Dissertations / Theses on the topic 'Differential equations, Nonlinear Data processing'

Solving nonlinear systems of equations is a central problem in numerical analysis, with enormous significance for science and engineering. A special case, sparse systems of equations, occurs frequently in various applications. Sparsity occurs in the analysis of many types of complex systems because of the local nature of the dependence or connectivity among system components. One such system which may be modeled by a nonlinear sparse set of equations is the power system load flow analysis. This is a mathematical study performed by electrical utilities to monitor the electrical power system. The data from system components are used to create a set of nonlinear equations. These equations are then solved to find the voltage profile of the power network. With these data, control and security of the power system are achieved. Solving problems of this type is very time consuming when the system is large. This dissertation proposes a highly parallel computer architecture for solving large sets of nonlinear sparse equations. The goal of this architecture is to reduce the processing time required to solve this type of problem. In particular, the load flow problem is analyzed and implemented on this architecture. For the FPL network, the speed is increased by a factor of about 2000.

This thesis is divided into five chapters. The scope of problems considered is defined in chapter I. Next, chapter II provides background material on image processing with partial differential equations and a review of prior work in the field. Chapter III covers the medical imaging portion of the research; the key contribution is a control-based algorithm for interactive image segmentation. Applications of the feedback-augmented level set method to fracture reconstruction and surgical planning are shown. Problems in vision-based control are considered in Chapters IV and V. A method of improving performance in closed-loop target tracking using level set segmentation is developed, with unmanned aerial vehicle or next-generation missile guidance being the primary applications of interest. Throughout this thesis, the two application types are connected into a unified viewpoint of open-loop systems that are augmented by exogenous data.

In this thesis we tackle two problems, namely, the data interpolation prob- lem in the context of depth computation both for images and for videos, and the problem of the estimation of the apparent movement of objects in image sequences. The rst problem deals with completion of depth data in a region of an image or video where data are missing due to occlusions, unreliable data, damage or lost of data during acquisition. In this thesis we tackle it in two ways. First, we propose a non-local gradient-based energy which is able to complete planes locally. We consider this model as an extension of the bilateral lter to the gradient domain. We have successfully evaluated our model to complete synthetic depth images and also incomplete depth maps provided by a Kinect sensor. The second approach to tackle the problem is an experimental study of the Biased Absolutely Minimizing Lipschitz Extension (biased AMLE in short) for anisotropic interpolation of depth data to big empty regions without informa- tion. The AMLE operator is a cone interpolator, but the biased AMLE is an exponential cone interpolator which makes it more addapted to depth maps of real scenes that usually present soft convex or concave surfaces. Moreover, the biased AMLE operator is able to expand depth data to huge regions. By con- sidering the image domain endowed with an anisotropic metric, the proposed method is able to take into account the underlying geometric information in order not to interpolate across the boundary of objects at di erent depths. We have proposed a numerical model to compute the solution of the biased AMLE which is based on the eikonal operators. Additionally, we have extended the proposed numerical model to video sequences. The second problem deals with the motion estimation of the objects in a video sequence. This problem is known as the optical ow computation. The Optical ow problem is one of the most challenging problems in computer vision. Traditional models to estimate it fail in presence of occlusions and non-uniform illumination. To tackle these problems we proposed a variational model to jointly estimate optical ow and occlusion. Moreover, the proposed model is able to deal with the usual drawback of variational methods in dealing with fast displacements of objects in the scene which are larger than the object it- self. The addition of a term that balance gradient and intensities increases the robustness to illumination changes of the proposed model. The inclusions of a supplementary matches given by exhaustive search in speci cs locations helps to follow large displacements.
En esta tesis se abordan dos problemas: interpolación de datos en el contexto del cálculo de disparidades tanto para imágenes como para video, y el problema de la estimación del movimiento aparente de objetos en una secuencia de imágenes. El primer problema trata de la completación de datos de profundidad en una región de la imagen o video dónde los datos se han perdido debido a oclusiones, datos no confiables, datos dañados o pérdida de datos durante la adquisición. En esta tesis estos problemas se abordan de dos maneras. Primero, se propone una energía basada en gradientes no-locales, energía que puede (localmente) completar planos. Se considera este modelo como una extensión del filtro bilateral al dominio del gradiente. Se ha evaluado en forma exitosa el modelo para completar datos sintéticos y también mapas de profundidad incompletos de un sensor Kinect. El segundo enfoque, para abordar el problema, es un estudio experimental del biased AMLE (Biased Absolutely Minimizing Lipschitz Extension) para interpolación anisotrópica de datos de profundidad en grandes regiones sin información. El operador AMLE es un interpolador de conos, pero el operador biased AMLE es un interpolador de conos exponenciales lo que lo hace estar más adaptado a mapas de profundidad de escenas reales (las que comunmente presentan superficies convexas, concavas y suaves). Además, el operador biased AMLE puede expandir datos de profundidad a regiones grandes. Considerando al dominio de la imagen dotado de una métrica anisotrópica, el método propuesto puede tomar en cuenta información geométrica subyacente para no interpolar a través de los límites de los objetos a diferentes profundidades. Se ha propuesto un modelo numérico, basado en el operador eikonal, para calcular la solución del biased AMLE. Adicionalmente, se ha extendido el modelo numérico a sequencias de video. El cálculo del flujo óptico es uno de los problemas más desafiantes para la visión por computador. Los modelos tradicionales fallan al estimar el flujo óptico en presencia de oclusiones o iluminación no uniforme. Para abordar este problema se propone un modelo variacional para conjuntamente estimar flujo óptico y oclusiones. Además, el modelo propuesto puede tolerar, una limitación tradicional de los métodos variacionales, desplazamientos rápidos de objetos que son más grandes que el tamaño objeto en la escena. La adición de un término para el balance de gradientes e intensidades aumenta la robustez del modelo propuesto ante cambios de iluminación. La inclusión de correspondencias adicionales (obtenidas usando búsqueda exhaustiva en ubicaciones específicas) ayuda a estimar grandes desplazamientos.

