A natural way to model the evolution of an object (growth of a leaf for instance) is to estimate a plausible deforming path between two observations. This interpolation process can generate deceiving results when the set of considered deformations is not relevant to the observed data. To overcome this issue, the framework of deformation modules allows to incorporate in the model structured deformation patterns coming from prior knowledge on the data. The goal of this article is twofold. First defining new deformation modules incorporating structures coming from the elastic properties of the objects. Second, presenting the IMODAL library allowing to perform registration through structured deformations. This library is modular: adapted priors can be easily defined by the user, several priors can be combined into a global one and various types of data can be considered such as curves, meshes or images. It can be downloaded at https://github.com/imodal.

Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences such as the total variation and the relative entropy only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. In sharp contrast, Maximum Mean Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of distances between measures that take into account the geometry of the underlying space and metrize the convergence in law. This paper studies the Sinkhorn divergences, a family of geometric divergences that interpolates between MMD and OT. Relying on a new notion of geometric entropy, we provide theoretical guarantees for these divergences: positivity, convexity and metrization of the convergence in law. On the practical side, we detail a numerical scheme that enables the large scale application of these divergences for machine learning: on the GPU, gradients of the Sinkhorn loss can be computed for batches of a million samples.

Deformations, and diffeormophisms, in particular, have played a tremendous role in the field of statistical shape analysis, as a proxy to measure and interpret differences between similar objects but with different shapes. Diffeomorphisms usually result from the integration of a flow of regular velocity fields, whose parameters have not enabled, so far, a full control of the local behavior of the deformation. In this work, we propose a new mathematical and computational framework, in which these velocity fields are constrained to be built through a combination of local deformation modules with few degrees of freedom. Deformation modules contribute to the global velocity field, and interact with it during integration so that the local modules are transported by the global diffeomorphic deformation under construction. Such modular diffeomorphisms are used to deform shapes and to provide the shape space with a sub-Riemannian metric. We then derive a method to estimate a Fréchet mean from a series of observations, and to decompose the variations in shape observed in the training samples into a set of elementary deformation modules encoding distinctive and interpretable aspects of the shape variability. We show how this approach brings new solutions to long lasting problems in the fields of computer vision and medical image analysis. For instance, the easy implementation of priors in the type of deformations offers a direct control to favor one solution over another in situations where multiple solutions may fit the observations equally well. It allows also the joint optimization of a linear and a nonlinear deformation between shapes, the linear transform simply being a particular type of module. The proposed approach generalizes previous methods for constructing diffeomorphisms and opens up new perspectives in the field of statistical shape analysis.

In the field of Computational Anatomy, biological form (including our focus, neuroanatomy) is studied quantitatively through the action of the diffeomorphism group on example anatomies - a technique called diffeomorphometry. Here we design an algorithm within this framework to pass from dense objects common in neuromaging studies (binary segmentations, structural images) to a sparse representation defined on the surface boundaries of anatomical structures, and embedded into the low dimensional coordinates of a parametric model. Our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points, as well as images through functional composition with the inverse. This allows us to index the diffeomorphisms with respect to two-dimensional surface geometries like subcortical gray matter structures, but explicitly map onto cost functions determined by noisy 3-dimensional measurements. We consider models generated from empirical covariance of training data, as well as bandlimited (Laplace-Beltrami eigenfunction) models when no such data is available. We show applications to noisy or anomalous segmentations, and other typical problems in neuroimaging studies. We reproduce statistical results detecting changes in Alzheimer's disease, despite dimensionality reduction. Lastly we apply our algorithm to the common problem of segmenting subcortical structures from T1 MR images.

In this article, we study the consistency of the template estimation with the Fréchet mean in quotient spaces. The Fréchet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm only allow for the study of a single shape. In this paper, we introduce and discuss both a theoretical and practical setting for the simultaneous study of multiple shapes that are either stitched to one another or slide along a submanifold. The method is described within the optimal control formalism, and optimality conditions are given, together with the equations that are needed to implement augmented Lagrangian methods. Experimental results are provided for stitched and sliding surfaces.

Receptor tyrosine kinase KIT controls many signal transduction pathways and represents a typical allosterically regulated protein. The mutation-induced deregulation of KIT activity impairs cellular physiological functions and causes serious human diseases. The impact of hotspots mutations (D816H/Y/N/V and V560G/D) localized in crucial regulatory segments, the juxtamembrane region (JMR) and the activation (A-) loop, on KIT internal dynamics was systematically studied by molecular dynamics simulations. The mutational outcomes predicted in silico were correlated with in vitro and in vivo activation rates and drug sensitivities of KIT mutants. The allosteric regulation of KIT in the native and mutated forms is described in terms of communication between the two remote segments, JMR and A-loop. A strong correlation between the communication profile and the structural and dynamical features of KIT in the native and mutated forms was established. Our results provide new insight on the determinants of receptor KIT constitutive activation by mutations and resistance of KIT mutants to inhibitors. Depiction of an intra-molecular component of the communication network constitutes a first step towards an integrated description of vast communication pathways established by KIT in physiopathological contexts.

