3D shape matching and registration : a probabilistic perspective

Dense correspondence is a key area in computer vision and medical image analysis. It has applications in registration and shape analysis. In this thesis, we develop a technique to recover dense correspondences between the surfaces of neuroanatomical objects over heterogeneous populations of individuals. We recover dense correspondences based on 3D shape matching. In this thesis, the 3D shape matching problem is formulated under the framework of Markov Random Fields (MRFs). We represent the surfaces of neuroanatomical objects as genus zero voxel-based meshes. The surface meshes are projected into a Markov random field space. The projection carries both geometric and topological information in terms of Gaussian curvature and mesh neighbourhood from the original space to the random field space. Gaussian curvature is projected to the nodes of the MRF, and the mesh neighbourhood structure is projected to the edges. 3D shape matching between two surface meshes is then performed by solving an energy function minimisation problem formulated with MRFs. The outcome of the 3D shape matching is dense point-to-point correspondences. However, the minimisation of the energy function is NP hard. In this thesis, we use belief propagation to perform the probabilistic inference for 3D shape matching. A sparse update loopy belief propagation algorithm adapted to the 3D shape matching is proposed to obtain an approximate global solution for the 3D shape matching problem. The sparse update loopy belief propagation algorithm demonstrates significant efficiency gain compared to standard belief propagation. The computational complexity and convergence property analysis for the sparse update loopy belief propagation algorithm are also conducted in the thesis. We also investigate randomised algorithms to minimise the energy function. In order to enhance the shape matching rate and increase the inlier support set, we propose a novel clamping technique. The clamping technique is realized by combining the loopy belief propagation message updating rule with the feedback from 3D rigid body registration. By using this clamping technique, the correct shape matching rate is increased significantly. Finally, we investigate 3D shape registration techniques based on the 3D shape matching result. Based on the point-to-point dense correspondences obtained from the 3D shape matching, a three-point based transformation estimation technique is combined with the RANdom SAmple Consensus (RANSAC) algorithm to obtain the inlier support set. The global registration approach is purely dependent on point-wise correspondences between two meshed surfaces. It has the advantage that the need for orientation initialisation is eliminated and that all shapes of spherical topology. The comparison of our MRF based 3D registration approach with a state-of-the-art registration algorithm, the first order ellipsoid template, is conducted in the experiments. These show dense correspondence for pairs of hippocampi from two different data sets, each of around 20 60+ year old healthy individuals

Learning distributions of shape trajectories from longitudinal datasets: a hierarchical model on a manifold of diffeomorphisms

International audienceWe propose a method to learn a distribution of shape tra-jectories from longitudinal data, i.e. the collection of individual objects repeatedly observed at multiple time-points. The method allows to compute an average spatiotemporal trajectory of shape changes at the group level, and the individual variations of this trajectory both in terms of geometry and time dynamics. First, we formulate a non-linear mixed-effects statistical model as the combination of a generic statistical model for manifold-valued longitudinal data, a deformation model defining shape trajectories via the action of a finite-dimensional set of diffeomorphisms with a man-ifold structure, and an efficient numerical scheme to compute parallel transport on this manifold. Second, we introduce a MCMC-SAEM algorithm with a specific approach to shape sampling, an adaptive scheme for proposal variances , and a log-likelihood tempering strategy to estimate our model. Third, we validate our algorithm on 2D simulated data, and then estimate a scenario of alteration of the shape of the hippocampus 3D brain structure during the course of Alzheimer's disease. The method shows for instance that hippocampal atrophy progresses more quickly in female subjects, and occurs earlier in APOE4 mutation carriers. We finally illustrate the potential of our method for classifying pathological trajectories versus normal ageing

Méthodes numériques et statistiques pour l'analyse de trajectoire dans un cadre de geométrie Riemannienne.

