112 research outputs found

    Left-invariant evolutions of wavelet transforms on the Similitude Group

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    Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

    Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging

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    Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well-defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations U:Rd⋊Sd−1→RU:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R} defined on the extended space of positions and orientations, which we relate to data on the roto-translation group SE(d)SE(d), d=2,3d=2,3. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in SE(d)SE(d). These curve fits minimize first or second order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on SE(d)SE(d). We include these gauge frames in differential invariants and crossing preserving PDE-flows acting on extended data representation UU and we show their advantage compared to the standard left-invariant frame on SE(d)SE(d). Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores

    T-folds, doubled geometry, and the SU(2) WZW model

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    The SU(2) WZW model at large level N can be interpreted semiclassically as string theory on S^3 with N units of Neveu-Schwarz H-flux. While globally geometric, the model nevertheless exhibits an interesting doubled geometry possessing features in common with nongeometric string theory compactifications, for example, nonzero Q-flux. Therefore, it can serve as a fertile testing ground through which to improve our understanding of more exotic compactifications, in a context in which we have a firm understanding of the background from standard techniques. Three frameworks have been used to systematize the study of nongeometric backgrounds: the T-fold construction, Hitchin's generalized geometry, and fully doubled geometry. All of these double the standard description in some way, in order to geometrize the combined metric and Neveu Schwarz B-field data. We present the T-fold and fully doubled descriptions of WZW models, first for SU(2) and then for general group. Applying the formalism of Hull and Reid-Edwards, we indeed recover the physical metric and H-flux of the WZW model from the doubled description. As additional checks, we reproduce the abelian T-duality group and known semiclassical spectrum of D-branes.Comment: 69 pages; uses amslatex; v4 minor revision

    Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

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    We introduce a data-driven version of the plus Cartan connection on the homogeneous space M2\mathbb{M}_2 of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on M2\mathbb{M}_{2}. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on M2\mathbb{M}_2 that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at GitHub (https://github.com/NickyvdBerg/DataDrivenTracking)

    A constitutive model of human esophagus tissue with application for the treatment of stenosis

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    This dissertation is a research about the mechanical behavior of the human esophagus. This work is intended to be applied in the treatment of stenosis and other esophageal diseases that frequently require a procedure of forced dilation, that involves high pressures on the esophagus wall. This study proposes a constitutive model to simulate forced dilatations. This study includes the experimental characterization of the mechanical behavior of human esophagus. In addition, some theoretical questions and experimental issues were addressed and solved. This dissertation summarizes the previous work on esophageal tissues by other authors. Additionally, a short explanation about the microcontinuum theory developed in the last decades is given, as well as a summary of the general theory of nonlinear hyperelastic constitutive models (with large deformation). This study required extensive testing of esophageal tissue in order to characterize the in vitro mechanical behavior. The testing included mainly tensile tests and complementary inflation tests. Optical motion track analysis was used for accurate computation of the strains in the tissue. The results of the tests were used for adjusting the mechanical properties that characterize the mechanical behavior of esophagus in the proposed models of the literature. The statistical analysis of the data revealed, some significant correlations between anthropometric factors, such as the body mass index, and some mechanical properties were found in the analysis of the data. The typical values of the mechanical properties were used to perform some numerical finite element simulations based on the proposed models. In addition, a number of theoretical results were obtained concerning the residual stress and the predictions of statistical mechanics for a system of collagenous fibers inside a soft tissue. The main result is a constitutive non-linear microstretch anisotropic hyperelastic constitutive model with large deformations (and with residual stresses) to characterize the multi-layered tissue of human esophagus. This model is suitable for numerical simulation.Esta tesis es una investigación sobre el comportamiento mecánico del esófago humano. Este trabajo se pensó para ser aplicado al tratamiento de la estenosis y otras afecciones esofágicas que, con frecuencia, requieren dilatación forzada, lo que supone altas presiones sobre la pared esofágica. Este estudio propone un modelo constitutivo para simular estas dilataciones forzadas. Este estudio incluye la caracterización experimental del comportamiento mecánico del esófago. Además, se han planteado algunas cuestiones teóricas y experimentales que han sido resueltas. Esta tesis resume el trabajo previo sobre tejido esofágico de otros investigadores. Además, se da una pequeña explicación sobre la teoría del medio microntinuo desarrollada en las últimas décadas y un breve resumen de la teoría general de modelos constitutivos hiperelásticos no-lineales (con grandes deformaciones). El estudio requirió experimentación de tejido para caracterizar el comportamiento in vitro. Los test incluyeron test de tracción y test de inflado. Se empleó rastreo óptico para un cálculo adecuado de la deformación. Los resultados de los test permitieron encontrar las propiedades mecánicas según dos modelos de esófago de la literatura. El análisis de los datos rebeló, algunas correlaciones significativas entre factores como el índice de masa corporal y algunas propiedades mecánicas. Los valores típicos de las propiedades fueron usados para algunas simulaciones numéricas basadas en los modelos propuestos. Además se han obtenido algunos resultados teóricos sobre la tensión residual y las predicciones de la mecánica estadística para un sistema de fibras de colágeno del tejido. El principal resultado es un modelo constitutivo de microestiramiento anisótropo no-lineal e hiperelástico para caracterizar el esófago Este modelo es adecuado para la computación numérica.Postprint (published version

    Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy - Geometrical and Statistical Methods for Modelling Biological Shape Variability

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    International audienceComputational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model the organ development across time (growth or aging), to establish their variability, and to correlate this variability information with other functional, genetic or structural information. The Mathematical Foundations of Computational Anatomy (MFCA) workshop aims at fostering the interactions between the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum for the exchange of the theoretical ideas and aims at being a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the successful rst edition of this workshop in 20061 and second edition in New-York in 20082, the third edition was held in Toronto on September 22 20113. Contributions were solicited in Riemannian and group theoretical methods, geometric measurements of the anatomy, advanced statistics on deformations and shapes, metrics for computational anatomy, statistics of surfaces, modeling of growth and longitudinal shape changes. 22 submissions were reviewed by three members of the program committee. To guaranty a high level program, 11 papers only were selected for oral presentation in 4 sessions. Two of these sessions regroups classical themes of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session of 5 papers addresses more application oriented works on computational anatomy

    Arithmetic and Hyperbolic Structures in String Theory

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    This monograph is an updated and extended version of the author's PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of a spacelike singularity (the "BKL-limit"). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be described in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of the theory. Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are described by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by automorphic forms on the double quotient G(Z)\G/K. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on non-holomorphic Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also show how these techniques can be applied to hypermultiplet moduli spaces in type II Calabi-Yau compactifications, and we provide a detailed analysis for the universal hypermultiplet.Comment: 346 pages, updated and extended version of the author's PhD thesi
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