In this paper, we address the problem of orientation that naturally arises when representing shapes such as curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient in modeling a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kinds of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by Almgren, allow the representation of any nonoriented manifold (more generally any nonoriented rectifiable set). In particular, we explain how varifolds can encode numerically nonoriented objects both from the discrete and the continuous points of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (Theorem 4.1), and we derive a formula for the variation of metrics with respect to the shape (Theorem 4.2). Finally, we propose a generalization to nonoriented shapes of registration algorithms in the context of large deformation diffeomorphic metric mapping (LDDMM), which we detail with a few examples in the last part of the paper.

Face recognition is a challenging problem, complicated by variations in pose, expression, lighting, and the passage of time. Significant work has been done to solve each of these problems separately. We consider the problems of lighting and expression variation together, proposing a method that accounts for both variabilities within a single model. We present a novel deformation and lighting insensitive metric to compare images, and we present a novel framework to optimize over this metric to calculate dense correspondences between images. Typical correspondence cost patterns are learned between face image pairs and a Naïve Bayes classifier is applied to improve recognition accuracy. Very promising results are presented on the AR Face Database, and we note that our method can be extended to a broad set of applications.

The alignment and normalization of individual brain structures is a prerequisite for group-level analyses of structural and functional neuroimaging data. The techniques currently available are either based on volume and/or surface attributes, with limited insight regarding the consistent alignment of anatomical landmarks across individuals. This article details a global, geometric approach that performs the alignment of the exhaustive sulcal imprints (cortical folding patterns) across individuals. This DIffeomorphic Sulcal-based COrtical (DISCO) technique proceeds to the automatic extraction, identification and simplification of sulcal features from T1-weighted Magnetic Resonance Image (MRI) series. These features are then used as control measures for fully-3-D diffeomorphic deformations. Quantitative and qualitative evaluations show that DISCO correctly aligns the sulcal folds and gray and white matter volumes across individuals. The comparison with a recent, iconic diffeomorphic approach (DARTEL) highlights how the absence of explicit cortical landmarks may lead to the misalignment of cortical sulci. We also feature DISCO in the automatic design of an empirical sulcal template from group data. We also demonstrate how DISCO can efficiently be combined with an image-based deformation (DARTEL) to further improve the consistency and accuracy of alignment performances. Finally, we illustrate how the optimized alignment of cortical folds across subjects improves sensitivity in the detection of functional activations in a group-level analysis of neuroimaging data.

The link between Bayesian and variational approaches is well known in the image analysis community in particular in the context of deformable models. However, true generative models and consistent estimation procedures are usually not available and the current trend is the computation of statistics mainly based on PCA analysis. We advocate in this paper a careful statistical modeling of deformable structures and we propose an effective and consistent estimation algorithm for the various parameters (geometric and photometric) appearing in the models

The problem of classification is so fundamental that it has been intensively investigated by many researchers from different domains. In this paper, we present a novel unsupervised clustering algorithm derived from the techniques of probabilistic modeling which is implemented by a stochastic gradient algorithm. Then its application to challenging landcover classification based on Daedalus data of the SMART project is explored by combining both spectral feature and spatial contextual information. Our first experiments show its potential usefulness in remote sensing.

Over the past several years our group has been studying biological shape in the emerging new discipline of computational anatomy (CA). CA consists of several components: (i) the construction of coordinatized anatomical manifolds, (ii) comparison of anatomical manifolds, and (iii) inference of morphometric change on anatomical manifolds. In this paper we focus on (ii) the comparison of anatomical shapes and structures in imagery via metric mapping. The purpose of this paper is to examine the generation of the geodesics associated with the metric from several points of view, the first the Euler equation describing the geodesic diffeomorphic flow, and the second the variational formulation of the geodesic in terms of the minimizing flow of vector fields which generate them.

One important problem in any retrieval system is the design of good features and of good similarity measures between features. Usually these similarity functions are defined through ad-hoc distances between features. We propose a new way to design such distances, based on non-rigid deformation of nonlinear principal components, in the framework of semi-parametric statistical regression. The proposed approach is applied to the construction of new rotation invariant distance between planar curves.

On images resulting from the observation of some physical experiments there is no sharp interface between two homogeneous regions, so that the detection of an unique edge is sometimes of little use for the physicist. The authors propose to substitute this detection the determination of a transition zone. This paper addresses the problem of finding the boundaries of such zones with a deformable model approach.>