Interactive Segmentation of 3D Medical Images with Implicit Surfaces

To cope with a variety of clinical applications, research in medical image processing has led to a large spectrum of segmentation techniques that extract anatomical structures from volumetric data acquired with 3D imaging modalities. Despite continuing advances in mathematical models for automatic segmentation, many medical practitioners still rely on 2D manual delineation, due to the lack of intuitive semi-automatic tools in 3D. In this thesis, we propose a methodology and associated numerical schemes enabling the development of 3D image segmentation tools that are reliable, fast and interactive. These properties are key factors for clinical acceptance. Our approach derives from the framework of variational methods: segmentation is obtained by solving an optimization problem that translates the expected properties of target objects in mathematical terms. Such variational methods involve three essential components that constitute our main research axes: an objective criterion, a shape representation and an optional set of constraints. As objective criterion, we propose a unified formulation that extends existing homogeneity measures in order to model the spatial variations of statistical properties that are frequently encountered in medical images, without compromising efficiency. Within this formulation, we explore several shape representations based on implicit surfaces with the objective to cover a broad range of typical anatomical structures. Firstly, to model tubular shapes in vascular imaging, we introduce convolution surfaces in the variational context of image segmentation. Secondly, compact shapes such as lesions are described with a new representation that generalizes Radial Basis Functions with non-Euclidean distances, which enables the design of basis functions that naturally align with salient image features. Finally, we estimate geometric non-rigid deformations of prior templates to recover structures that have a predictable shape such as whole organs. Interactivity is ensured by restricting admissible solutions with additional constraints. Translating user input into constraints on the sign of the implicit representation at prescribed points in the image leads us to consider inequality-constrained optimization

Anatomical Modeling of Cerebral Microvascular Structures: Application to Identify Biomarkers of Microstrokes

