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Location-aware devices are one of the examples of variety of systems that can provide trajectory data. The formal definition of a trajectory is the path of a moving object in space as a function of time. Surveillance systems can now automatically detect moving objects and provide a useful dataset for further analysis. Clustering moving objects in a given scene can provide vital information about the trajectory patterns and outliers. The trajectory of an object may contain extended data at each position where the object was detected such as size, colour, etc. The focus of this work is to find an efficient trajectory clustering solution given the most fundamental trajectory data, namely position and time. The main challenge of clustering trajectory data is to handle the length of a single trajectory. The length of a trajectory can be extremely long in some cases. Hence it may cause problems to keep trajectories in main memory or it may be very inefficient to process them. Preprocessing trajectories and simplifying them will help minimize the effects of such issues. We will use some algorithms taken from literature in conjunction with some of our own algorithms in order to cluster trajectories in an efficient manner. In an attempt to accomplish this, we have designed a representation of a trajectory Furthermore, we have designed and implemented algorithms to simplify and evaluate distances between these trajectories. Moreover, we proved that our distance function obeys triangulation properties which is beneficial for clustering algorithms. Our distance function is a variation of the Fréchet distance proposed in 1906 by Maurice René Fréchet. Additionally, we will illustrate how our work can be integrated with an incremental clustering algorithm to cluster trajectories.

Existence of basis in locally convex space has been an important problem in functional analysis for more than 40 years. In this thesis the conditions for the existence of basis are examined. These thesis consist of three parts. The first part is about the exterior interpolative conditions. The second part deals with the inner interpolative conditions on nuclear frechet space. These are sufficient conditions on existence of basis. In the last part, it is shown that for a regular nuclear K&ouml / the space the inner interpolative conditions are satisfied and moreover another type of inner interpolative conditions are introduced.

In this thesis I present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method and does not employ the constant elements and require the mesh generator to match the earth’s surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. It is shown that it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissae number) and the sub-domain size. The higher order or smaller the subdomain size employed, the more accurate the solution. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography are presented and found to yield results comparable to finite element solutions involving a dense mesh. The thesis also presents for the first time explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and is developed both from a formal perturbation analysis and by means of a numerical (finite element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry; these are given analytically. The Fréchet derivative expressions are given for both the 2.5-D and the 3-D problem using both constant point and constant block model parameterisations. Special cases like the isotropic earth and tilted transversely isotropic (TTI) media are shown to emerge from the general solutions. Numerical examples are presented for the various sensitivities as functions of the dip angle and strike of the plane of stratification in uniform TTI media. In addition, analytic solutions are derived for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic homogeneous medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ₁,dG/dσ₁,dG/dθ₀ ) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2009

Orientadores: Daniela Mariz Silva Vieira, Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T00:22:09Z (GMT). No. of bitstreams: 1 Nachtigall_Cicero_D.pdf: 625263 bytes, checksum: 7dc27b9956ffbb6e717aa95776b7b8c8 (MD5) Previous issue date: 2011 / Resumo: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital. / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations. / Doutorado / Matematica / Doutor em Matemática

This thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Frechet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.

En anatomie computationnelle, on suppose que les formes d'organes sont issues des déformations d'un template commun. Les données peuvent être des images ou des surfaces d'organes, les déformations peuvent être des difféomorphismes. Pour estimer le template, on utilise souvent un algorithme appelé «max-max» qui minimise parmi tous les candidats, la somme des carrées des distances après recalage entre les données et le template candidat. Le recalage est l'étape de l'algorithme qui trouve la meilleure déformation pour passer d'une forme à une autre. Le but de cette thèse est d'étudier cet algorithme max-max d'un point de vue mathématique. En particulier, on prouve que cet algorithme est inconsistant à cause du bruit. Cela signifie que même avec un nombre infini de données et avec un algorithme de minimisation parfait, on estime le template original avec une erreur non nulle. Pour prouver l'inconsistance, on formalise l'estimation du template. On suppose que les déformations sont des éléments aléatoires d'un groupe qui agit sur l'espace des observations. L'algorithme étudié est interprété comme le calcul de la moyenne de Fréchet dans l'espace des observations quotienté par le groupe des déformations. Dans cette thèse, on prouve que l'inconsistance est dû à la contraction de la distance quotient par rapport à la distance dans l'espace des observations. De plus, on obtient un équivalent de biais de consistance en fonction du niveau de bruit. Ainsi, l'inconsistance est inévitable quand le niveau de bruit est suffisamment grand. / In computational anatomy, organ shapes are assumed to be deformation of a common template. The data can be organ images but also organ surfaces, and the deformations are often assumed to be diffeomorphisms. In order to estimate the template, one often uses the max-max algorithm which minimizes, among all the prospective templates, the sum of the squared distance after registration between the data and a prospective template. Registration is here the step of the algorithm which finds the best deformation between two shapes. The goal of this thesis is to study this template estimation method from a mathematically point of view. We prove in particular that this algorithm is inconsistent due to the noise. This means that even with an infinite number of data, and with a perfect minimization algorithm, one estimates the original template with an error. In order to prove inconsistency, we formalize the template estimation: deformations are assumed to be random elements of a group which acts on the space of observations. Besides, the studied algorithm is interpreted as the computation of the Fréchet mean in the space of observations quotiented by the group of deformations. In this thesis, we prove that the inconsistency comes from the contraction of the distance in the quotient space with respect to the distance in the space of observations. Besides, we obtained a Taylor expansion of the consistency bias with respect to the noise level. As a consequence, the inconsistency is unavoidable when the noise level is high.

