Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Computer Analysis of Visual Properties of Curved Objects

A method is presented for the visual analysis of objects by computer. It is particularly well suited for opaque objects with smoothly curved surfaces. The method extracts information about the object's surface properties, including measures of its specularity, texture, and regularity. It also aids in determining the object's shape. The application of this method to a simple recognition task ??e recognition of fruit ?? discussed. The results on a more complex smoothly curved object, a human face, are also considered.

Surface Reflectance Recognition and Real-World Illumination Statistics

Humans distinguish materials such as metal, plastic, and paper effortlessly at a glance. Traditional computer vision systems cannot solve this problem at all. Recognizing surface reflectance properties from a single photograph is difficult because the observed image depends heavily on the amount of light incident from every direction. A mirrored sphere, for example, produces a different image in every environment. To make matters worse, two surfaces with different reflectance properties could produce identical images. The mirrored sphere simply reflects its surroundings, so in the right artificial setting, it could mimic the appearance of a matte ping-pong ball. Yet, humans possess an intuitive sense of what materials typically "look like" in the real world. This thesis develops computational algorithms with a similar ability to recognize reflectance properties from photographs under unknown, real-world illumination conditions. Real-world illumination is complex, with light typically incident on a surface from every direction. We find, however, that real-world illumination patterns are not arbitrary. They exhibit highly predictable spatial structure, which we describe largely in the wavelet domain. Although they differ in several respects from the typical photographs, illumination patterns share much of the regularity described in the natural image statistics literature. These properties of real-world illumination lead to predictable image statistics for a surface with given reflectance properties. We construct a system that classifies a surface according to its reflectance from a single photograph under unknown illuminination. Our algorithm learns relationships between surface reflectance and certain statistics computed from the observed image. Like the human visual system, we solve the otherwise underconstrained inverse problem of reflectance estimation by taking advantage of the statistical regularity of illumination. For surfaces with homogeneous reflectance properties and known geometry, our system rivals human performance.

Representation and Detection of Shapes in Images

We present a set of techniques that can be used to represent and detect shapes in images. Our methods revolve around a particular shape representation based on the description of objects using triangulated polygons. This representation is similar to the medial axis transform and has important properties from a computational perspective. The first problem we consider is the detection of non-rigid objects in images using deformable models. We present an efficient algorithm to solve this problem in a wide range of situations, and show examples in both natural and medical images. We also consider the problem of learning an accurate non-rigid shape model for a class of objects from examples. We show how to learn good models while constraining them to the form required by the detection algorithm. Finally, we consider the problem of low-level image segmentation and grouping. We describe a stochastic grammar that generates arbitrary triangulated polygons while capturing Gestalt principles of shape regularity. This grammar is used as a prior model over random shapes in a low level algorithm that detects objects in images.

Convergence Rates of Approximation by Translates

In this paper we consider the problem of approximating a function belonging to some funtion space Φ by a linear comination of n translates of a given function G. Ussing a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the erro is 0(1/n) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev tpe, in which the number of weak derivatives is required to be larger than the number of dimensions. We give results both for approximation in the L2 norm and in the Lc norm. The interesting feature of these results is that, thanks to the constructive nature of Jones" and Barron"s lemma, an iterative procedure is defined that can achieve this rate.

Exercise sheet 7 pdf

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Exercise sheet 7

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MA348exam 1999

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Exercise sheet 3 pdf

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Exercise sheet 3

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MA348exam 2000 pdf

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MA348exam 2000

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MA348exam 1999 pdf

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Factores asociados a recuperación visual deficiente después de cirugía exitosa para desprendimiento de retina regmatógeno

Introducción: Varias características pueden afectar el pronóstico visual después de resolver quirúrgicamente el desprendimiento de retina. Existen características no observables por el ojo humano por si solo pero si por tomografía óptica coherente que se relacionan con la recuperación visual. Objetivo: Describir las características clínicas y topográfica en los periodos pre y postquirúrgico de ojos que ha sufrido DR regmatógeno con compromiso macular y su relación con la calidad de recuperación visual después de una cirugía considerada exitosa desde el punto de vista anatómico. Materiales y métodos: Estudio descriptivo en el que se comparan algunas características en tres periodos perioeperatorios, uno antes y dos después de cirugía (3 y 6 meses) de 24 ojos con DRregmatógeno y compromiso macular intervenidos mediante retinopexia combinada con vitrectomía pars plana. Resultados: La recuperación visual mejor o igual que logMAR 0,397 (20/50) se dió en 41,7% de ojos y 16,7%. alcanzaron agudeza visual logMAR 0,301 (20/40). Cinco ojos no alcanzaron una ganancia de líneas de visión mayor a cinco. El líquido submacular ausente se observó en la mayoría de ojos que recuperaron más de cinco líneas al igual que aquellos con elipsoide conservado. La regularidad del neuroepitelio y el edema en el periodo posquirúrgico no mostraron comportamientos claros respecto a recuperación visual al igual que la altura del desprendimiento y el número de cuadrantes afectados. Una mejor recuperación visual fue más frecuente en aquellos con menos de cinco semanas de desprendimiento de retina. Conclusiones: El retraso menor a cinco semanas en la resolución del desprendimiento de retina, la conservación del elipsoide y la ausencia de líquido submacular en el periodo postquirúrgico se observó más frecuentemente en ojos con mejor recuperación visual.