Joan Alexis Glaunès

Surface matching via currents

We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data.

Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching

In the paper, we study the problem of optimal matching of two generalized functions (distributions) via a diffeomorphic transformation of the ambient space. In the particular case of discrete distributions (weighted sums of Dirac measures), we provide a new algorithm to compare two arbitrary unlabelled sets of points, and show that it behaves properly in limit of continuous distributions on sub-manifolds. As a consequence, the algorithm may apply to various matching problems, such as curve or surface matching (via a sub-sampling), or mixings of landmark and curve data. As the solution forbids high energy solutions, it is also robust towards addition of noise and the technique can be used for nonlinear projection of datasets. We present 2D and 3D experiments.

Large Deformation Diffeomorphic Metric Curve Mapping

We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space ℝd. Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in ℝ2 and ℝ3. Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and experiments with 3D curves extracted from brain cortical surfaces are presented.

Landmark Matching via Large Deformation Diffeomorphisms on the Sphere

We consider signal and image restoration using convex cost-functions composed of a non-smooth data-fidelity term and a smooth regularization term. We provide a convergent method to minimize such cost-functions. In order to restore data corrupted with outliers and impulsive noise, we focus on cost-functions composed of an ℓ1 data-fidelity term and an edge-preserving regularization term. The analysis of the minimizers of these cost-functions provides a natural justification of the method. It is shown that, because of the ℓ1 data-fidelity, these minimizers involve an implicit detection of outliers. Uncorrupted (regular) data entries are fitted exactly while outliers are replaced by estimates determined by the regularization term, independently of the exact value of the outliers. The resultant method is accurate and stable, as demonstrated by the experiments. A crucial advantage over alternative filtering methods is the possibility to convey adequate priors about the restored signals and images, such as the presence of edges. Our variational method furnishes a new framework for the processing of data corrupted with outliers and different kinds of impulse noise.

Diffeomorphic metric surface mapping in subregion of the superior temporal gyrus

This paper describes the application of large deformation diffeomorphic metric mapping to cortical surfaces based on the shape and geometric properties of subregions of the superior temporal gyrus in the human brain. The anatomical surfaces of the cortex are represented as triangulated meshes. The diffeomorphic matching algorithm is implemented by defining a norm between the triangulated meshes, based on the algorithms of Vaillant and Glaunès. The diffeomorphic correspondence is defined as a flow of the extrinsic three dimensional coordinates containing the cortical surface that registers the initial and target geometry by minimizing the norm. The methods are demonstrated in 40 high-resolution MRI cortical surfaces of planum temporale (PT) constructed from subsets of the superior temporal gyrus (STG). The effectiveness of the algorithm is demonstrated via the Euclidean positional distance, distance of normal vectors, and curvature before and after the surface matching as well as the comparison with a landmark matching algorithm. The results demonstrate that both the positional and shape variability of the anatomical configurations are being represented by the diffeomorphic maps.

Diffeomorphic Brain Registration Under Exhaustive Sulcal Constraints

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.

Vestibular Asymmetry as the Cause of Idiopathic Scoliosis: A Possible Answer from Xenopus

Human idiopathic scoliosis is characterized by severe deformations of the spine and skeleton. The occurrence of vestibular-related deficits in these patients is well established but it is unclear whether a vestibular pathology is the common cause for the scoliotic syndrome and the gaze/posture deficits or if the latter behavioral deficits are a consequence of the scoliotic deformations. A possible vestibular origin was tested in the frog Xenopus laevis by unilateral removal of the labyrinthine endorgans at larval stages. After metamorphosis into young adult frogs, X-ray images and three-dimensional reconstructed micro-computer tomographic scans of the skeleton showed deformations similar to those of scoliotic patients. The skeletal distortions consisted of a curvature of the spine in the frontal and sagittal plane, a transverse rotation along the body axis and substantial deformations of all vertebrae. In terrestrial vertebrates, the initial postural syndrome after unilateral labyrinthectomy recovers over time and requires body weight-supporting limb proprioceptive information. In an aquatic environment, however, this information is absent. Hence, the lesion-induced asymmetric activity in descending spinal pathways and the resulting asymmetric muscular tonus persists. As a consequence the mostly cartilaginous skeleton of the frog tadpoles progressively deforms. Lack of limb proprioceptive signals in an aquatic environment is thus the element, which links the Xenopus model with human scoliosis because a comparable situation occurs during gestation in utero. A permanently imbalanced activity in descending locomotor/posture control pathways might be the common origin for the observed structural and behavioral deficits in humans as in the different animal models of scoliosis.

