vision November 2002 L1.5© 2002 by Davi Geiger

Two different parameterizations:Two different parameterizations:

counterclockwisecounterclockwise

clockwiseclockwise

withwith

When the curve is closed we haveWhen the curve is closed we have

Parameterized Shapes

}10:)({1 ssx

}10:)(~{2 ttx

)1()(~ txtx

(1)x~(0)x~ and )1()0( xx

Computer Vision November 2002 L1.6© 2002 by Davi Geiger

Cost Functional/Energy Density

dF

stE

JumpCost )()similarity-self(

))(),((1

0

Co-Circularity

Computer Vision November 2002 L1.7© 2002 by Davi Geiger

Structural propertiesStructural propertieso SymmetricSymmetric

Geometrical propertiesGeometrical propertieso Translation invariantTranslation invarianto Rotation invariantRotation invariant

Self-Similarity propertiesSelf-Similarity properties

Cost Functional/Energy Density

Computer Vision November 2002 L1.8© 2002 by Davi Geiger

A Dynamic Programming Solution

Computer Vision November 2002 L1.9© 2002 by Davi Geiger

A Dynamic Programming Solution

Computer Vision November 2002 L1.10© 2002 by Davi Geiger

Experimental Results for Closed Shapes

Shape Axis

SA-Tree

Shape

Computer Vision November 2002 L1.11© 2002 by Davi Geiger

Experimental Results for Open Shapes

Shape Axis (SA)

SA-Tree

Shape

Computer Vision November 2002 L1.12© 2002 by Davi Geiger

Shape Axis (SA)

SA-Forest

Experimental Results for Open Shapes

Computer Vision November 2002 L1.13© 2002 by Davi Geiger

Matching Trees with deletions and merges for articulation and occlusions

Computer Vision November 2002 L1.14© 2002 by Davi Geiger

Convexity v.s. Symmetry

White Convex Region Black Convex Region