Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces

The human cerebral cortex is marked by great complexity as well as substantial dynamic changes during early postnatal development. To obtain a fairly comprehensive picture of its age-induced and/or disorder-related cortical changes, one needs to match cortical surfaces to one another, while maximizing their anatomical alignment. Methods that geodesically shoot surfaces into one another as currents (a distribution of oriented normals) and varifolds (a distribution of non-oriented normals) provide an elegant Riemannian framework for generic surface matching and reliable statistical analysis. However, both conventional current and varifold matching methods have two key limitations. First, they only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the orientations of the inherently convoluted cortical sulcal and gyral folds. Second, the ‘conversion’ of a surface into a current or a varifold operates at a fixed scale under which geometric surface details will be neglected, which ignores the dynamic scales of cortical foldings. To overcome these limitations and improve varifold-based cortical surface registration, we propose two different strategies. The first strategy decomposes each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information of the orientation of cortical folding and better characterization of the complex cortical geometry. The second strategy explores the informative cortical geometric features to perform a dynamic-scale measurement of the cortical surface that depends on the local surface topography (e.g., principal curvature), thereby we introduce the concept of a topography-based dynamic-scale varifold. We tested the proposed varifold variants for registering 12 pairs of dynamically developing cortical surfaces from 0 to 6 months of age. Both variants improved the matching accuracy in terms of closeness to the target surface and the goodness of alignment with regional anatomical boundaries, when compared with three state-of-the-art methods: (1) diffeomorphic spectral matching, (2) conventional current-based surface matching, and (3) conventional varifold-based surface matching

ANALYSIS OF GEOMETRIC SHAPES WITH VARIFOLD REPRESENTATION

This thesis is concerned with the theory and applications of varifolds to the representation, approximation, and diffeomorphic registration of shapes. Originating from geometric measure theory, the theory of varifolds provides a convenient way to represent geometric shapes like curves, surfaces or, submanifolds both in continuous and discrete settings. Previous works in shape analysis have made use of this representation as a surrogate to design numerically tractable fidelity terms for curve and surface registration problems. So far, these approaches have primarily focused on processing submanifold data and were not designed to handle more general structures. The varifold representation however provides a very flexible framework that is not restricted to submanifolds but its generality has not yet been exploited to its full extent in shape analysis. In this work, we take a step in this direction by considering deformations acting on general varifolds, and propose a mathematical model for diffeomorphic registration of varifolds under a natural group action which we formulate as an optimal control problem. This new framework allows us to tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. Varifold matching frameworks heavily rely on the kernel metrics defined on the varifolds spaces. However, the properties of this type of metrics and their relationships with the classical metrics/topologies on measure spaces have not been investigated thoroughly yet. In this work, we study in detail the construction of kernel metrics on the space of varifold and the resulting topological properties of those metrics. Based on these results, we address the problem of optimal finite approximations (quantization) for kernel metrics, propose a projection-based approach for varifold representation, and show a $\Gamma$-convergence property for the discrete registration functionals. In the last part of this thesis, we tackle the imbalanced shape matching problems, namely the situation in which the source and target shapes involve considerable variations of mass or density which cannot be entirely described by diffeomorphic transformations. We extend our varifold matching model by augmenting the diffeomorphic component with a global or local density changes. Based on the optimality conditions provided by the Pontryagin maximum principle, we derive a shooting algorithm to numerically estimate solutions and illustrate the practical interest of this model for several types of geometric data such as fiber bundles with inconsistent fiber densities or partially observed and incomplete surfaces

Elastic shape analysis of geometric objects with complex structures and partial correspondences

