Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / A structure-from-motion implementation for enclosed environments is presented. The various aspects covered include a discussion on optimised luminance computations—a technique to compute an optimally weighted luminance that maintains a greatest amount of data fidelity. Furthermore a visual engine is created that forms the basis of data input for reconstruction purposes; such an inexpensive solution is found to offer realistic environments along with precise control of scene and camera elements. A motion estimation system provides tracking information of scene elements and an unscented Kalman filter is used as depth estimator. The elements are combined into an accurate reconstructor for enclosed environments.

Thesis (PhD (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In many applications, an accurate, robust and fast tracker is needed, for example in surveillance, gesture recognition, tracking lips for lip-reading and creating an augmented reality by embedding a tracked object in a virtual environment. In this dissertation we investigate the viability of a tracker that combines the accuracy of active appearancemodels with the robustness of the particle lter (a stochastic process)—we call this combination the PFAAM. In order to obtain a fast system, we suggest local optimisation as well as using active appearance models tted with non-linear approaches. Active appearance models use both contour (shape) and greyscale information to build a deformable template of an object. ey are typically accurate, but not necessarily robust, when tracking contours. A particle lter is a generalisation of the Kalman lter. In a tutorial style, we show how the particle lter is derived as a numerical approximation for the general state estimation problem. e algorithms are tested for accuracy, robustness and speed on a PC, in an embedded environment and by tracking in ìD. e algorithms run real-time on a PC and near real-time in our embedded environment. In both cases, good accuracy and robustness is achieved, even if the tracked object moves fast against a cluttered background, and for uncomplicated occlusions. / AFRIKAANSE OPSOMMING: ’nAkkurate, robuuste en vinnige visuele-opspoorderword in vele toepassings benodig. Voorbeelde van toepassings is bewaking, gebaarherkenning, die volg van lippe vir liplees en die skep van ’n vergrote realiteit deur ’n voorwerp wat gevolg word, in ’n virtuele omgewing in te bed. In hierdie proefskrif ondersoek ons die lewensvatbaarheid van ’n visuele-opspoorder deur die akkuraatheid van aktiewe voorkomsmodellemet die robuustheid van die partikel lter (’n stochastiese proses) te kombineer—ons noem hierdie kombinasie die PFAAM. Ten einde ’n vinnige visuele-opspoorder te verkry, stel ons lokale optimering, sowel as die gebruik van aktiewe voorkomsmodelle wat met nie-lineêre tegnieke gepas is, voor. Aktiewe voorkomsmodelle gebruik kontoer (vorm) inligting tesamemet grysskaalinligting om ’n vervormbaremeester van ’n voorwerp te bou. Wanneer aktiewe voorkomsmodelle kontoere volg, is dit normaalweg akkuraat,maar nie noodwendig robuust nie. ’n Partikel lter is ’n veralgemening van die Kalman lter. Ons wys in tutoriaalstyl hoe die partikel lter as ’n numeriese benadering tot die toestand-beramingsprobleem afgelei kan word. Die algoritmes word vir akkuraatheid, robuustheid en spoed op ’n persoonlike rekenaar, ’n ingebedde omgewing en deur volging in ìD, getoets. Die algoritmes loop intyds op ’n persoonlike rekenaar en is naby intyds op ons ingebedde omgewing. In beide gevalle, word goeie akkuraatheid en robuustheid verkry, selfs as die voorwerp wat gevolg word, vinnig, teen ’n besige agtergrond beweeg of eenvoudige okklusies ondergaan.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2004. / Suppose a lottery scheme consists of randomly selecting an unordered winning n-subset from a universal set of m numbers, while a player participates in the scheme by purchasing a playing set of any number of unordered n-subsets from the same universal set prior to a winning draw, and is awarded a prize if ...

