Alternative 2D Shape Representations using the Symmetry Set

Abstract

Abstract. Among the many attempts made to represent families of 2D shapes in a simpler way, the Medial Axis (MA) takes a prominent place. Its graphical representation is intuitively appealing and can be computed efficiently. Small perturbations of the shape can have large impact on the MA and are regarded as instabilities, although these changes are mathematically known from the investigations on a super set, the Symmetry Set (SS). This set has mainly been in a mathematical research stage, partially due to computational aspects, and partially due to its unattractive representation in the plane.In this paper novel methods are introduced to overcome both aspects. As a result, it is possible to represent the SS as a string is presented. The advantage of such a structure is that it allows fast and simple query algorithms for comparisons.Second, alternative ways to visualize the SS are presented. They use the distances from the shape to the set as extra dimension as well as the so-called pre-Symmetry Set and antiSymmetry Set. Information revealed by these representations can be used to calculate the linear string representation structure.Example shapes from a data base are shown and their data structures derived.

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Contour Grouping Based on Contour-Skeleton Duality

“…Shapes of objects and their symmetries have several dual geometric properties (Giblin and Kimia 2003b;Kuijper and Olsen 2004). Shapes can be represented using signatures defined using locally symmetric curves (Kuijper et al 2006b) and shapes can be reconstructed from symmetries (Giblin and Kimia 2003a). Symmetry for rigid and non-rigid objects can be captured in several ways: for example using ribbons (Mohan and Nevatia 1992), using planar reflections (Podolak et al 2006), using intrinsic self-similarities (Raviv et al 2007) or more classically using medial axes (Blum 1967(Blum , 1973 and symmetry sets (Bruce et al 1985;Giblin 2000).…”