Cette thèse présente des travaux sur l’étude et la calibration de modèles d’équations aux dérivées partielles pour la croissance tumorale. La première partie porte sur l’analyse d’un modèle de croissance tumorale pour le cas de métastases au foie de tumeurs gastro-intestinales (GIST). Le modèle est un système d’équations aux dérivées partielles couplées et prend en compte plusieurs traitements dont un traitement anti-angiogénique. Le modèle permet de reproduire des données cliniques. La première partie de ce travail concerne la preuve d’existence/unicité de la solution du modèle. La seconde partie du travail porte sur l’étude du comportement asymptotique de la solution du modèle lorsqu’un paramètre du modèle, décrivant la capacité de la tumeur à évacuer la nécrose, converge vers 0. La seconde partie de la thèse concerne le développement d’un modèle de croissance pour des sphéroïdes tumoraux ainsi que sur la calibration de ce modèle à partir de données expérimentales in vitro. L’objectif est de développer un modèle permettant de reproduire quantitativement la distribution des cellules proliférantes à l’intérieur d’un sphéroïde en fonction de la concentration en nutriments. Le travail de modélisation et de calibration du modèle a été effectué à partir de données expérimentales permettant d’obtenir la répartition spatiale de cellules proliférantes dans un sphéroïde tumoral
In this thesis, we present several works on the study and the calibration of partial differential equations models for tumor growth. The first part is devoted to the mathematical study of a model for tumor drug resistance in the case of gastro-intestinal tumor (GIST) metastases to the liver. The model we study consists in a coupled partial differential equations system and takes several treatments into account, such as a anti-angiogenic treatment. This model is able to reproduce clinical data. In a first part, we present the proof of the existence/uniqueness of the solution to this model. Then, in a second part, we study the asymptotic behavior of the solution when a parameter of this model, describing the capacity of the tumor to evacuate the necrosis, goes to 0. In the second part of this thesis, we present the development of model for tumor spheroids growth. We also present the model calibration thanks to in vitro experimental data. The main objective of this work is to reproduce quantitatively the proliferative cell distribution in a spheroid, as a function of the concentration of nutrients. The modeling and calibration of this model have been done thanks to experimental data consisting of proliferative cells distribution in a spheroid

CAPES
Em ensaios não destrutivos por ultrassom, o sinal obtido a partir de um sistema de aquisição de dados real podem estar contaminados por ruído e os ecos podem ter atrasos de tempo subamostrais. Em alguns casos, esses aspectos podem comprometer a informação obtida de um sinal por um sistema de aquisição. Para lidar com essas situações, podem ser utilizadas técnicas de estimativa de atraso temporal (Time Delay Estimation ou TDE) e também técnicas de reconstrução de sinais, para realizar aproximações e obter mais informações sobre o conjunto de dados. As técnicas de TDE podem ser utilizadas com diversas finalidades na defectoscopia, como por exemplo, para a localização precisa de defeitos em peças, no monitoramento da taxa de corrosão em peças, na medição da espessura de um determinado material e etc. Já os métodos de reconstrução de dados possuem uma vasta gama de aplicação, como nos NDT, no imageamento médico, em telecomunicações e etc. Em geral, a maioria das técnicas de estimativa de atraso temporal requerem um modelo de sinal com precisão elevada, caso contrário, a localização dessa estimativa pode ter sua qualidade reduzida. Neste trabalho, é proposto um esquema alternado que estima de forma conjunta, uma referência de eco e atrasos de tempo para vários ecos a partir de medições ruidosas. Além disso, reinterpretando as técnicas utilizadas a partir de uma perspectiva probabilística, estendem-se suas funcionalidades através de uma aplicação conjunta de um estimador de máxima verossimilhança (Maximum Likelihood Estimation ou MLE) e um estimador máximo a posteriori (MAP). Finalmente, através de simulações, resultados são apresentados para demonstrar a superioridade do método proposto em relação aos métodos convencionais.
Abstract (parágrafo único): In non-destructive testing (NDT) with ultrasound, the signal obtained from a real data acquisition system may be contaminated by noise and the echoes may have sub-sample time delays. In some cases, these aspects may compromise the information obtained from a signal by an acquisition system. To deal with these situations, Time Delay Estimation (TDE) techniques and signal reconstruction techniques can be used to perform approximations and also to obtain more information about the data set. TDE techniques can be used for a number of purposes in the defectoscopy, for example, for accurate location of defects in parts, monitoring the corrosion rate in pieces, measuring the thickness of a given material, and so on. Data reconstruction methods have a wide range of applications, such as NDT, medical imaging, telecommunications and so on. In general, most time delay estimation techniques require a high precision signal model, otherwise the location of this estimate may have reduced quality. In this work, an alternative scheme is proposed that jointly estimates an echo model and time delays for several echoes from noisy measurements. In addition, by reinterpreting the utilized techniques from a probabilistic perspective, its functionalities are extended through a joint application of a maximum likelihood estimator (MLE) and a maximum a posteriori (MAP) estimator. Finally, through simulations, results are presented to demonstrate the superiority of the proposed method over conventional methods.

