Affine invariant medial axis and skew symmetry

Affine invariant medial axes and symmetry sets of planar shapes are introduced and studied in this paper. Two different approaches are presented. The first one is based on affine invariant distances, and defines the symmetry set, a set containing the medial axis; as the closure of the locus of points on (at least) two affine normals an affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) three-point contact with two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set (medial axis) are equivalent, this is not the case for the affine group. We then show how to use the symmetry set to detect affine skew symmetry, proving that the contact based symmetry set is a straight line if and only if the given shape is the affine transformation of a symmetric object.

[1]  J. W. Bruce,et al.  Growth, motion and 1-parameter families of symmetry sets , 1986, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  Guillermo Sapiro,et al.  Affine-Invariant Distances, Envelopes and Symmetry Sets , 1998 .

[3]  P. Olver,et al.  Affine Geometry, Curve Flows, and Invariant Numerical Approximations , 1996 .

[4]  Tat-Jen Cham,et al.  Symmetry detection through local skewed symmetries , 1995, Image Vis. Comput..

[5]  Robert L. Ogniewicz,et al.  Discrete Voronoi skeletons , 1992 .

[6]  Peter Giblin,et al.  Perpendicular bisectors, duality and local symmetry of plane curves , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  Luc Van Gool,et al.  The Characterization and Detection of Skewed Symmetry , 1995, Comput. Vis. Image Underst..

[8]  G. Sapiro,et al.  On affine plane curve evolution , 1994 .

[9]  M. Leyton.  Symmetry, Causality, Mind , 1999 .

[10]  P. Giblin,et al.  Curves and Singularities , 1984 .

[11]  B. Julesz,et al.  Perceptual sensitivity maps within globally defined visual shapes , 1994, Nature.

[12]  Jean Ponce,et al.  On characterizing ribbons and finding skewed symmetries , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[13]  Takeo Kanade,et al.  Recovery of the Three-Dimensional Shape of an Object from a Single View , 1981, Artif. Intell..

[14]  M. Brady,et al.  Smoothed Local Symmetries and Their Implementation , 1984 .

[15]  Alfred M. Bruckstein,et al.  Skeletonization via Distance Maps and Level Sets , 1995, Comput. Vis. Image Underst..

[16]  P. Giblin,et al.  Curves and singularities : a geometrical introduction to singularity theory , 1992 .