Computer Vision and Image Understanding 143 (2016) 92–103 Contents lists available at ScienceDirect Computer Vision and Image Understanding journal homepage: www.elsevier.com/locate/cviu Oriented boundary graph: An efficient structuring model for segmentation of 3D images Fabien Baldacci ∗ , Achille Braquelaire University of Bordeaux, LaBRI, UMR CNRS 5800, 351 cours de la Libration, 33405 Talence, France article info Article history: Received 15 October 2014 Accepted 2 October 2015 Keywords: 3D segmentation Topological structuring Split and merge abstract From a theoretical point of view, most of image segmentation methods that have been developed for 2D im- ages can be generalized to higher dimensions. In actual practice, the cost in space to encode 3D data structure and the cost in time to run 3D algorithms does not allow to conveniently implement those classical segmen- tation algorithms in the 3D case. In this article, we describe a new model to efficiently represent and update both the topological and the geometrical structure of the regions of a 3D segmented image. This model has been defined from a pragmatical approach that consists in specifying a basic region-based segmentation framework, and then in building a minimal model that encodes all the relationships needed for an efficient implementation of this framework. This approach leads to a model suitable for a wide range of segmenta- tion methods and allowing an efficient computation of most of the segmentation criteria involved in image segmentation. © 2016 Elsevier Inc. All rights reserved. 1. Introduction The segmentation process aims to define a partition of the image into relevant regions according to given criteria. Different approaches exist to achieve this segmentation like split and merge methods [1–5], which are based on regions, active contour methods [6–9] based on region boundaries, and Markov methods [10,11] using a probabilistic approach. Split and merge methods consists in alterna- tively splitting and merging the regions of the partition. Since those segmentation methods update the partition many times, it is neces- sary to efficiently encode the partition and the main primitives. Such methods are currently widely used to solve 2D segmentation prob- lems [12–14], but their extension in 3D suffers from the lack of an efficient 3D partition representation model. Several topological models have been proposed in 2D. One pop- ular model is the model of Region Adjacency Graph (RAG) [15] which is efficiently built, but does not capture the whole topology of the segmented image. Other popular and more sophisticated models are based on combinatorial maps [16,17]. Those models are based on both a geometrical and topological representation of an image parti- tion and could efficiently be updated considering partition modifica- tions. They provide a general image analysis framework [18] on which a general segmentation platform has been developed 1 allowing to ∗ Corresponding author. E-mail addresses: fabien.baldacci@labri.fr (F. Baldacci), achille@labri.fr (A. Braquelaire). 1 http://girl.labri.fr/. design complex region based segmentation algorithms [19]. Combi- natorial maps can be defined in 3D [20] and two models using 3D combinatorial maps have been proposed: the Hierarchical Local Em- bedding (HLE) [21,22] and the Geometrical Embedding (GE) [23]. Com- binatorial maps capture the whole topology of the partition but re- quire more computation time than the RAG, and these two models have some lacks according to the implementation of a basic segmen- tation framework. The lack of efficiency is due to the 3D combinato- rial map which cannot easily be built from an image partition descrip- tion. A combinatorial map requires extra information to be added and maintained because the boundary of a face has to be connected, and because some elements are duplicated. The resulting structures are not optimized according to the geometrical elements provided by the objects obtained from a segmentation. We have extended the 2D framework [18] to the 3D case and defined a new topological struc- turing model designed for this framework that overcomes the draw- backs of the previous models. This new model, based on the study of advantages and drawbacks of existing ones, aims to have an opti- mal time complexity to implements the usual functions composing an image analysis framework. After presenting the main definition and notation in use in this paper in Section 2, we define the specification of the functionalities required by a segmentation framework in Section 3. Then the exist- ing topological structuring models are presented and discussed in Section 4. Our new model is defined in Section 5, its implementa- tion is detailed and analysed in Section 6, and the implementation of the segmentation framework is given in Section 7. Section 8 presents examples of image analysis and segmentation based on our model. http://dx.doi.org/10.1016/j.cviu.2015.10.003 1077-3142/© 2016 Elsevier Inc. All rights reserved.