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

Earth resources: A continuing bibliography with indexes (issue 51)

This bibliography lists 382 reports, articles and other documents introduced into the NASA scientific and technical information system between July 1 and September 30, 1986. Emphasis is placed on the use of remote sensing and geophysical instrumentation in spacecraft and aircraft to survey and inventory natural resources and urban areas. Subject matter is grouped according to agriculture and forestry, environmental changes and cultural resources, geodesy and cartography, geology and mineral resources, hydrology and water management, data processing and distribution systems, instrumentation and sensors, and economic analysis

Geometric rigidity estimates for isometric and conformal maps from S^(n-1) to R^n

In this thesis we study qualitative as well as quantitative stability aspects of isometric and conformal maps from S^(n-1) to R^n, when n is greater or equal to 2 or 3 respectively. Starting from the classical theorem of Liouville, according to which the isometry group of S^(n-1) is the group of its rigid motions and the conformal group of S^(n-1) is the one of its Möbius transformations, we obtain stability results for these classes of mappings among maps from S^(n-1) to R^n in terms of appropriately defined deficits. Unlike classical geometric rigidity results for maps defined on domains of R^n and mapping into R^n, not only an isometric\ conformal deficit is necessary in this more flexible setting, but also a deficit measuring how much the maps in consideration distort S^(n-1) in a generalized sense. The introduction of the latter is motivated by the classical Euclidean isoperimetric inequality

Singular Sets of Generalized Convex Functions

In the first part of the dissertation we prove that, under quite general conditions on a cost function $c$ in \RR^n, the Hausdorff dimension of the singular set of a $c$-concave function has dimension at most $n-1$. Our result applies for non-semiconcave cost functions and has applications in optimal mass transportation. The purpose of the second part of the thesis is to extend a result of Alberti and Ambrosio about singularity sets of monotone multivalued maps to the sub-Riemannian setting of Heisenberg groups. We prove that the $k$-th horizontal singular set of a $H$-monotone multivalued map of the Heisenberg group \HH^n, with values in \RR^{2n}, has Hausdorff dimension at most $2n+2-k$

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an N-marginal optimal transport problem with N equal marginals supported on Rd and with cost of the form ∑i≠j|xi−xj|−s. In this setting we determine the second-order term in the N→∞ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents 0<s<d. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman–Gregg decomposition [20], [26], extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer [36], who dealt with the second-order term for the Coulomb case s=1,d=3

Glosarium Matematika

273 p.; 24 cm

4D imaging in tomography and optical nanoscopy

Diese Dissertation gehört zu den Gebieten mathematische Bildverarbeitung und inverse Probleme. Ein inverses Problem ist die Aufgabe, Modellparameter anhand von gemessenen Daten zu berechnen. Solche Probleme treten in zahlreichen Anwendungen in Wissenschaft und Technik auf, z.B. in medizinischer Bildgebung, Biophysik oder Astronomie. Wir betrachten Rekonstruktionsprobleme mit Poisson Rauschen in der Tomographie und optischen Nanoskopie. Bei letzterer gilt es Bilder ausgehend von verzerrten und verrauschten Messungen zu rekonstruieren, wohingegen in der Positronen-Emissions-Tomographie die Aufgabe in der Visualisierung physiologischer Prozesse eines Patienten besteht. Standardmethoden zur 3D Bildrekonstruktion berücksichtigen keine zeitabhängigen Informationen oder Dynamik, z.B. Herzschlag oder Atmung in der Tomographie oder Zellmigration in der Mikroskopie. Diese Dissertation behandelt Modelle, Analyse und effiziente Algorithmen für 3D und 4D zeitabhängige inverse Probleme. This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics or astronomy. We mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In the latter case, the task is to reconstruct images from blurred and noisy measurements, whereas in positron emission tomography the task is to visualize physiological processes of a patient. In 3D static image reconstruction standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in tomography or cell motion in microscopy. This thesis is a treatise on models, analysis and efficient algorithms to solve 3D and 4D time-dependent inverse problems