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Ordinary differential equations (ODEs) play an important role in mathematics. Although intrinsically, the setting for describing ODEs is the continuous framework, where differential operators are considered as maps from one function space to another, common numerical algorithms for ODEs discretise problems early on in the solution process. This thesis is about continuous analogues of such discrete algorithms for the numerical solution of ODEs. This thesis shows how Newton's method for finite dimensional system can be generalised to function spaces, where it is known as Newton-Kantorovich iteration. It presents affine invariant damping strategies for increasing the chance of convergence for the Newton-Kantorovich iteration. The derivatives required in this continuous setting are Fréchet derivatives, the continuous analogue of Jacobian matrices. In this work, we present how automatic differentiation techniques can be applied to compute Fréchet derivatives. We introduce chebop, a Matlab solver for nonlinear boundary-value problems, which combines damped Newton iteration in function space and automatic Fréchet differentiation. By proving that affine operators have constant Fréchet derivatives, it is demonstrated how automatic linearity detection of computed quantities can be implemented. This is valuable for black-box solvers, which can use the information to determine whether an iteration scheme has to be employed for solving a problem. Like nonlinear systems of equations, nonlinear boundary-value problems can have multiple solutions. This thesis present two techniques for obtaining multiple solutions of operator equations: deflation and path-following. An algorithm combining the two techniques is proposed.

Le but initial de cette thèse était d’étudier les espaces de superlacets, version géométrique des espaces de supercordes en Physique. Le point de départ était alors d’étendre les résultats de classifications de l’article de Oleg Mokhov : Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems au cadre de la supergéométrie. Dans cet article l’auteur établit une classification des formes symplectiques locales homogènes d’ordre 0, 1 et 2 sur l’espace des lacets LM = C1(S1;M) à partir d’objets géométriques sur la variété différentiable M. Dans cette thèse, on remplace la variété M par une supervariété Mpjq et le cercle S1 par un supercercle S1jn et l’on étudie l’espace des morphismes de supervariétésMor(S1jn;Mpjq). Dans les deux premières parties, l’on définit les structures géométriques classiques et super des espaces de superlacets. Pour ce faire, l’on se restreint aux deux supercercles S1j1 et en s’inspirant des travaux sur LM, l’on détermine une structure de variété de Fréchet des espaces de superlacets SLM = Mor(S1j1;M). Puis l’on introduit la structure super qui nous a semblé la plus naturelle sur SLM en terme de faisceaux. Afin de pouvoir travailler en coordonnées, l’on introduit la structure super par un autre point de vue en considérant l’espace de superlacets SLM comme le foncteur de points SLM. De plus, en interprétant les calculs de Mokhov en terme de jets, ceci nous permet d’une part d’apporter une justification rigoureuse aux-dits calculs et d’autre part, d’obtenir une généralisation directe des méthodes de calculs en coordonnées ("à la physicienne"). Le troisième chapitre expose les résultats de classification obtenus. Comme dans le cas classique, on obtient un théorème de dépendance limitée de l’ordre des jets qui interviennent dans les formes d’ordre 0 et 1. Puis, on obtient une classification des formes d’ordre 0 au moyen de formes différentielles sur la supervariété Mpjq. Une classification des formes homogènes d’ordre 1 et 2 au moyen de métriques Riemaniennes et de connexions sur Mpjq. Enfin le quatrième chapitre est consacré à la généralisation des résultats d’un autre article de O. Mokhov : Complex homogeneous forms on loop spaces of smooth manifolds and their cohomology groups. De par la présence de la variable impaire, on précise tout d’abord la définition des formes homogènes locales sur SLM, puis on démontre que muni de la différentielle extérieure, l’espace des formes homogènes sur SLM d’ordre m 2 N donné définit un complexe. On calcule alors complètement les espaces de cohomologie pour les ordres m = 0 et 1, partiellement pour les ordres 2 et 3 et on explicite ainsi les formes symplectiques exactes obtenues au troisième chapitre. / The goal of the thesis was to study superloopspaces, the geometric version of superstrings in Physics, by extending the classification results contained in Oleg Mokov’s paper : Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems to the supergeometric setting. In it, lies the classification of local homogeneous symplectic forms of order 0, 1 and 2 on the loopspace LM = C1(S1;M) by means of geometric objects on the manifold M. In this thesis, the manifold M becomes a supermanifold Mpjq, the circle S1 becomes a supercircle S1jn and we consider the superloopspace as the space of morphisms of supermanifolds Mor(S1jn;Mpjq). In the two first chapters, we look at the classical and super geometric structures of the superloopspaces. To do this, we restrict ourselves to the two supercircles S1j1 and using the previous works on LM, we define a Fréchet manifold structure on the superloopspaces SLM = Mor(S1j1;M). Then we bring in what we consider as the most natural superstructure on SLM by means of sheaves. In order to work with coordinates, we adopt another point of view considering SLM as the functor of points SLM. Moreover, rewriting Mokhov results in terms of jets allows us to give a rigorous proof of those calculations and also to extend right away the methods of calculations in coordinates. The third chapter contains the new classification results we obtained. Similarly to the classical case, we first show that the order of the jets in the forms of order 0 and 1 is bounded. Then we give the complete classification of the symplectics forms of order 0 by means of differential forms on the manifold Mpjq and of homogeneous symplectics forms of order 1 and 2 using Riemannian metrics and connections on Mpjq. Finally, the fourth chapter is devoted to extending the cohomology results of an other Mokhov’s article : Complex homogeneous forms on loop spaces of smooth manifolds and their cohomology groups. We first discuss the dependance of the odd variable in the homogeneous forms on SLM, and show that with the exterior derivative, the space of homogeneous forms on SLM of a given order m 2 N is a complex. We then calculate the cohomological spaces, completely for the order m = 0 and 1, partially for the order 2 and 3 and we identify the exact forms amongst those of the third chapter.