A simplified fold number detector for shapes with monotonic radii

-This paper concentrates on the fold number detection problem for the shapes with monotonic radii. The proposed method is extremely simple. Two monotonicity conditions are derived to ensure that the smallest positive integer I making SStr.o~sr2elt°dr dO nonzero is exactly the fold number of the given shape S. The fold numbers of regular polygons, roses, bolt nuts, and other kinds of shapes discussed in the present paper, can therefore be detected quite easily. Note especially that the proposed method uses no matching procedure, a procedure essential in many reported methods. Theoretical properties, mathematical proofs, illustrative figures, and experimental results, are all included in this paper. Copyright ,~) 1996 Pattern Recognition Society. Published by Elsevier Science Ltd. Rotational symmetry Mirror symmetry Monotonicity Rotation-matching Regular polygons Bolt nuts Roses 1. I N T R O D U C T I O N An important shape feature, which is widely used in shape recovery, Ix) shape registration, 12'a) shape storage space reduction, 141 etc., is the so-called symmetry property. At least two kinds of symmetry can be identified, namely, mirror symmetry 15'61 and rotation symmetry. Ivl To make use of the shape symmetry property, it is necessary to compute either the number of symmetry axes or the number of folds of a given shape. Several methods ~5'6's) have been proposed to find the number of the symmetry axes of a given mirror symmetric shape (i.e., symmetric about certain lines called symmetry axes). On the other hand, the number of folds n of a given rotationally symmetric shape can be detected by the methods described in References (1), (2) and (5). Highnam ~s~ used a string matching technique to detect n, whereas Leou and Tsai ~21 counted as n the frequency that the boundary of the given rotationally symmetric shape runs across a special circle related to the given shape. Recently, Lin et al. proposed a simpler method in Reference (1) to detect n based on a simple mathematical property of rotationally symmetric shapes. Note that these methods require the use of certain matching techniques: string matching is used in Reference (5), whereas shape matching is used in both References (1) and (2). L ine t al. had pointed out in Reference (1) that their method has an extra benefit that once the fold number is obtained, the shape orientation can also be obtained immediately by a time-negligible step: a simple call of the arc-tangent function followed by an arithmetic division. We therefore concentrate the study on the method of Lin et al., and try to discover some more mathematical properties of the method. More precisely, we will investigate what will happen if the method is applied to a special branch of the rotationally symmetric shapes, namely, the rotationally symmetric shapes of which the polar radii are monotonic in a fold (or in a half-fold if the fold itself is also mirror-symmetric). This special branch includes regular polygons, bolt nuts, roses and so on. It will be proved that many procedures used by Lin et al., including the matching procedure, can be discarded. As a result, if the image processing environment (also called the processing domain) is limited to the collection of all regular polygons, bolt nuts, roses and so on, then their method, which is already very attractive due to its certain properties [see the final section of Reference (1)], can be simplified further. The remainder of the paper is organized as follows: in Section 2, the definitions of rotationally symmetric shapes and polar radii function are introduced, and the rotation matching procedure of Linet aL is reviewed. The theory needed to simplify the method of Lin et al. is introduced in Sections 3 and 4. Experimental results are provided in Section 5, and a summary and some concluding discussion are given in Section 6. Most of the proofs are collected in an appendix to make the paper easier to read. 2~ D E F I N I T I O N S AND LIN et aL's R O T A T I O N M A T C H I N G