Les réseaux microvasculaires corticaux sont responsables du transport de l’oxygène et des substrats énergétiques vers les neurones. Ces réseaux réagissent dynamiquement aux demandes énergétiques lors d’une activation neuronale par le biais du couplage neurovasculaire. Afin d’élucider le rôle de la composante microvasculaire dans ce processus de couplage, l’utilisation de la modélisation in-formatique pourrait se révéler un élément clé. Cependant, la manque de méthodologies de calcul appropriées et entièrement automatisées pour modéliser et caractériser les réseaux microvasculaires reste l’un des principaux obstacles. Le développement d’une solution entièrement automatisée est donc important pour des explorations plus avancées, notamment pour quantifier l’impact des mal-formations vasculaires associées à de nombreuses maladies cérébrovasculaires. Une observation courante dans l’ensemble des troubles neurovasculaires est la formation de micro-blocages vascu-laires cérébraux (mAVC) dans les artérioles pénétrantes de la surface piale. De récents travaux ont démontré l’impact de ces événements microscopiques sur la fonction cérébrale. Par conséquent, il est d’une importance vitale de développer une approche non invasive et comparative pour identifier leur présence dans un cadre clinique. Dans cette thèse,un pipeline de traitement entièrement automatisé est proposé pour aborder le prob-lème de la modélisation anatomique microvasculaire. La méthode de modélisation consiste en un réseau de neurones entièrement convolutif pour segmenter les capillaires sanguins, un générateur de modèle de surface 3D et un algorithme de contraction de la géométrie pour produire des mod-èles graphiques vasculaires ne comportant pas de connections multiples. Une amélioration de ce pipeline est développée plus tard pour alléger l’exigence de maillage lors de la phase de représen-tation graphique. Un nouveau schéma permettant de générer un modèle de graphe est développé avec des exigences d’entrée assouplies et permettant de retenir les informations sur les rayons des vaisseaux. Il est inspiré de graphes géométriques déformants construits en respectant les morpholo-gies vasculaires au lieu de maillages de surface. Un mécanisme pour supprimer la structure initiale du graphe à chaque exécution est implémenté avec un critère de convergence pour arrêter le pro-cessus. Une phase de raffinement est introduite pour obtenir des modèles vasculaires finaux. La modélisation informatique développée est ensuite appliquée pour simuler les signatures IRM po-tentielles de mAVC, combinant le marquage de spin artériel (ASL) et l’imagerie multidirectionnelle pondérée en diffusion (DWI). L’hypothèse est basée sur des observations récentes démontrant une réorientation radiale de la microvascularisation dans la périphérie du mAVC lors de la récupéra-tion chez la souris. Des lits capillaires synthétiques, orientés aléatoirement et radialement, et des angiogrammes de tomographie par cohérence optique (OCT), acquis dans le cortex de souris (n = 5) avant et après l’induction d’une photothrombose ciblée, sont analysés. Les graphes vasculaires informatiques sont exploités dans un simulateur 3D Monte-Carlo pour caractériser la réponse par résonance magnétique (MR), tout en considérant les effets des perturbations du champ magnétique causées par la désoxyhémoglobine, et l’advection et la diffusion des spins nucléaires. Le pipeline graphique proposé est validé sur des angiographies synthétiques et réelles acquises avec différentes modalités d’imagerie. Comparé à d’autres méthodes effectuées dans le milieu de la recherche, les expériences indiquent que le schéma proposé produit des taux d’erreur géométriques et topologiques amoindris sur divers angiogrammes. L’évaluation confirme également l’efficacité de la méthode proposée en fournissant des modèles représentatifs qui capturent tous les aspects anatomiques des structures vasculaires. Ensuite, afin de trouver des signatures de mAVC basées sur le signal IRM, la modélisation vasculaire proposée est exploitée pour quantifier le rapport de perte de signal intravoxel minimal lors de l’application de plusieurs directions de gradient, à des paramètres de séquence variables avec et sans ASL. Avec l’ASL, les résultats démontrent une dif-férence significative (p <0,05) entre le signal calculé avant et 3 semaines après la photothrombose. La puissance statistique a encore augmenté (p <0,005) en utilisant des angiogrammes capturés à la semaine suivante. Sans ASL, aucun changement de signal significatif n’est trouvé. Des rapports plus élevés sont obtenus à des intensités de champ magnétique plus faibles (par exemple, B0 = 3) et une lecture TE plus courte (<16 ms). Cette étude suggère que les mAVC pourraient être carac-térisés par des séquences ASL-DWI, et fournirait les informations nécessaires pour les validations expérimentales postérieures et les futurs essais comparatifs.----------ABSTRACT Cortical microvascular networks are responsible for carrying the necessary oxygen and energy substrates to our neurons. These networks react to the dynamic energy demands during neuronal activation through the process of neurovascular coupling. A key element in elucidating the role of the microvascular component in the brain is through computational modeling. However, the lack of fully-automated computational frameworks to model and characterize these microvascular net-works remains one of the main obstacles. Developing a fully-automated solution is thus substantial for further explorations, especially to quantify the impact of cerebrovascular malformations associ-ated with many cerebrovascular diseases. A common pathogenic outcome in a set of neurovascular disorders is the formation of microstrokes, i.e., micro occlusions in penetrating arterioles descend-ing from the pial surface. Recent experiments have demonstrated the impact of these microscopic events on brain function. Hence, it is of vital importance to develop a non-invasive and translatable approach to identify their presence in a clinical setting. In this thesis, a fully automatic processing pipeline to address the problem of microvascular anatom-ical modeling is proposed. The modeling scheme consists of a fully-convolutional neural network to segment microvessels, a 3D surface model generator and a geometry contraction algorithm to produce vascular graphical models with a single connected component. An improvement on this pipeline is developed later to alleviate the requirement of water-tight surface meshes as inputs to the graphing phase. The novel graphing scheme works with relaxed input requirements and intrin-sically captures vessel radii information, based on deforming geometric graphs constructed within vascular boundaries instead of surface meshes. A mechanism to decimate the initial graph struc-ture at each run is formulated with a convergence criterion to stop the process. A refinement phase is introduced to obtain final vascular models. The developed computational modeling is then ap-plied to simulate potential MRI signatures of microstrokes, combining arterial spin labeling (ASL) and multi-directional diffusion-weighted imaging (DWI). The hypothesis is driven based on recent observations demonstrating a radial reorientation of microvasculature around the micro-infarction locus during recovery in mice. Synthetic capillary beds, randomly- and radially oriented, and op-tical coherence tomography (OCT) angiograms, acquired in the barrel cortex of mice (n=5) before and after inducing targeted photothrombosis, are analyzed. The computational vascular graphs are exploited within a 3D Monte-Carlo simulator to characterize the magnetic resonance (MR) re-sponse, encompassing the effects of magnetic field perturbations caused by deoxyhemoglobin, and the advection and diffusion of the nuclear spins. The proposed graphing pipeline is validated on both synthetic and real angiograms acquired with different imaging modalities. Compared to other efficient and state-of-the-art graphing schemes, the experiments indicate that the proposed scheme produces the lowest geometric and topological error rates on various angiograms. The evaluation also confirms the efficiency of the proposed scheme in providing representative models that capture all anatomical aspects of vascular struc-tures. Next, searching for MRI-based signatures of microstokes, the proposed vascular modeling is exploited to quantify the minimal intravoxel signal loss ratio when applying multiple gradient di-rections, at varying sequence parameters with and without ASL. With ASL, the results demonstrate a significant difference (p<0.05) between the signal-ratios computed at baseline and 3 weeks after photothrombosis. The statistical power further increased (p<0.005) using angiograms captured at week 4. Without ASL, no reliable signal change is found. Higher ratios with improved significance are achieved at low magnetic field strengths (e.g., at 3 Tesla) and shorter readout TE (<16 ms). This study suggests that microstrokes might be characterized through ASL-DWI sequences, and provides necessary insights for posterior experimental validations, and ultimately, future transla-tional trials

