Reconstruction from Medial Representations

Abstract

In the analysis of planar shapes there are a number of ways of reducing the shape to a one-dimensional “skeleton”: the medial axis, the symmetry set, and the “smoothed local symmetry,” which we call here the midpoint locus. All depend on circles which are twice-tangent to the boundary curve of the shape. In addition there is a “pre-symmetry set” which underlies all these constructions. We describe how a shape can be reconstructed from its medial axis or symmetry set, that is, the centers of the twice-tangent circles, and a knowledge of their radii. Then we ask the question: can the shape be reconstructed similarly from its midpoint locus—this amounts to giving the midpoints of chords of twice-tangent circles instead of their centers—and the radii? The first answer is “yes, given an initial condition,” but further analysis shows that this is so sensitive to that initial condition as to make the reconstruction difficult. We also suggest other avenues of investigation.