This PhD proposes new Riemannian geometry tools for the analysis of longitudinal observations of neuro-degenerative subjects. First, we propose a numerical scheme to compute the parallel transport along geodesics. This scheme is efficient as long as the co-metric can be computed efficiently. Then, we tackle the issue of Riemannian manifold learning. We provide some minimal theoretical sanity checks to illustrate that the procedure of Riemannian metric estimation can be relevant. Then, we propose to learn a Riemannian manifold so as to model subject's progressions as geodesics on this manifold. This allows fast inference, extrapolation and classification of the subjects.Cette thèse porte sur l'élaboration d'outils de géométrie riemannienne et de leur application en vue de la modélisation longitudinale de sujets atteints de maladies neuro-dégénératives. Dans une première partie, nous prouvons la convergence d'un schéma numérique pour le transport parallèle. Ce schéma reste efficace tant que l'inverse de la métrique peut être calculé rapidement. Dans une deuxième partie, nous proposons l'apprentissage une variété et une métrique riemannienne. Après quelques résultats théoriques encourageants, nous proposons d'optimiser la modélisation de progression de sujets comme des géodésiques sur cette variété

A Bayesian Approach to Sparse Model Selection in Statistical Shape Models

Groupwise registration of point sets is the fundamental step in creating statistical shape models (SSMs). When the number of points on the sets varies across the population, each point set is often regarded as a spatially transformed Gaussian mixture model (GMM) sample, and the registration problem is formulated as the estimation of the underlying GMM from the training samples. Thus, each Gaussian in the mixture specifies a landmark (or model point), which is probabilistically corresponded to a training point. The Gaussian components, transformations, and probabilistic matches are often computed by an expectation-maximization (EM) algorithm. To avoid over- and under-fitting errors, the SSM should be optimized by tuning the required number of components. In this paper, rather than manually setting the number of components before training, we start from a maximal model and prune out the negligible points during the registration by a sparsity criterion. We show that by searching over the continuous space for optimal sparsity level, we can reduce the fitting errors (generalization and specificities), and thereby help the search process for a discrete number of model points. We propose an EM framework, adopting a symmetric Dirichlet distribution as a prior, to enforce sparsity on the mixture weights of Gaussians. The negligible model points are pruned by a quadratic programming technique during EM iterations. The proposed EM framework also iteratively updates the estimates of the rigid registration parameters of the point sets to the mean model. Next, we apply the principal component analysis to the registered and equal-length training point sets and construct the SSMs. This method is evaluated by learning of sparse SSMs from 15 manually segmented caudate nuclei, 24 hippocampal, and 20 prostate data sets. The generalization, specificity, and compactness of the proposed model favorably compare to a traditional EM based model

Quantitative MRI correlates of hippocampal and neocortical pathology in intractable temporal lobe epilepsy

Intractable or drug-resistant epilepsy occurs in over 30% of epilepsy patients, with many of these patients undergoing surgical excision of the affected brain region to achieve seizure control. Advances in MRI have the potential to improve surgical treatment of epilepsy through improved identification and delineation of lesions. However, validation is currently needed to investigate histopathological correlates of these new imaging techniques. The purpose of this work is to investigate histopathological correlates of quantitative relaxometry and DTI from hippocampal and neocortical specimens of intractable TLE patients. To achieve this goal I developed and evaluated a pipeline for histology to in-vivo MRI image registration, which finds dense spatial correspondence between both modalities. This protocol was divided in two steps whereby sparsely sectioned histology from temporal lobe specimens was first registered to the intermediate ex-vivo MRI which is then registered to the in-vivo MRI, completing a pipeline for histology to in-vivo MRI registration. When correlating relaxometry and DTI with neuronal density and morphology in the temporal lobe neocortex, I found T1 to be a predictor of neuronal density in the neocortical GM and demonstrated that employing multi-parametric MRI (combining T1 and FA together) provided a significantly better fit than each parameter alone in predicting density of neurons. This work was the first to relate in-vivo T1 and FA values to the proportion of neurons in GM. When investigating these quantitative multimodal parameters with histological features within the hippocampal subfields, I demonstrated that MD correlates with neuronal density and size, and can act as a marker for neuron integrity within the hippocampus. More importantly, this work was the first to highlight the potential of subfield relaxometry and diffusion parameters (mainly T2 and MD) as well as volumetry in predicting the extent of cell loss per subfield pre-operatively, with a precision so far unachievable. These results suggest that high-resolution quantitative MRI sequences could impact clinical practice for pre-operative evaluation and prediction of surgical outcomes of intractable epilepsy

Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results