© 2003 by Davi GeigerComputer Vision November 2003 L1.1 Shape Representation and Similarity OcclusionArticulation Shape similarities should be preserved under occlusions and articulation. Two equal stretching along contours can affect shape similarity differently. Shapes can be represented by contours, or by a set of interior points, or other ways. What is the shape representation that can be best used for shape similarity? Stretching

© 2003 by Davi GeigerComputer Vision November 2003 L1.2 Shape Axis (SA) SA-Tree Shape Symmetry Axis Representation

© 2003 by Davi GeigerComputer Vision November 2003 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 Based on work of Liu, Kohn and Geiger

© 2003 by Davi GeigerComputer Vision November 2003 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 of symmetry. The cost functional is decided by the self-similarity criteria of symmetry. The Insight counterclockwise clockwise

© 2003 by Davi GeigerComputer Vision November 2003 L1.5 Two different parameterizations: Two different parameterizations: counterclockwise counterclockwise clockwise clockwise When the curve is closed we have When the curve is closed we have Matching the curves is described as the match of functions Matching the curves is described as the match of functions Parameterized Shapes

© 2003 by Davi GeigerComputer Vision November 2003 L1.6 A Global Optimization Approach We seek “good” matches over the possible correspondences We seek “good” matches over the possible correspondences

© 2003 by Davi GeigerComputer Vision November 2003 L1.7 Co-Circularity Mirror Symmetry Similarity criteria: Symmetry

© 2003 by Davi GeigerComputer Vision November 2003 L1.8 A Global Optimization Approach

© 2003 by Davi GeigerComputer Vision November 2003 L1.9 Constraints on the form of F

© 2003 by Davi GeigerComputer Vision November 2003 L1.10 Constraints on the form of F (cont.)

© 2003 by Davi GeigerComputer Vision November 2003 L1.11 Similarity criteria: Symmetry

© 2003 by Davi GeigerComputer Vision November 2003 L1.12 Similarity criteria: Symmetry II (a) (b) (c) (d)

© 2003 by Davi GeigerComputer Vision November 2003 L1.13 Structural properties Structural properties o Symmetric o Parameterization Independent Geometrical properties Geometrical properties o Translation invariant o Rotation invariant o Scale “almost” invariance Self-Similarity properties Self-Similarity properties Summary: Cost Functional/Energy Density

© 2003 by Davi GeigerComputer Vision November 2003 L1.14 A Dynamic Programming Solution

© 2003 by Davi GeigerComputer Vision November 2003 L1.15 A Dynamic Programming Solution The recurrence is on the “square boxes”. Each box is either linked to one inside square box or linked to a pair of inside and well aligned square boxes (leading to bifurcations). Assumption, that will not be needed later: The final state is a match (0,0). … ? ! ! ! ! ! ! bifurcation t s 1-s 1-t tt 1-tt s+1/N t+1/N ! ! ! ! ! !

© 2003 by Davi GeigerComputer Vision November 2003 L1.16 t 1-s tt t+1/N … ? ! ! ! ! !! ! s 1-t 1-tt s+1/N A Dynamic Programming Solution s t

© 2003 by Davi GeigerComputer Vision November 2003 L1.17 A Dynamic Programming Solution A few more steps … iteration direction s t Starting states (matching candidates). All corresponding to “self-matches”

© 2003 by Davi GeigerComputer Vision November 2003 L1.18 A General Dynamic Programming Solution No need to force a match at (0,0) or any particular node/match. s t Extended Graph Starting states (matching candidates). All corresponding to “self-matches” Ending states Extend the parameters s and t such that negative values x  s,t=1+x. x x

© 2003 by Davi GeigerComputer Vision November 2003 L1.19 A Dynamic Programming Solution

© 2003 by Davi GeigerComputer Vision November 2003 L1.20 Experimental Results for Closed Shapes Shape Axis SA-Tree Shape

© 2003 by Davi GeigerComputer Vision November 2003 L1.21 Experimental Results for Open Shapes (the first and last points are assumed to be a match) Shape Axis (SA) SA-Tree Shape

© 2003 by Davi GeigerComputer Vision November 2003 L1.22 Shape Axis (SA) SA-Forest Experimental Results for Open Shapes

© 2003 by Davi GeigerComputer Vision November 2003 L1.23 Matching Trees with deletions and merges for articulation and occlusions

© 2003 by Davi GeigerComputer Vision November 2003 L1.24 Experimental Results for Open Shapes

© 2003 by Davi GeigerComputer Vision November 2003 L1.25 Convexity v.s. Symmetry White Convex RegionBlack Convex Region