Benjamin Charlier

The Fshape Framework for the Variability Analysis of Functional Shapes

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. Analyzing variability of fshapes’ ensembles requires the modeling 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 geometric-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 thoroughly the numerical implementation of the tangential simplified metamorphosis model 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.

Atlas-based shape analysis and classification of retinal optical coherence tomography images using the functional shape (fshape) framework

&NA; We propose a novel approach for quantitative shape variability analysis in retinal optical coherence tomography images using the functional shape (fshape) framework. The fshape framework uses surface geometry together with functional measures, such as retinal layer thickness defined on the layer surface, for registration across anatomical shapes. This is used to generate a population mean template of the geometry‐function measures from each individual. Shape variability across multiple retinas can be measured by the geometrical deformation and functional residual between the template and each of the observations. To demonstrate the clinical relevance and application of the framework, we generated atlases of the inner layer surface and layer thickness of the Retinal Nerve Fiber Layer (RNFL) of glaucomatous and normal subjects, visualizing detailed spatial pattern of RNFL loss in glaucoma. Additionally, a regularized linear discriminant analysis classifier was used to automatically classify glaucoma, glaucoma‐suspect, and control cases based on RNFL fshape metrics.

Optimal Transport for Diffeomorphic Registration

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.

A General Framework for Curve and Surface Comparison and Registration with Oriented Varifolds

This paper introduces a general setting for the construction of data fidelity metrics between oriented or non-oriented geometric shapes like curves, curve sets or surfaces. These metrics are based on the representation of shapes as distributions of their local tangent or normal vectors and the definition of reproducing kernels on these spaces. The construction, that combines in one common setting and extends the previous frameworks of currents and varifolds, provides a very large class of kernel metrics which can be easily computed without requiring any kind of parametrization of shapes and which are smooth enough to give robustness to certain imperfections that could result e.g. from bad segmentation. We then give a sense, with synthetic examples, of the versatility and potentialities of such metrics when used in various problems like shape comparison, clustering and diffeomorphic registration.

A Fanning Scheme for the Parallel Transport Along Geodesics on Riemannian Manifolds

Parallel transport on Riemannian manifolds allows one to connect tangent spaces at different points in an isometric way and is therefore of importance in many contexts, such as statistics on manifolds. The existing methods for computing parallel transport require either the computation of Riemannian logarithms, such as Schilds ladder, or the Christoffel symbols. The logarithm is rarely given in closed form, and therefore expensive to compute, whereas the Christoffel symbols are in general hard and costly to compute. From an identity between parallel transport and Jacobi fields, we propose a numerical scheme to approximate parallel transport along a geodesic. We find and prove an optimal convergence rate for the scheme, which is equivalent to Schilds ladders. We investigate potential variations of the scheme and give experimental results on the Euclidean 2-sphere and on the manifold of symmetric positive definite matrices.

Parallel transport in shape analysis: a scalable numerical scheme

The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this paper, we adapt a generic numerical scheme recently introduced for computing parallel transport along geodesics in a Riemannian manifold to finite-dimensional manifolds of diffeomorphisms. We provide a qualitative and quantitative analysis of its behavior on high-dimensional manifolds, and investigate an application with the prediction of brain structures progression.

Metamorphoses of Functional Shapes in Sobolev Spaces

In this paper, we describe in detail a model of geometric-functional variability between fshapes. These objects were introduced for the first time by Charlier et al. (J Found Comput Math, 2015. arXiv:1404.6039) and are basically the combination of classical deformable manifolds with additional scalar signal map. Building on the aforementioned work, this paper’s contributions are several. We first extend the original

White Matter Fiber Segmentation Using Functional Varifolds

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