Construction and analysis of a head CT-scan database for craniofacial reconstruction

This paper is devoted to the construction of a complete database which is intended to improve the implementation and the evaluation of automated facial reconstruction. This growing database is currently composed of 85 head CT-scans of healthy European subjects aged 20-65 years old. It also includes the triangulated surfaces of the face and the skull of each subject. These surfaces are extracted from CT-scans using an original combination of image-processing techniques which are presented in the paper. Besides, a set of 39 referenced anatomical skull landmarks were located manually on each scan. Using the geometrical information provided by triangulated surfaces, we compute facial soft-tissue depths at each known landmark positions. We report the average thickness values at each landmark and compare our measures to those of the traditional charts of [J. Rhine, C.E. Moore, Facial Tissue Thickness of American Caucasoïds, Maxwell Museum of Anthropology, Albuquerque, New Mexico, 1982] and of several recent in vivo studies [M.H. Manhein, G.A. Listi, R.E. Barsley, et al., In vivo facial tissue depth measurements for children and adults, Journal of Forensic Sciences 45 (1) (2000) 48-60; S. De Greef, P. Claes, D. Vandermeulen, et al., Large-scale in vivo Caucasian facial soft tissue thickness database for craniofacial reconstruction, Forensic Science International 159S (2006) S126-S146; R. Helmer, Schödelidentifizierung durch elektronische bildmischung, Kriminalistik Verlag GmbH, Heidelberg, 1984].

Spatial and Anatomical Regularization of SVM: A General Framework for Neuroimaging Data

This paper presents a framework to introduce spatial and anatomical priors in SVM for brain image analysis based on regularization operators. A notion of proximity based on prior anatomical knowledge between the image points is defined by a graph (e.g., brain connectivity graph) or a metric (e.g., Fisher metric on statistical manifolds). A regularization operator is then defined from the graph Laplacian, in the discrete case, or from the Laplace-Beltrami operator, in the continuous case. The regularization operator is then introduced into the SVM, which exponentially penalizes high-frequency components with respect to the graph or to the metric and thus constrains the classification function to be smooth with respect to the prior. It yields a new SVM optimization problem whose kernel is a heat kernel on graphs or on manifolds. We then present different types of priors and provide efficient computations of the Gram matrix. The proposed framework is finally applied to the classification of brain Magnetic Resonance (MR) images (based on Gray Matter (GM) concentration maps and cortical thickness measures) from 137 patients with Alzheimers Disease (AD) and 162 elderly controls. The results demonstrate that the proposed classifier generates less-noisy and consequently more interpretable feature maps with high classification performances.

Craniofacial reconstruction as a prediction problem using a Latent Root Regression model

In this paper, we present a computer-assisted method for facial reconstruction. This method provides an estimation of the facial shape associated with unidentified skeletal remains. Current computer-assisted methods using a statistical framework rely on a common set of extracted points located on the bone and soft-tissue surfaces. Most of the facial reconstruction methods then consist of predicting the position of the soft-tissue surface points, when the positions of the bone surface points are known. We propose to use Latent Root Regression for prediction. The results obtained are then compared to those given by Principal Components Analysis linear models. In conjunction, we have evaluated the influence of the number of skull landmarks used. Anatomical skull landmarks are completed iteratively by points located upon geodesics which link these anatomical landmarks, thus enabling us to artificially increase the number of skull points. Facial points are obtained using a mesh-matching algorithm between a common reference mesh and individual soft-tissue surface meshes. The proposed method is validated in term of accuracy, based on a leave-one-out cross-validation test applied to a homogeneous database. Accuracy measures are obtained by computing the distance between the original face surface and its reconstruction. Finally, these results are discussed referring to current computer-assisted reconstruction facial techniques.