Symmetry-curvature duality

Several studies have shown the importance of two very different descriptors for shape: symmetry structure and curvature extrema. The main theorem proved by this paper, i.e. the Symmetry-Curvature Duality Theorem, states that there is an important relationship between symmetry and curvature extrema: If we say that curvature extrema are of two opposite types, either maxima or minima, then the theorem states: Any segment of a smooth planar curve, bounded by two consecutive curvature extrema of the same type, has a unique symmetry axis, and the axis terminates at the curvature extremum of the opposite type. The theorem is initially proved using Brady’s SLS as the symmetry analysis. However, the theorem is then generalized for any differential symmetry analysis. In order to prove the theorem, a number of results are established concerning the symmetry structure of Hoffman’s and Richards’ codons. All results are obtained first by observing that any codon is a string of two, three, or four spirals, and then by reducing the theory of codons to that of spirals. We show that the SLS of a codon is either (1) an SAT, which is a more restricted symmetry analysis that was introduced by Blum, or (2) an ESAT, which is a symmetry analysis that is introduced in the present paper and is dual to Blum’s SAT. c © 1987 Academic Press, Inc. Perceptual studies in both psychology and computer vision have revealed the importance of two very different structural characteristics for shape: (1) Symmetry. Since the German gestalt school first showed that symmetry is a crucial organizing principle of shape (Wertheimer [24]; Goldmeier [8]), many studies have corroborated and extended their results. For example, Psotka [18] presented subjects with figures, such as the outline of a man, and asked the subjects to place a dot at the first place that came to mind. Pooling the results, he found that the dots were distributed along the local symmetry axes. Leyton [12, 13, 15, 16] gave subjects a set of twenty-two complex abstract and natural shapes; e.g. of animals, birds and plants, and asked them to place lines along the directions of greatest shape flexibility. The subjects consistently chose local symmetry axes. Richards & Kaufman [21] placed cutouts of figures against a TV screen exhibiting random "snow" and found that subjects saw