This article introduces a full mathematical and numerical framework for treating functional shapes (or fshapes) following the landmarks of shape spaces and shape analysis. Functional shapes can be described as signal functions supported on varying geometrical supports. Analysing variability of fshapes' ensembles require the modelling and quantification of joint variations in geometry and signal, which have been treated separately in previous approaches. Instead, building on the ideas of shape spaces for purely geometrical objects, we propose the extended concept of fshape bundles and define Riemannian metrics for fshape metamorphoses to model geometrico-functional transformations within these bundles. We also generalize previous works on data attachment terms based on the notion of varifolds and demonstrate the utility of these distances. Based on these, we propose variational formulations of the atlas estimation problem on populations of fshapes and prove existence of solutions for the different models. The second part of the article examines the numerical implementation of the models by detailing discrete expressions for the metrics and gradients and proposing an optimization scheme for the atlas estimation problem. We present a few results of the methodology on a synthetic dataset as well as on a population of retinal membranes with thickness maps

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This paper introduces the use of unbalanced optimal transport methods as a similarity measure for diffeomorphic matching of imaging data. The similarity measure is a key object in diffeomorphic registration methods that, together with the regularization on the deformation, defines the optimal deformation. Most often, these similarity measures are local or non local but simple enough to be computationally fast. We build on recent theoretical and numerical advances in optimal transport to propose fast and global similarity measures that can be used on surfaces or volumetric imaging data. This new similarity measure is computed using a fast generalized Sinkhorn algorithm. We apply this new metric in the LDDMM framework on synthetic and real data, fibres bundles and surfaces and show that better matching results are obtained.

Alzheimer’s disease (AD) is characterized by the progressive alterations seen in brain images which give rise to the onset of various sets of symptoms. The variability in the dynamics of changes in both brain images and cognitive impairments remains poorly understood. This paper introduces AD Course Map a spatiotemporal atlas of Alzheimer’s disease progression. It summarizes the variability in the progression of a series of neuropsychological assessments, the propagation of hypometabolism and cortical thinning across brain regions and the deformation of the shape of the hippocampus. The analysis of these variations highlights strong genetic determinants for the progression, like possible compensatory mechanisms at play during disease progression. AD Course Map also predicts the patient’s cognitive decline with a better accuracy than the 56 methods benchmarked in the open challenge TADPOLE. Finally, AD Course Map is used to simulate cohorts of virtual patients developing Alzheimer’s disease. AD Course Map offers therefore new tools for exploring the progression of AD and personalizing patients care.

A new class of statistical deformable models is introduced to study high-dimensional curves or images. In addition to the standard measurement error term, these deformable models include an extra error term modeling the individual variations in intensity around a mean pattern. It is shown that an appropriate tool for statistical inference in such models is the notion of sample Fréchet means, which leads to estimators of the deformation parameters and the mean pattern. The main contribution of this paper is to study how the behavior of these estimators depends on the number n of design points and the number J of observed curves (or images). Numerical experiments are given to illustrate the finite sample performances of the procedure.Acknowledgements -We would like to thank Dominique Bakry for fruitful discussions. Both authors would like to thank the Center for Mathematical Modeling and the CNRS for financial support and excellent hospitality while visiting Santiago where part of this work was carried out. We are very much indebted to the referees and the Associate Editor for their constructive comments and remarks that resulted in a major revision of the original manuscript.consistency of the proposed estimators in various asymptotic settings: either when both the number n of design points and the number J of curves (or images) tend to infinity, or when n (resp. J) remains fixed while J (resp. n) tends to infinity.In many situations, data sets of curves or images exhibit a source of geometric variations in time or shape. In such settings, the usual Euclidean meanȲ ℓ = 1 J J j=1 Y ℓ j in model (1.1) cannot be used to recover a meaningful mean pattern. Indeed, consider the following simple model of randomly shifted curves (with d = 1) which is commonly used in many applied areas such as neuroscience [TIR10] where f : Ω −→ R is the mean pattern of the observed curves, and the θ * j 's are i.i.d. random variables in R with density g and independent of the ε ℓ j 's. In model (1.2), the shifts θ * j represent a source of variability in time. However, in (1.2) the Euclidean mean is not a consistent estimator of the mean pattern f since by the law of large numbersThe randomly shifted curves model (1.2) is close to the perturbation model introduced by [Goo91] in shape analysis for the study of consistent estimation of a mean pattern from a set of random planar shapes. The mean pattern to estimate in [Goo91] is called a population mean, but to stress the fact that it comes from a perturbation model [Huc10] uses the term perturbation mean. To achieve consistency in such models, a Procrustean procedure is used in [Goo91], which leads to the statistical analysis of sample Fréchet means [Fré48] which are extensions of the usual Euclidean mean to non-linear spaces using non-Euclidean metrics. For random variables belonging to a nonlinear manifold, a well-known example is the computation of the mean of a set of planar shapes in the Kendall's shape space [Ken84] which leads to the Procrustean means studied in [Goo91]. Consistent ...

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Abstract. Parallel transport on Riemannian manifolds allows one to connect tangent spaces at 12 different points in an isometric way and is therefore of importance in many contexts, such as statistics 13 on manifolds. The existing methods to compute parallel transport require either the computation 14 of Riemannian logarithms, such as the Schild's ladder, or the Christoffel symbols. The Logarithm is 15 rarely given in closed form, and therefore costly to compute whereas the number of Christoffel symbols 16 explodes with the dimension of the manifold, making both these methods intractable. From an 17 identity between parallel transport and Jacobi fields, we propose a numerical scheme to approximate 18 the parallel transport along a geodesic. We find and prove an optimal convergence rate for the 19 scheme, which is equivalent to Schild's ladder's. We investigate potential variations of the scheme 20 and give experimental results on the Euclidean two-sphere and on the manifold of symmetric positive 21 definite matrices. 22 This manuscript is for review purposes only.

The analysis of longitudinal trajectories is a longstanding problem in medical imaging which is often tackled in the context of Riemannian geometry: the set of observations is assumed to lie on an a priori known Riemannian manifold. When dealing with high-dimensional or complex data, it is in general not possible to design a Riemannian geometry of relevance. In this paper, we perform Riemannian manifold learning in association with the statistical task of longitudinal trajectory analysis. After inference, we obtain both a submanifold of observations and a Riemannian metric so that the observed progressions are geodesics. This is achieved using a deep generative network, which maps trajectories in a low-dimensional Euclidean space to the observation space.