ENGLISH ABSTRACT: Determining the (classical) chromatic number of a graph (i.e. finding the smallest number of colours with which the vertices of a graph may be coloured so that no two adjacent vertices receive the same colour) is a well known combinatorial optimization problem and is widely encountered in scheduling problems. Since the late 1960s the notion of the chromatic number has been generalized in several ways by relaxing the restriction of independence of the colour classes. / AFRIKAANSE OPSOMMING: Die bepaling van die (klassieke) chromatiese getal van ’n grafiek (naamlik die kleinste aantal kleure waarmee die punte van ’n grafiek gekleur kan word sodat geen twee naasliggende punte dieselfde kleur ontvang nie) is ’n bekende kombinatoriese optimeringsprobleem wat wyd in skeduleringstoepassings te¨egekom word. Sedert die laat 1960s is die definisie van die chromatiese getal op verskeie maniere veralgemeen deur die vereiste van onafhanklikheid van die kleurklasse te verslap. / Thesis (DPhil)--Stellenbosch University, 2007.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Research on transfer processes in various types of porous media has become important for the optimization of high technology engineering devices and processes. In this study the micro-structural parameters of different types of porous media, namely granular media, foamlike media and fibre beds, are characterized and quantified. Existing analytical modelling procedures for the three different types of porous media have been unified and refined to improve their predictive capabilities. Deterministic equations are proposed for predicting the streamwise pressure gradient, permeability and inertial coefficient of each type of porous medium. The equations are applicable over the entire porosity range and steady laminar flow regime and well suited as drag models in numerical computations. It is shown that the improved granular model can be regarded as qualitative and quantitative proof of the extensively used semi-empirical Ergun equation. The proposed model is used to provide physical meaning to the empirical coefficients. An Ergun-type equation is also proposed for foamlike media by remodelling the interstitial geometric configuration and accompanying flow conditions. The range of applicability of the existing foam model has been extended by incorporating the effect of developing flow in the pressure drop prediction. An equation is proposed in which the variation in the cross-sectional shape of the fibres can be incorporated into the interstitial form drag coefficient used in the foam model. This serves as an improvement on the constant value previously used. The existing foam model is also adapted to account for anisotropy resulting from compression. Two case studies are considered, namely compression of a non-woven glass fibre filter and compression of a soft polyester fibre material. The significant effect of compression on permeability is illustrated. In each case study the permeability values range over more than an order of magnitude for the narrow porosity ranges involved. The pressure drop prediction of the foam model is furthermore adapted to account for the combined effects of compression and developing flow. The newly proposed model diminishes the significant over-prediction of the existing foam model. An equation is furthermore proposed for predicting the permeability of Fontainebleau sandstones in which the effect of blocked throats is accounted for. Lastly, equations are proposed for predicting diffusivity ratios of unconsolidated arrays of squares and cubes. The prediction of the diffusivity ratio proposed in the present study, as opposed to model predictions from the literature, takes into account diffusion that may take place in stagnant fluid volumes. It is shown that a specific weighted average model proposed in the literature is not adequate to predict the diffusivity ratio of fully staggered arrays of squares, since it is shown not to be applicable to rectangular unit cells. Instead a new weighted average model is proposed which is applicable over the entire porosity range and for both staggered and non-staggered arrays of solid squares and cubes. The proposed weighted average model provides satisfactory agreement with experimental data from the literature and numerical data generated in the present study. / AFRIKAANSE OPSOMMING: Navorsing op oordragsprosesse in verskeie tipes poreuse media het belangrik geword vir die optimisering van ho¨e-tegnologie ingenieurstoestelle- en prosesse. In hierdie studie word die mikro-struktuur parameters van verskillende tipes poreuse media, naamklik korrelagtige media, sponsatige media en veselbeddens gekarakteriseer en gekwantifiseer. Bestaande analitiese modelleringsprosedures vir die drie verskillende tipes poreuse media is verenig en verfyn om die voorspelbare bekwaamheid daarvan te verbeter. Deterministiese vergelykings is voorgestel vir die voorspelling van die stroomsgewyse gradi¨ent, permeabiliteit en inersi¨ele ko¨effisi¨ent van elke tipe poreuse medium. Die vergelykings is geldig oor die hele porositeitsgrens en gestadigde laminˆere vloeigrens en goed geskik as weerstandsmodelle in numeriese berekeninge. Dit is aangetoon dat die verbeterde korrelmodel beskou kan word as kwalitatiewe en kwantitatiewe bewys van die ekstensiewe gebruikte semi-empiriese Ergun vergelyking. Die voorgestelde model is gebruik om fisiese betekenis aan die empiriese ko¨effisi¨ente te gee. ’n Ergun-tipe vergelyking is ook voorgestel vir sponsagtige media deur hermodellering van die tussenruimtelike geometriese konfigurasie en gepaardgaande vloeikondisies. Die grense van toepaslikheid van die bestaande sponsmodel is uitgebrei deur die inkorporering van die effek van ontwikkelende vloei in die voorspelling van die drukval. ’n Vergelyking is voorgestel waarin die variasie in die deursnit vorm van die vesels ingesluit is in die sponsmodel. Dit dien as verbetering op die konstante waarde wat voorheen gebruik is. Die bestaande sponsmodel is ook aangepas om voorsiening te maak vir anisotropie as gevolg van kompressie. Twee gevallestudies is oorweeg, naamlik kompressie van ’n nie-geweefde glasvesel filter en kompressie van ’n sagte polyester veselmateriaal. Die beduidende effek van kompressie op permeabiliteit is aangetoon. In elke gevallestudie strek die permeabiliteit waardes oor meer as ’n grootte orde vir die skrale porositeitgrense betrokke. Die drukvalvoorspelling van die sponsmodel