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Geometric Variational Models for Inverse Problems in Imaging

This dissertation develops geometric variational models for different inverse problems in imaging that are ill-posed, designing at the same time efficient numerical algorithms to compute their solutions. Variational methods solve inverse problems by the following two steps: formulation of a variational model as a minimization problem, and design of a minimization algorithm to solve it. This dissertation is organized in the same manner. It first formulates minimization problems associated with geometric models for different inverse problems in imaging, and it then designs efficient minimization algorithms to compute their solutions. The minimization problem summarizes both the data available from the measurements and the prior knowledge about the solution in its objective functional; this naturally leads to the combination of a measurement or data term and a prior term. Geometry can play a role in any of these terms, depending on the properties of the data acquisition system or the object being imaged. In this context, each chapter of this dissertation formulates a variational model that includes geometry in a different manner in the objective functional, depending on the inverse problem at hand. In the context of compressed sensing, the first chapter exploits the geometric properties of images to include an alignment term in the sparsity prior of compressed sensing; this additional prior term aligns the normal vectors of the level curves of the image with the reconstructed signal, and it improves the quality of reconstruction. A two-step recovery method is designed for that purpose: first, it estimates the normal vectors to the level curves of the image; second, it reconstructs an image matching the compressed sensing measurements, the geometric alignment of normals, and the sparsity constraint of compressed sensing. The proposed method is extended to non-local operators in graphs for the recovery of textures. The harmonic active contours of Chapter 2 make use of differential geometry to interpret the segmentation of an image as a minimal surface manifold. In this case, geometry is exploited in both the measurement term, by coupling the different image channels in a robust edge detector, and in the prior term, by imposing smoothness in the segmentation. The proposed technique generalizes existing active contours to higher dimensional spaces and non-flat images; in the plane, it improves the segmentation of images with inhomogeneities and weak edges. Shape-from-shading is investigated in Chapter 3 for the reconstruction of a silicon wafer from images of printed circuits taken with a scanning electron microscope. In this case, geometry plays a role in the image acquisition system, that is, in the measurement term of the objective functional. The prior term involves a smoothness constraint on the surface and a shape prior on the expected pattern in the circuit. The proposed reconstruction method also estimates a deformation field between the ideal pattern design and the reconstructed surface, substituting the model of shape variability necessary in shape priors with an elastic deformation field that quantifies deviations in the manufacturing process. Finally, the techniques used for the design of efficient numerical algorithms are explained with an example problem based on the level set method. To this purpose, Chapter 4 develops an efficient algorithm for the level set method when the level set function is constrained to remain a signed distance function. The distance function is preserved by the introduction of an explicit constraint in the minimization problem, the minimization algorithm is efficient by the adequate use of variable-splitting and augmented Lagrangian techniques. These techniques introduce additional variables, constraints, and Lagrange multipliers in the original minimization problem, and they decompose it into sub-optimization problems that are simple and can be efficiently solved. As a result, the proposed algorithm is five to six times faster than the original algorithm for the level set method

Développement d’outils de vectorisation d’angiographies obtenues par microscopie 2-photons dans le contexte du vieillissement du cerveau

