THEORETICAL ADVANCES Fabien Feschet Canonical representations of discrete curves Received: 9 June 2004 / Accepted: 4 February 2005 / Published online: 10 June 2005 Ó Springer-Verlag London Limited 2005 Abstract A new representation of digital curves is introduced. It has the property of being unique and canonical when computed on closed curves. The repre- sentation is based on the discrete notion of tangents and is complete in the sense that it contains all discrete seg- ments and all polygonalizations which can be con- structed with connected subsets of the original curve. This representation is extended for dealing with noisy curves and we also propose a multi-scale extension. An application is given to curve decomposition into con- cave–convex parts and with application in syntactical based methods. 1 Introduction Digital objects, curves, surfaces are commonly acquired by physical sensors. Shapes can be obtained as the outer boundary of the digital acquisition of a planar object and are thus usually described by a discrete curve. Two problems arise when considering the digi- tization of a real objects. One concerns the problem of characterizing digitizations which preserve topological or differential properties of real objects [15, 18] and another concerns the manipulation of the discrete data using discrete methods [20, 21]. The first problem is a very challenging one but is not concerned by the pres- ent study; that is, we do not wonder how the discrete data were acquired and we only focus on their repre- sentation and manipulation. We thus suppose that we have a list of points of Z 2 : It is obviously the case when considering shapes. Describing discrete curves by their list of points is a common approach but lacks geometrical information. It is a straightforward and simple idea to try to use discrete lines and segments to represent discrete curves with less geometrical primi- tives [23]. One of the standard approaches is the rep- resentation of discrete curves by discrete segments connected by their endpoints leading to the notion of polygonalization [25]. However, this representation is extremely sensitive to the starting point of the compu- tation. This is not a drawback for open curves since it is sufficient to take one of the extremities of the curve as a starting point. However, this behavior becomes an important drawback when considering closed curves, since for closed curves no point can be privileged. The main goal of this paper is to introduce a new repre- sentation, called the tangential cover, of closed or open discrete curves and to show that this representation has the property of being canonical meaning that no starting point is privileged. This decomposition might be used in shape matching or shape classification but we do not study its application to those problems in the present paper. Our goal is to propose a new way of representing discrete curves, on which new methods for shape analysis will be presented in future papers. As will be explained later, the tangential cover has two interpretations: one is global and concerns the geometrical structure of discrete segments built from connected portions of the curve, while the second is local through the notion of discrete tangents. The most interesting property of the tangential cover is its com- pleteness meaning that it contains all possible discrete segments. As a consequence, this decomposition con- tains all the polygonalizations of the curve meaning that if curves are described by our representation, classical methods remain valid with a simple linear time preprocessing. An application of this representation is given for a decomposition of curves into homogeneous parts-in a sense formally described further on in the paper. We also provide ways to extend the decompo- sition into a multi-scale representation to deal with noisy curves. We end this paper with some conclusions and perspectives after introducing the Farey arcs as F. Feschet LLAIC1 - IUT Clermont-Ferrand 1 - Campus des Ce´zeaux, BP 86 - 63172, Aubie`re, France E-mail: feschet@llaic3.u-clermont1.fr Pattern Anal Applic (2005) 8: 84–94 DOI 10.1007/s10044-005-0246-5