• We have recently presented  a novel framework based on wave propagation which combines
• analytic curve evolution
• bisectors of computational geometry approaches.
• On the one hand, curve evolution methods are attractive because computations are local. However,

these approaches cannot often operate on edge maps, are computationally expensive, and
most importantly mixes the detection and regularization stage due to the errors produced evolution process.
• On the other hand, computational geometry approaches  compute bisector curves of the curve segments representing edge map as the symmetry set. These approaches can produce exact symmetries since

computations are analytic. However, these approaches are global and often require  excessive
computations due to (1) detecting unnecessary symmetry branches and (2) removing them.
• The proposed framework  effectively combines these two approaches via explicit wavefront propagation

on a dual grid:
• Eulerian grid (fixed grid), which simulates discrete waves
• Lagrangian grid (dynamic grid) where corresponds to the bisectors of the curve segments

• Examples:
• three-lines
• simple shape
• more complex shape

• Main contributions:
• exact construction ofsymmetry representation of free-form curves
• low order complexity
• based on local operations
• tree-like graph representation of shapes without any post-processing
• applicable to curve segments with possible intersections with other curve segments,
• extensible to 3D
•  exact distance transforms.

Results: