Piecewise-diffeomorphic image registration: Application to the motion estimation between 3D CT lung images with sliding conditions Laurent Risser a,⇑ , François-Xavier Vialard b , Habib Y. Baluwala a , Julia A. Schnabel a a Institute of Biomedical Engineering, Department of Engineering Science, University of Oxford, UK b CEREMADE, Université de Paris-Dauphine, France article info Article history: Received 26 January 2012 Received in revised form 24 October 2012 Accepted 26 October 2012 Available online xxxx Keywords: Diffeomorphic registration Sliding motion LDDMM LogDemons Respiratory motion abstract In this paper, we propose a new strategy for modelling sliding conditions when registering 3D images in a piecewise-diffeomorphic framework. More specifically, our main contribution is the development of a mathematical formalism to perform Large Deformation Diffeomorphic Metric Mapping registration with sliding conditions. We also show how to adapt this formalism to the LogDemons diffeomorphic registra- tion framework. We finally show how to apply this strategy to estimate the respiratory motion between 3D CT pulmonary images. Quantitative tests are performed on 2D and 3D synthetic images, as well as on real 3D lung images from the MICCAI EMPIRE10 challenge. Results show that our strategy estimates accu- rate mappings of entire 3D thoracic image volumes that exhibit a sliding motion, as opposed to conven- tional registration methods which are not capable of capturing discontinuous deformations at the thoracic cage boundary. They also show that although the deformations are not smooth across the loca- tion of sliding conditions, they are almost always invertible in the whole image domain. This would be helpful for radiotherapy planning and delivery. Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved. 1. Introduction Until recently, the main efforts to estimate the respiratory mo- tion in lung imaging focused on image registration techniques based on B-splines (Mattes et al., 2003; McClelland et al., 2006), thin-plate splines (Coselmon et al., 2004; Li et al., 2003), hybrid feature- and intensity-based registration (Stewart et al., 2004), optical flow or diffusion (demons) type methods (Werner et al., 2010) with a spatially homogeneous regularisation of the deforma- tions. A review of these techniques is given in (Sluimer et al., 2006). The MICCAI Evaluation of Methods for Pulmonary Image Registra- tion 2010 (EMPIRE10) challenge (Murphy et al., 2011) allowed the comparison of state-of-the art algorithms for this purpose using a common dataset. None of the above mentioned methods is how- ever designed to capture non-smooth, discontinuous sliding mo- tion due to organ slippage adequately. Slippage between the lungs and surrounding structures is illustrated in Fig. 1, where we show the displacement of a vessel within the lungs, relative to two ribs. Sliding motion of the lungs against the chest wall nat- urally takes place across the pleura during breathing. The pleural cavity is a body cavity that surrounds the lungs and allows the pleurae to slide effortlessly against each other during ventilation. For this reason, standard methods, which do not explicitly model sliding conditions, typically register only the lungs extracted from the background. For instance, the most successful methods of (Murphy et al., 2011) first affinely register the lung masks before performing a non-rigid registration of the masked lung CT images. Moreover, due to the sparse distribution of features within the lungs, a large amount of regularisation is required to properly reg- ister the lungs. This obviously leads to the estimation of non- natural deformations in the thoracic cage’s neighbourhood, in particular, in the ribs and the spine, which exhibit a locally rigid motion. For a simultaneous registration of the lungs and their sur- roundings, models taking into account more information about the thoracic physiology, in particular sliding conditions, are conse- quently required. Clinical applications would benefit from incorpo- rating sliding conditions when registering lung images, for instance in radiotherapy planning or in image guided surgery. A recent approach to biomechanical patient-specific respiratory motion modelling is described in (Werner et al., 2009). This method is however computationally too complex to be used for automated image registration tasks. Several approaches for direc- tion-dependent regularisation in diffusion- or spline-based regis- tration frameworks were also recently introduced (Wu et al., 2008; Ruan et al., 2009; Schmidt-Richberg et al., 2012; Yin et al., 2010; Pace et al., 2011; Delmon et al., 2011; Vandemeulebroucke et al., 2012) for modelling sliding conditions. In particular (Wu et al., 2008; Vandemeulebroucke et al., 2012) use masked spline deformations and (Delmon et al., 2011) uses an additional spline 1361-8415/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.media.2012.10.001 ⇑ Corresponding author. E-mail address: laurent.risser@gmail.com (L. Risser). Medical Image Analysis xxx (2012) xxx–xxx Contents lists available at SciVerse ScienceDirect Medical Image Analysis journal homepage: www.elsevier.com/locate/media Please cite this article in press as: Risser, L., et al. Piecewise-diffeomorphic image registration: Application to the motion estimation between 3D CT lung images with sliding conditions. Med. Image Anal. (2012), http://dx.doi.org/10.1016/j.media.2012.10.001