In this dissertation, we address the development of elastic shape analysis frameworks for the registration, comparison and statistical shape analysis of geometric objects with complex topological structures and partial correspondences. In particular, we introduce a variational framework and several numerical algorithms for the estimation of geodesics and distances induced by higher-order elastic Sobolev metrics on the space of parametrized and unparametrized curves and surfaces. We extend our framework to the setting of shape graphs (i.e., geometric objects with branching structures where each branch is a curve) and surfaces with complex topological structures and partial correspondences. To do so, we leverage the flexibility of varifold fidelity metrics in order to augment our geometric objects with a spatially-varying weight function, which in turn enables us to indirectly model topological changes and handle partial matching constraints via the estimation of vanishing weights within the registration process. In the setting of shape graphs, we prove the existence of solutions to the relaxed registration problem with weights, which is the main theoretical contribution of this thesis. In the setting of surfaces, we leverage our surface matching algorithms to develop a comprehensive collection of numerical routines for the statistical shape analysis of sets of 3D surfaces, which includes algorithms to compute Karcher means, perform dimensionality reduction via multidimensional scaling and tangent principal component analysis, and estimate parallel transport across surfaces (possibly with partial matching constraints). Moreover, we also address the development of numerical shape analysis pipelines for large-scale data-driven applications with geometric objects. Towards this end, we introduce a supervised deep learning framework to compute the square-root velocity (SRV) distance for curves. Our trained network provides fast and accurate estimates of the SRV distance between pairs of geometric curves, without the need to find optimal reparametrizations. As a proof of concept for the suitability of such approaches in practical contexts, we use it to perform optical character recognition (OCR), achieving comparable performance in terms of computational speed and accuracy to other existing OCR methods. Lastly, we address the difficulty of extracting high quality shape structures from imaging data in the field of astronomy. To do so, we present a state-of-the-art expectation-maximization approach for the challenging task of multi-frame astronomical image deconvolution and super-resolution. We leverage our approach to obtain a high-fidelity reconstruction of the night sky, from which high quality shape data can be extracted using appropriate segmentation and photometric techniques

Mechanics of the Developing Brain: From Smooth-walled Tube to the Folded Cortex

Over the course of human development, the brain undergoes dramatic physical changes to achieve its final, convoluted shape. However, the forces underlying every cinch, bulge, and fold remain poorly understood. This doctoral research focuses on the mechanical processes responsible for early (embryonic) and late (preterm) brain development. First, we examine early brain development in the chicken embryo, which is similar to human at these stages. Research has primarily focused on molecular signals to describe morphogenesis, but mechanical analysis can also provide important insights. Using a combination of experiments and finite element modeling, we find that actomyosin contraction is responsible for initial segmentation of the forebrain. By considering mechanical forces from the internal and external environment, we propose a role for mechanical feedback in maintaining these segments during subsequent inflation and bending. Next, we extend our analysis to division of right and left cerebral hemispheres. In this case, we discover that morphogen signals and mechanical feedback act synergistically to shape the hemispheres. In human, cerebral hemispheres go on to form complex folds through a mechanical process that involves rapid expansion of the cortical surface. However, the spatiotemporal dynamics of cortical growth remain unknown in human. Here, we develop a novel strain energy minimization approach to measure regional growth in complex surfaces. By considering brain surfaces of preterm subjects, reconstructed from magnetic resonance imaging (MRI), this analysis reveals distinct patterns of cortical growth that evolve over the third trimester. This information provides a comprehensive view of cortical growth and folding, connecting what is known about patterns of development at the cellular and folding scales. Abnormal brain morphogenesis can lead to serious structural defects and neurological disorders such as epilepsy and autism. By integrating mechanics, biology, and neuroimaging, we gain a more complete understanding of brain development. By studying physical changes from the simple, microscopic embryo to the macroscopic, folded cortex, we gain insight into relevant biological and physical mechanisms across developmental stages

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

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

White Matter Fiber Segmentation Using Functional Varifolds

International audienceThe extraction of fibers from dMRI data typically produces a large number of fibers, it is common to group fibers into bundles. To this end, many specialized distance measures, such as MCP, have been used for fiber similarity. However, these distance based approaches require point-wise correspondence and focus only on the geometry of the fibers. Recent publications have highlighted that using microstructure measures along fibers improves tractography analysis. Also, many neurodegenerative diseases impacting white matter require the study of microstructure measures as well as the white matter geometry. Motivated by these, we propose to use a novel computational model for fibers, called functional varifolds, characterized by a metric that considers both the geometry and microstructure measure (e.g. GFA) along the fiber pathway. We use it to cluster fibers with a dictionary learning and sparse coding-based framework, and present a preliminary analysis using HCP data