%0 Journal Article
%T A Variational Model Dedicated to Joint Segmentation, Registration, and Atlas Generation for Shape Analysis
%+ Institut Pascal (IP)
%+ University of Cambridge [UK] (CAM)
%+ Informatique, BioInformatique, Systèmes Complexes (IBISC)
%+ National Physical Laboratory [Teddington] (NPL)
%+ Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI)
%+ Department of Applied Mathematics and Theoretical Physics [Cambridge] (DAMTP)
%A Debroux, Noémie
%A Aston, John
%A Bonardi, Fabien
%A Forbes, Alistair
%A Le Guyader, Carole
%A Romanchikova, Marina
%A Schönlieb, Carola-Bibiane
%< avec comité de lecture
%J SIAM Journal on Imaging Sciences
%I Society for Industrial and Applied Mathematics
%V 13
%N 1
%P 351-380
%8 2020-01
%D 2020
%R 10.1137/19M1271907
%K Segmentation
%K registration
%K nonlinear elasticity
%K Ogden materials
%K Potts model
%K atlas generation
%K asymptotic results
%K D m -splines
%K geometric PCA AMS subject classifications. 68U10
%K 49
%K 65D18
%K 41A15
%K 62H25
%Z Computer Science [cs]/Medical ImagingJournal articles
%X In 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.
%G English
%2 https://uca.hal.science/hal-03034316/document
%2 https://uca.hal.science/hal-03034316/file/ex_article.pdf
%L hal-03034316
%U https://uca.hal.science/hal-03034316
%~ PRES_CLERMONT
%~ CNRS
%~ UNIV-BPCLERMONT
%~ UNIV-EVRY
%~ INSA-ROUEN
%~ IBISC
%~ INSTITUT_PASCAL
%~ COMUE-NORMANDIE
%~ LMI-ROUEN
%~ UNIV-PARIS-SACLAY
%~ UNIV-EVRY-SACLAY
%~ INSA-GROUPE
%~ IBISC-SIAM
%~ TEST-HALCNRS
%~ UNIVERSITE-PARIS-SACLAY
%~ GS-ENGINEERING
%~ GS-COMPUTER-SCIENCE
%~ GS-LIFE-SCIENCES-HEALTH
%~ TEST3-HALCNRS