A new representation of polygonal curves based on piecewise-linear functions with potential application in shape identification

Abstract

In this paper, a new representation of polygonal curves is proposed. The motivation behind this proposal is to find a descriptive model with potential application in shape identification. In particular, our work addresses the problem of identifying the shape of a given polygonal curve from a set of different ones. In accordance with the proposed representation, a curve is originally described as a series of consecutive points whose connection by straight lines sketches its characteristic shape. In order to define a proper identification scheme, each series of points is mapped to a one-dimensional piecewise-linear function that assigns to each point its corresponding angle. Depending on the case under study, this assignation is performed by following two possible alternatives: (i) in the case where the shape to be identified exactly corresponds with one of the previously stored ones, a sorted sequence beginning from the minimum value of angle and ending with the maximum one is considered, and (ii) in the case where the shape to be identified is similar to one of the previously stored ones, the assignation is performed by following the sequence of points as they appear in the polygonal curve (without sorting). Under this scheme of representation, by a cyclic comparative process between the computed functions, after several steps it is decided whether the graph of any input piecewise-linear function matches the polygonal curve to be recognized within a certain tolerance. In particular, in the case of the identification of two equal shapes, this proposal exploits the well-known principle of similarity geometry, which allows a polygon to be recognized independently of scale, translation and rotation. In order to validate this representation, a comparative analysis between two different shape identification methods (Fourier descriptors and canonical representation) and the piecewise-linear proposal is performed.