https://hal.uca.fr/hal-03034316Debroux, NoémieNoémieDebrouxIP - Institut Pascal - SIGMA Clermont - SIGMA Clermont - UCA [2017-2020] - Université Clermont Auvergne [2017-2020] - CNRS - Centre National de la Recherche ScientifiqueAston, JohnJohnAstonStatslab, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of CambridgeBonardi, FabienFabienBonardiIBISC - Informatique, BioInformatique, Systèmes Complexes - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-SaclayForbes, AlistairAlistairForbesNPL - National Physical Laboratory [Teddington]Le Guyader, CaroleCaroleLe GuyaderLMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie - INSA Rouen Normandie - Institut national des sciences appliquées Rouen Normandie - INSA - Institut National des Sciences Appliquées - NU - Normandie UniversitéRomanchikova, MarinaMarinaRomanchikovaNPL - National Physical Laboratory [Teddington]Schönlieb, Carola-BibianeCarola-BibianeSchönliebDAMTP - Department of Applied Mathematics and Theoretical Physics [Cambridge] - CMS - Faculty of mathematics Centre for Mathematical Sciences [Cambridge] - CAM - University of Cambridge [UK]A Variational Model Dedicated to Joint Segmentation, Registration, and Atlas Generation for Shape AnalysisHAL CCSD2020Segmentationregistrationnonlinear elasticityOgden materialsPotts modelatlas generationasymptotic resultsD m -splinesgeometric PCA AMS subject classifications. 68U104965D1841A1562H25[INFO.INFO-IM] Computer Science [cs]/Medical ImagingDEBROUX, Noémie2020-12-01 16:47:332022-03-16 03:53:362020-12-02 09:40:14enJournal articleshttps://hal.uca.fr/hal-03034316/document10.1137/19M1271907application/pdf1In medical image analysis, constructing an atlas, i.e. a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterise and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the L ∞ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e. more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which existence of minimisers, analysis of a numerical method, asymptotic results and a PCA analysis, as well as numerical simulations demonstrating the ability of the model to produce an atlas exhibiting sharp edges, high contrast and a consistent shape.