is verder aangepas om voorsiening te maak vir die gekombineerde effekte van kompressie en ontwikkelende vloei. Die nuwe voorgestelde model verminder die beduidende oor-voorspelling van die bestaande sponsmodel. ’n Vergelyking is verder voorgestel vir die voorspelling van die permeabiliteit van Fontainebleau sandsteen waarin die effek van geblokte porie¨e in ag geneem is. Laastens is vergelykings voorgestel vir die voorspelling van die diffusiwiteitsverhoudings van nie-konsoliderende rangskikkings van vierkante en kubusse. Die diffusiwiteitsverhouding voorspel in die huidige studie, teenoor modelvoorspellings vanaf die literatuur, neem diffusie in ag wat plaasvind in die stagnante vloeistofvolumes. Dit is aangetoon dat ’n geweegde gemiddelde model, voorgestel in die literatuur, nie in staat is om die diffusiwiteitsverhouding van ten volle verspringende rangskikkings van vierkante te voorspel nie, aangesien dit nie toepaslik is vir reghoekige eenheidselle nie. ’n Nuwe geweegde model is in plaas daarvan voorgestel wat toepaslik is oor die hele porositeitsgrens en vir beide verspringende en nieverspringende rangskikkings van soliede vierkante en kubusse. Die voorgestelde geweegde gemiddelde model bied bevredigende ooreenstemming met eksperimentele data uit die literatuur en numeriese data gegenereer in die huidige studie.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008. / In this paper, tracking tools are developed based on object features to robustly track the object using particle filtering. Automatic on-line initialisation techniques use motion detection and dynamic background modelling to extract features of moving objects. Automatically adapting the feature models during tracking is implemented and tested.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--Stellenbosch University, 2008. / The NLS is a universal equation of the class of nonlinear integrable systems. The aim of this thesis is to study the NLS numerically. More speci cally, an algorithm is developed to calculate its nonlinear spectrum. The nonlinear spectrum is then used as a diagnostic for numerical studies of the NLS. The spectrum consists of a discrete part, further subdivided into the main part, the auxiliary part, and the continuous spectrum. Two algorithms are developed for calculating the main spectrum. One is based on Floquet theory, rst implemented by Overman [12]. The other is a direct calculation of the eigenvalues by Herbst and Weideman [16]. These algorithms are combined through the marching squares algorithm to calculate the continuous spectrum. All ideas are illustrated by numerical examples.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / The face is one of the most characteristic parts of the human body and has been used by people for personal identification for centuries. In this thesis an automatic process for frontal face recognition from 2–dimensional images is presented based on principal component analysis. The goal is to use these concepts in eventual face–recognizing login software. The first step is detecting faces in images that are allowed a certain degree of clutter. This is achieved by skin colour detection in the HSV colourspace. This process indicates the area of the image most likely corresponding to the face. Extracting the face is achieved by morphological processing of this area of the image. The face is then normalized by a transformation that uses the eye coordinates as input. Automatic eye detection is implemented based on colour analysis of the facial images and a 91.1% success rate is achieved. Recognition of the normalized faces is achieved using eigenfaces. To calculate these, a large enough database of facial images is needed. The xm2vts database is used in this thesis as the images have very constant lighting conditions throughout – an important factor affecting the accuracy of the recognition stage. Distinction is also made between identification and verification of faces. For identification, up to 80.1% accuracy is achieved, while for verification, the equal error rate is approximately 3.5%.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2009. / In this thesis, we investigate two numerical methods to price nancial options. We look at two types of options, namely European options and Asian options. The numerical methods we use are the nite di erence method and numerical inversion of the Laplace transform. We apply nite di erence methods to partial di erential equations with both uniform and non-uniform spatial grids. The Laplace inversion method we use is due to Talbot. It is based on the midpoint-type approximation of the Bromwich integral on a deformed contour. When applied to Asian options, we have the problem of computing the hypergeometric function of the rst kind. We propose a new method for numerically calculating the hypergeometric function. This method too is based on using Talbot contours. Throughout the thesis, we use the Black-Scholes equation as our benchmark problem.

Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / The Korteweg-de Vries (KdV) hierarchy is an important class of nonlinear evolution equa- tions with various applications in the physical sciences and in engineering. In this thesis analytical solution methods were used to ¯nd exact solutions of the third and ¯fth order KdV equations, and numerical methods were used to compute numerical solutions of these equations. Analytical methods used include the Fan sub-equation method for constructing exact trav- eling wave solutions, and the simpli¯ed Hirota method for constructing exact N-soliton solutions. Some well known cases were considered. The Fourier spectral method and the ¯nite di®erence method with Runge-Kutta time dis- cretisation were employed to solve the third and the ¯fth order KdV equations with periodic boundary conditions. The one soliton and the two soliton solutions were used as initial conditions. The numerical solutions are obtained and compared with the exact solutions. The propagation of a single soliton as well as the interaction of double soliton solutions is modeled well by both numerical methods, although the Fourier spectral method performs better. The stability, consistency and convergence of these numerical methods were investigated. Error propagation is studied. The theoretically predicted quadratic convergence of the ¯nite di®erence method as well as the exponential convergence of the Fourier spectral method is con¯rmed in numerical experiments.