A method for representing 3D tree objects using chain coding Ernesto Bribiesca * Departmento de Ciencias de la Computacio ´ n, Instituto de Investigaciones en Matema ´ ticas Aplicadas y en Sistemas, Universidad Nacional Auto ´noma de Me ´xico, Apdo. 20-726, Mexico, DF 01000, Mexico Received 23 March 2007; accepted 4 January 2008 Available online 15 January 2008 Abstract We describe a method for representing 3D (three-dimensional) tree objects by means of a chain code. These 3D tree objects correspond to natural existing 3D tree structures, such as: blood vessels, plants, live trees, and so on. Thus, trees are digitalized and represented by a notation called the unique tree descriptor. The unique tree descriptor is invariant under translation and rotation. Furthermore, this descriptor is starting vertex normalized via the unique path in the tree. Also, it is possible to obtain the mirror image of any tree with ease. This unique tree descriptor preserves the shape of trees (and the shape of their branches), allows us to know their geometrical and topological properties. To determine if two 3D tree objects have the same shape, it is only necessary to see if their descriptors are equal. In this manner, graph comparisons and tree searches are eliminated. Also, the proposed tree descriptor is a good tool for storing of 3D tree objects. Finally, in order to prove our method for representing 3D tree objects, we obtain some tree descriptors of objects on real images. Ó 2008 Published by Elsevier Inc. Keywords: 3D tree objects; 3D tree structures; Unique tree descriptor; 3D discrete branches; Chain coding; 3D tree representation 1. Introduction Representation and comparison of the shape of natural existing 3D tree structures (such as: vascular system, plants, natural trees, and so on) are important topics in image representation, computer vision, and pattern recog- nition. Tree structures cover a wide variety of applications [1]. This paper deals with 3D tree representation by means of a chain code. Thus, 3D tree objects are digitalized and represented by constant orthogonal straight-line segments. These 3D tree objects are described by means of a chain of base-five digit strings suitably combined by means of parentheses. The proposed notation represents trees of maximum degree six in three dimensions. This is due to the fact that the above-mentioned trees are only repre- sented using orthogonal straight-line segments of the same length. Closson et al. [2] have represented graphs in a 3D orthogonal drawing in a fully dynamic form, they describe a 3D orthogonal drawings strategy for graphs containing no vertex of degree seven or higher. Di Battista et al. [3] presented a method for constructing orthogonal drawings of graphs of maximum degree six in three dimensions. The method is based on generating the final drawing through a sequence of steps. In graph theory there is no unique way of drawing a graph; the relative positions of points representing vertices and lines representing edges have no significance [1]. How- ever, in the proposed method for representing 3D tree objects, the geometry and topology of 3D tree structures should be preserved. In order to generate the unique 3D tree descriptor, we use chain coding for representing the geometry of the tree. Chain codes are the standard input format for numerous shape analysis algorithms [4]. Chain-code methods are widely used because they preserve information and allow considerable data reduction. The first approach for representing 3D digital curves using chain code was introduced by Freeman in 1974 [5]. Free- man chains are not invariant under rotation, this is due to the fact that absolute directions are used. Guzma ´n 1047-3203/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.jvcir.2008.01.001 * Corresponding author. Fax: +52 555622 3620. E-mail address: ernesto@leibniz.iimas.unam.mx www.elsevier.com/locate/jvci Available online at www.sciencedirect.com J. Vis. Commun. Image R. 19 (2008) 184–198