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In this paper we discuss techniques for 3D data modelling and processing where the data are usually provided as point clouds which arise from 3D scanning devices. The particular approaches we adopt in modelling 3D data involves the use of Partial Differential Equations (PDEs). In particular we show how the continuous and discrete versions of elliptic PDEs can be used for data modelling. We show that using PDEs it is intuitively possible to model data corresponding to complex scenes. Furthermore, we show that data can be stored in compact format in the form of PDE boundary conditions. In order to demonstrate the methodology we utlise several examples of practical nature.

Yan, Kan.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 64-[66]).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstracts also in Chinese.

In this thesis we analyse particular differential equations that arise in physical situations. This is achieved with the aid of the computer software package called Mathematica. We first describe the basic features of Mathematica highlighting its capabilities in performing calculations in mathematics. Then we consider a first order Newtonian equation representing the trajectory of a particle around a spherical object. Mathematica is used to solve the Newtonian equation both analytically and numerically. Graphical plots of the trajectories of the planetary bodies Mercury, Earth and Jupiter are presented. We attempt a similar analysis for the corresponding relativistic equation governing the orbits of gravitational objects. Only numerical results are possible in this case. We also perform a perturbative analysis of the relativistic equation and determine the amount of perihelion shift. The second equation considered is the Emden-Fowler equation of order two which arises in many physical problems, including certain inhomogeneous cosmological applications. The analytical features of this equation are investigated using Mathematica and the Lie analysis of differential equations. Different cases of the related autonomous form of the Emden-Fowler equation are investigated and graphically represented. Thereafter, we generate a number of profiles of the energy density and the pressure for a particular solution which demonstrates that a numerical approach for studying inhomogeneity, in cosmological models in general relativity, is feasible.
Thesis (M.Sc.)-University of Natal, Durban, 2001.

M. Tech. Mathematical Technology.
Aims of this research project is to mathematically analyse froth patterns and build a database of the images at different stages of the fermentation process, so that a decision-making procedure can be developed, which enables a computer to react according to what has been observed. This would allow around-the-clock observation which is not possible with humans. In addition, mechanised decision-making would minimize errors usually associated with human actions. Different mathematical algorithms for image processing will be considered and compared. These algorithms have been designed for different image processing situations. In this dissertation the algorithms will be applied to froth images in particular and will be used to simulate the human eye for decision-making in the fermentation process. The preamble of the study will be to consider algorithms for the detection of edges and then analyse these edges. MATLAB will be used to do the pre-processing of the images and to write and test any new algorithms designed for this project.

Indiana University-Purdue University Indianapolis (IUPUI)
As our population ages, treating bone and joint ailments is becoming increasingly important. Both osteoporosis, a bone disease characterized by a decreased density of mineral in bone, and osteoarthritis, a joint disease characterized by the degeneration of cartilage on the ends of bones, are major causes of decreased movement ability and increased pain. To combat these diseases, many treatments are offered, including drugs and exercise, and much biomedical research is being conducted. However, how can we get the most out of the research we perform and the treatment we do have? One approach is through computational analysis and mathematical modeling. In this thesis, quantitative methods of analysis are applied in different ways to two systems: osteoporosis and osteoarthritis. A mouse model simulating osteoporosis is treated with salubrinal and knee loading. The bone and cell data is used to formulate a system of differential equations to model the response of bone to each treatment. Using Particle Swarm Optimization, optimal treatment regimens are found, including a consideration of budgetary constraints. Additionally, an in vitro model of osteoarthritis in chondrocytes receives RNA silencing of Lrp5. Microarray analysis of gene expression is used to further elucidate the mode of regulation of ADAMTS5, an aggrecanase associated with cartilage degradation, by Lrp5, including the development of a mathematical model. The math model of osteoporosis reveals a quick response to salubrinal and a delayed but substantial response to knee loading. Consideration of cost effectiveness showed that as budgetary constraints increased, treatment did not start until later. The quantitative analysis of ADAMTS5 regulation suggested the involvement of IL1B and p38 MAPK. This research demonstrates the application of quantitative methods to further the usefulness of biomedical and biomolecular research into treatment and signaling pathways. Further work using these techniques can help uncover a bigger picture of osteoarthritis's mode of action and ideal treatment regimens for osteoporosis.