© 2002 by Davi GeigerComputer Vision November 2002 L1.1 Symmetry Axis Stretching OcclusionArticulation Stretching Symmetry Axis

© 2002 by Davi GeigerComputer Vision November 2002 L1.2 Shape Axis (SA) SA-Tree Shape

© 2002 by Davi GeigerComputer Vision November 2002 L1.3  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-Similarity work of Liu, Kohn and Geiger

© 2002 by Davi GeigerComputer Vision November 2002 L1.4 Use two different parameterizations to compute the represention of a shape. Use two different parameterizations to compute the represention of a shape. Construct a cost functional to measure the goodness of a match between the two parameterizations. Construct a cost functional to measure the goodness of a match between the two parameterizations. The cost functional is decided by the self-similarity criteria. The cost functional is decided by the self-similarity criteria. Useful self-similarity criteria include symmetry, parallelism (translation), convexity and distance. Useful self-similarity criteria include symmetry, parallelism (translation), convexity and distance. The Insight counterclockwiseclockwise

© 2002 by Davi GeigerComputer Vision November 2002 L1.5 Two different parameterizations: Two different parameterizations: counterclockwise counterclockwise clockwise clockwise with with When the curve is closed we have When the curve is closed we have Parameterized Shapes

© 2002 by Davi GeigerComputer Vision November 2002 L1.6 Cost Functional/Energy Density Co-Circularity

© 2002 by Davi GeigerComputer Vision November 2002 L1.7 Structural properties Structural properties o Symmetric Geometrical properties Geometrical properties o Translation invariant o Rotation invariant Self-Similarity properties Self-Similarity properties Cost Functional/Energy Density

© 2002 by Davi GeigerComputer Vision November 2002 L1.8 A Dynamic Programming Solution

© 2002 by Davi GeigerComputer Vision November 2002 L1.9 A Dynamic Programming Solution

© 2002 by Davi GeigerComputer Vision November 2002 L1.10 Experimental Results for Closed Shapes Shape Axis SA-Tree Shape

© 2002 by Davi GeigerComputer Vision November 2002 L1.11 Experimental Results for Open Shapes Shape Axis (SA) SA-Tree Shape

© 2002 by Davi GeigerComputer Vision November 2002 L1.12 Shape Axis (SA) SA-Forest Experimental Results for Open Shapes

© 2002 by Davi GeigerComputer Vision November 2002 L1.13 Matching Trees with deletions and merges for articulation and occlusions

© 2002 by Davi GeigerComputer Vision November 2002 L1.14 Convexity v.s. Symmetry White Convex RegionBlack Convex Region