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International audience; Incorporating prior knowledge into a segmentation process, whether it be geometrical constraints such as volume penalisation, (partial) convexity enforcement, or topological prescriptions to preserve the contextual relations between objects, proves to improve accuracy in medical image segmentation, in particular when addressing the issue of weak boundary definition. Motivated by this observation, the proposed contribution aims to include geometrical constraints in the training of convolutional neural networks in the form of a penalty in the loss function. These geometrical constraints take several forms and encompass level curve alignment through the weighted total variation component, an area penalisation phrased as a hard constraint in the modelling, and an intensity homogeneity criterion based on a combination of the standard Dice loss with the piecewise constant Mumford-Shah model. The mathematical formulation yields a non-smooth non-convex optimisation problem, which rules out conventional smooth optimisation techniques and leads us to adopt a Lagrangian setting. The application falls within the scope of organ-at-risk segmentation in CT (Computed Tomography) images, in the context of radiotherapy planning. Experiments demonstrate that our method provides significant improvements over existing non-constrained approaches.

Motion degradation is a central problem in Magnetic Resonance Imaging (MRI). This work addresses the problem of how to obtain higher quality, super-resolved motion-free reconstructions from highly undersampled MRI data. In this work, we present for the first time a variational multi-task framework that allows joining three relevant tasks in MRI: reconstruction, registration and super-resolution. Our framework takes a set of multiple undersampled MR acquisitions corrupted by motion into a novel multi-task optimisation model, which is composed of an L2 fidelity term that allows sharing representation between tasks, super-resolution foundations and hyperelastic deformations to model biological tissue behaviors. We demonstrate that this combination yields significant improvements over sequential models and other bi-task methods. Our results exhibit fine details and compensate for motion producing sharp and highly textured images compared to state of the art methods while keeping low CPU time. Our improvements are appraised on both clinical assessment and statistical analysis.

In the seminal paper The Gradient Theory of Phase Transitions and the Minimal Interface Criterion, 1987, L. Modica ( [13]) proves some conjectures related to the Van der Waals-Cahn-Hilliard theory of phase transitions.This theory intends to overcome the issue of lack of uniqueness of the solution of the initially considered minimization problem —which aims to minimize the total energy of a fluid confined to a bounded container Ω ⊂ℝn and with Gibbs free energy per unit volume, a prescribed function W of the density distribution u —by enforcing that the interface has minimal surface. In that purpose, a family of functionals parameterized by e > 0 and including a dependency on the density gradient modelling this interfacial energy is introduced, and the asymptotic behavior as e→0+ of the solutions u e of the related minimization problem is analyzed through a Γ-convergence result. Motivated by this work, this paper addresses the question of extending this result to the weighted case. It is shown that this new model inherits the fine properties of the original unweighted one with in particular, the validity of the Γ-convergence result, useful for the minimization of weighted perimeter, relevant for image segmentation.

While segmentation consists in partitioning a given image into meaningful constituents in order to identify relevant structures such as homogeneous regions or edges, registration, given two images, aims at finding an optimal orientationpreserving one-to-one deformation aligning the structures visible in an image into the corresponding ones in the other. Recently, intertwining both tasks into a single framework has proven to yield better results in terms of accuracy —in particular when the images exhibit weak boundary definition —and increase of reliability of the encoded structure matching —since now, not only based on intensity distribution comparison but also on geometrical and topological features —. In line with this idea, we propose going a step further by adding explicitly some dynamics in the modelling, i.e., by making the minimization problem both space and time-dependent so that the correlation between both tasks is achieved through the process, connecting thus the problem to an interpolation one. The shapes to be matched are viewed as Saint Venant-Kirchhoff materials, a special instance of hyperelastic ones, and are implicitly modelled by level-set functions. These are evolved in order to minimize a functional including both a nonlinear-elasticity-based regularizer prescribing the physical nature of the deformation and a term penalizing the shape misalignment, thus promoting structure matching rather than intensity pairing. Theoretical results emphasizing the mathematical soundness of the model are provided, among which the existence of minimizers and the existence of a weak viscosity solution to the related evolution problem. The model is then applied to the longitudinal registration of hepatic dynamic contrast-enhanced MRI sequences and shows good performance. This application has an important impact on the computer-aided follow-up of patients suffering from liver cancers.

We study a spline-based approximation of vector fields in the conservative case. Its motivation comes from for instance when approximating current or wind velocity fields, the data deriving in those cases from a potential (pressure for the wind, etc‥). To model the problem, we introduce a minimization problem on a Hilbert space for which the existence and uniqueness of the solution are provided. A convergence result in Sobolev metric spaces is established using norm equivalences and compactness arguments, as well as an approximation error estimate of the involved smoothing $D^{m}$ splines. Possible applications include estimation of wind energy potential and prediction of wind power or marine current simulation.