Copyright (c) 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users by sending a request to pubs-permissions@ieee.org, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. Please, cite this article in Press: Castro-Mateos, Isaac et al. ‘Statistical Interspace Models (SIMs): Application to Robust 3D Spine Segmentation,” Medical Imaging, IEEE Transactions on, vol.34, no.8, pp.1663–1675, Aug. 2015 doi: 10.1109/TMI.2015.2443912. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7122305&isnumber=7172567. Statistical Interspace Models (SIMs): Application to Robust 3D Spine Segmentation Isaac Castro-Mateos, Student member, IEEE, Jose M. Pozo, Marco Perea˜ nez, Karim Lekadir, Aron Lazary, Alejandro F. Frangi, Fellow, IEEE Abstract—Statistical shape models (SSM) are used to introduce shape priors in the segmentation of medical images. However, such models require large training datasets in the case of multi- object structures, since it is required to obtain not only the individual shape variations but also the relative position and orientation among objects. A solution to overcome this limitation is to model each individual shape independently. However, this approach does not take into account the relative position, orientations and shapes among the parts of an articulated object, which may result in unrealistic geometries, such as with object overlaps. In this article, we propose a new Statistical Model, the Statistical Interspace Model (SIM), which provides information about the interaction of all the individual structures by modeling the interspace between them. The SIM is described using relative position vectors between pair of points that belong to different objects that are facing each other. These vectors are divided into their magnitude and direction, each of these groups modeled as independent manifolds. The SIM was included in a segmentation framework that contains an SSM per individual object. This framework was tested using three distinct types of datasets of CT images of the spine. Results show that the SIM completely eliminated the inter-process overlap while improving the segmentation accuracy. Index Term—Vertebral segmentation, statistical shape models, multi-object, inter-process overlap. I. I NTRODUCTION Statistical Shape Models (SSM) [1] are commonly em- ployed to extract the shape characteristics of a certain popu- lation. However, the direct use of such models to characterize multi-object or complex structures requires a large training dataset to properly represent new cases. The reason is that not only the shape of each individual substructure must be modeled, but also the relative positions and orientations among all of them. A possible solution is to use an individual SSM per object. However, this strategy ignores the inter- relationships between them, which can involve both shape and pose, specially for the neighbour structures. This problem may be solved by the use of conditional or hierarchical models [2], [3] that contain information about the shape inter- relationship between objects. However, these approaches do I. Castro-Mateos, J. Pozo, A.F. Frangi are with the Center for Computational Imaging and Simulation Technologies in Biomedicine (CISTIB), Electronic and Electrical Engineering Department, The University of Sheffield, Sheffield, United Kingdom (e-mail: isaac.casm@sheffield.ac.uk, j.pozo@sheffield.ac.uk, a.frangi@sheffield.ac.uk). M. Perea˜ nez and Karim Lekadir are with the Center for Computational Imaging and Simulation Technologies in Biomedicine (CISTIB), Universitat Pompeu Fabra, Barcelona, Spain. Aron Lazary is with the National Center for Spinal Disorders, Kiralyhago u. 1-3., Budapest, Hungary, H-1126 not specifically model the variability of the regions and relative positions between near-contact structures in neighbour objects. This is reflected by the appearance of: • Overlaps. • Excessive separation of neighbouring structures. • Unrealistic orientations between neighbouring objects. In this article, we introduce new statistical model type coined statistical interspace model (SIM), introduced in Sec- tion IV, embedded into a new segmentation framework, Sec- tion V. SIM directly addresses the neighboring relationship between regions in different objects (or even in the same) of a multi-object structure by learning the statistical distribution of the interspace between them. SIM completely avoids overlaps, and imposes soft constrains on the acceptable poses and shapes, solving the three aforementioned issues. This model may be applied in different applications con- cerning segmentation, registration, reconstruction, etc. For example, it may be employed to model the interspace between the two hemisphere of the brain, to restrain the separation between articulated bones, to control the relative position of the annulus and the nucleus of the Intervertebral discs (IVDs) or to maintain under control the separation between the two lungs to achieve realistic configurations, among possible applications. As proof of concept, we tested the proposed framework on vertebra segmentation, since they are complex bodies (Fig. 1) that form a multi-object structure when combined to constitute the vertebral column. We aim to not only provide high accurate segmentations but also to handle the problem of inter-process or vertebral body overlap (Fig. 2) as well as to avert unrealistic configurations. This issue is important for a range of appli- cation such as patient-specific biomechanical modeling [4], [5] or computer-assisted spine surgery [6]. For biomechanical models the correct definition of the processes and its correct geometry are essentials to perform realistic simulations; even high accurate segmentation will not be acceptable, if they have overlaps. In spinal fusion surgery, the correct delineation of processes and correct definition of the interspace between them is crucial for the precise placement of the screws. The remaining of the paper is structured as followed. Section II aims at providing a literature review of methods that employ overlap control or could be used for this purpose. Section III introduces the type of shape parametrization that will be used along this paper. This shapes will have an important role in each of the energies that will be used in the segmentation framework ( Section V). The SIM is introduced