RÉSUMÉ Les pathologies affectant les petits vaisseaux de la neurovasculature sont-elles à l’origine des effets cognitifs qui apparaissent au cours du vieillissement? Pour répondre à cette question, il faut d’abord posséder des outils permettant d’extraire la microvasculature du néocortex à partir d’angiographies acquises par microscopie à fluorescence deux-photons. Une meilleure compréhension de l’évolution de la microvasculature du cerveau avec l’âge et de l’effet de ces modifications sur les fonctions des aires corticales constitue en effet une étape essentielle vers la mise en place de nouveaux biomarqueurs du vieillissement du cerveau. Des modèles réalistes de la vasculature du cerveau peuvent servir de base à des modélisations du débit sanguin et à la simulation de signal IRM. En utilisant des vasculatures vectorisées, de nouvelles voies de recherche pourront donc être explorées, dont l’effet de différents types de pathologies des petits vaisseaux sur le signal IRM dépendant du niveau d’oxygène sanguin (BOLD). L’objectif de ce projet de recherche est donc le développement d’une méthode de segmentation des vaisseaux sanguins et d’un outil d’interaction permettant de corriger et de modifier le réseau vasculaire extrait. Ces outils sont utilisés pour comparer la microvasculature du néocortex de rats provenant de deux cohortes d’âges différents formés de 12 jeunes rats (âge = 11-15 semaines) et 12 rats âgés (âge = 23-25 mois) de type Long-Evans. Ces méthodes ont été développées en utilisant principalement la plateforme de programmation MATLAB, le module de gestion de pipeline de traitement PSOM et l’outil de traitement d’image FIJI. La méthode est semi-automatique, nécessitant une correction manuelle des graphes extraits des angiographies après la segmentation. L’approche modulable adoptée permet l’ajout de nouvelles fonctions et de nouveaux outils, ce qui pourra améliorer sa robustesse et l’automatisation de l’extraction des vaisseaux sanguins. En analysant les masques des vasculatures issus du prétraitement des données, il a été montré que la densité des capillaires dans le néocortex sensorimoteur de rats Long-Evans diminue avec l’âge, en passant de pour les jeunes rats à pour les rats âgés, ce qui représente une baisse statistiquement significative de 20 %. Une analyse utilisant les graphes nettoyés semble également aller dans ce sens en montrant que la densité linéaire des vaisseaux décroît au cours du vieillissement. Cette mesure est liée aux capacités de perfusion de la vasculature, et pourrait indiquer que l’efficacité d’apport en nutriment et en oxygène décroît dans le néocortex sensorimoteur de rats au cours du vieillissement.----------ABSTRACT Are the conditions affecting the small vessels of the neurovasculature the cause of cognitive impairments that appear with aging? To answer this question, we must have tools to extract the neocortex microvasculature from angiograms acquired by two-photon fluorescence microscopy. A better understanding of the brain microvasculature evolution with age and the effect of those changes on the cortical areas functions is indeed an essential step towards the development of new biomarkers of brain aging. Realistic models of the brain vasculature can be used as a basis of blood flow modeling and to simulate MRI signal originating from these vessels. Using vectorized vasculatures, new research avenues can be explored, including the effect of different types of small vessels diseases on blood oxygen level-dependent (BOLD) MRI signal. The main objective of this research project is the development of a blood vessel segmentation method and of an interface to correct and modify the extracted vascular networks. These tools are used to compare the neocortex microvasculature of rats from two different age cohorts. These cohorts consist of 12 young (age = 11-15 weeks) and 12 old Long-Evans rats (age = 23-25 months). The tools have been developed using the MATLAB programming platform, the pipeline processing module PSOM and the image processing tool FIJI. The method is semi-automatic, requiring manual correction of the extracted angiogram graphs after segmentation. The modular approach allows the addition of new features and tools, which can improve the robustness and automation of the blood vessels extraction technique. By analyzing the vasculature masks obtained by the initial data preprocessing, it is found that the density of capillaries in the sensorimotor neocortex of Long-Evans rats decreases with age, from ρ = 6.8 ± 0.3 [%] in young rats to ρ = 5.4 ± 0.3 [%] in aged rats, which represents a statistically significant decrease of 20%. An analysis using the cleaned graphs also seems to go in this direction by showing that the linear density of vessels decreases with aging. This density is linked to the perfusion capacity of the vasculature, and may indicate that the efficiency of nutrient and oxygen distribution decreases with aging in rat’s sensorimotor neocortex

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Generalization of geometrical flux maximizing flow on Riemannian manifolds for improved volumetric blood vessel segmentation

Geometric flux maximizing flow (FLUX) is an active contour based method which evolves an initial surface to maximize the flux of a vector field on the surface. For blood vessel segmentation, the vector field is defined as the vectors specified by vascular edge strengths and orientations. Hence, the segmentation performance depends on the quality of the detected edge vector field. In this paper, we propose a new method for level set based segmentation of blood vessels by generalizing the FLUX on a Riemannian manifold (R-FLUX). We consider a 3D scalar image I(x) as a manifold embedded in the 4D space (x, I(x)) and compute the image metric by pullback from the 4D space, whose metric tensor depends on the vessel enhancing diffusion (VED) tensor. This allows us to devise a non-linear filter which both projects and normalizes the original image gradient vectors under the inverse of local VED tensors. The filtered gradient vectors pertaining to the vessels are less sensitive to the local image contrast and more coherent with the local vessel orientation. The method has been applied to both synthetic and real TOF MRA data sets. Comparisons are made with the FLUX and vesselsness response based segmentations, indicating that the R-FLUX outperforms both methods in terms of leakage minimization and thiner vessel delineation