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# Doctoral Degrees (Applied Mathematics)

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- ItemAnalysing retinal fundus images with deep learning models(Stellenbosch : Stellenbosch University, 2023-12) Ofosu Mensah, Samuel; Bah, Bubacarr; Brink, Willie; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division.
Show more ENGLISH ABSTRACT: Convolutional neural networks (CNNs) have successfully been used to classify diabetic retinopathy but they do not provide immediate explanations for their decisions. Explainability is relevant, especially for clinicians. To make results explainable, we use a post-attention technique called gradient-weighted class activation mapping (Grad- CAM) on the penultimate layer of deep learning models to produce localisation maps on retinal fundus images after using them to classify diabetic retinopathy. Moreover, the models were initialised using pre-trained weights obtained from training models on the ImageNet dataset. The results of this are fewer training epochs and improved performance. Next, we predict cardiovascular risk factors (CVFs) using retinal fundus images. In detail, we use a multi-task learning (MTL) model since there are several CVFs. The impact of using an MTL model is the advantage of simultaneously training for and predicting several CVFs rather than doing so individually. Also, we investigate the performance of the fundus cameras used to capture the retinal fundus images. We notice a superior performance of the desktop fundus cameras to the handheld fundus camera. Finally, we propose a hybrid model that fuses convolutions and Transformer encoders. This is done to harness the benefits of convolutions and Transformer encoders. We compare the performance of the proposed model with other attention-based models and observe on-par performance.Show more - ItemA conceptual framework for the development of intelligent, learning style- and computer-based educational software for topics from operations research(Stellenbosch : Stellenbosch University, 1999-11) Du Plessis, S. A. (Sameul Altenstadt); De Kock, H. C.; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences.
Show more ENGLISH ABSTRACT: The purpose of this study was to construct a conceptual framework for the development of intelligent, learning style- and computer-based educational software and to apply it to linear programming (LP). A secondary goal was to extend the framework to also include other topics from Operations Research. The system that resulted from this study was named GEORGE, in honor of the inventor of the simplex method George Dantzig. GEORGE utilizes fuzzy interpretations of learning style inventories and models of teaching and learning to determine a student's learning and teaching style preferences. An individualized tutoring strategy is then computed and used to present the course material to the student. A whole range of modes of presentation can be included in such a strategy, from drill-and-practice exercises, demonstrations and step-by-step tutorials to flow- and step charts and point-and-query interfaces. GEORGE keeps a practical and effective student model and controls the tutoring with a domain- and motivational based planner. The models of teaching and learning, mentioned above, are based on the results of fuzzy interpretations of Kolb's learning style inventory (experimental learning), a Myers Briggs Type Indicator Test (personality), La Haye's temperament test, a visualizer-verbalizer questionnaire, a study preference guide (sequential/global preferences), the model of teaching and learning of Felder and Silverman (for engineering education), Neethling's Brain Profile Test, a model of teaching and learning that is based on left and right brain preferences, and Sternberg's model of thinking styles. GEORGE consists of six modules, namely a problem solving or domain expert module, a generic questioning module, a presentation module, an "artificial psychologist" module, a student model module and a tutorial module. The generic questioning module is used to generate tutoring and testing material for GEORGE and the "artificial psychologist" module is used primarily to supply students with individualized psychological help, from study techniques and emotional matters to motivation and goal setting. The remaining four modules correspond more or less with the four modules of a traditional intelligent tutoring system. A number of artificial intelligence techniques i.e. natural language understanding, fuzzy expert and fuzzy decision making systems, induction and neural networks, are used in the implementation of different components of GEORGE. Applications of De Bono's thinking skills also play an important role in a number of components (teaching students how to think), the presentation of various personal development or self improvement techniques are very prominent (in the "artificial psychologist" module), and the accommodation of differences among users (especially learning style preferences) receives high priority. The implementation of the various components of GEORGE resulted in a number of useable computer-based learning modules. These demonstration programs illustrate the various concepts within the suggested general framework. The system was developed in Turbo Pascal and integrated within the "Windows"-environment with the help of the authoring system, EasyTutor. GEORGE will eventually be extended to become not only a computer-based tutor of LP topics, but also a Resourceful Environment for the Clever Tutoring of other Operations Research techniques (RECTOR). Guidelines regarding the transformation of GEORGE into RECTOR are provided. RECTOR, and parts thereof, should be used in a very similar way as in GEORGE, to supply computer support of lectures, to provide computer-assisted learning, to conduct computer-based learning, to create a computer environment for calculations and as a source of self-paced and open access material.Show more - ItemContributions to the theory of near-vector spaces, their geometry, and hyperstructures(Stellenbosch : Stellenbosch University, 2022-12) Rabie, Jacques; Howell, Karin-Therese; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
Show more ENGLISH ABSTRACT: This thesis expands on the theory and application of near-vector spaces — in particular, the underlying geometry of near-vector spaces is studied, and the theory of near-vector spaces is applied to hyperstructures. More specifically, a near-linear space is defined and some properties of these spaces are proved. It is shown that by adding some axioms, the nearaffine space, as defined by André, i s obtained. A correspondence is shown between subspaces of nearaffine spaces generated by near-vector spaces, and the cosets of subspaces of the corresponding near-vector space. As a highlight, some of the geometric results are used to prove an open problem in near-vector space theory, namely that a non-empty subset of a near-vector space that is closed under addition and scalar multiplication is a subspace of the near-vector space. The geometric work of this thesis is concluded with a first look into the projections of nearaffine s paces, a branch of the geometry that contains interesting avenues for future research. Next the theory of hyper near-vector spaces is developed. Hyper near-vector spaces are defined having similar properties to André’s near-vector space. Important concepts, including independence, the notion of a basis, regularity, and subhyperspaces are defined, and an analogue of the Decomposition Theorem, an important theorem in the study of near-vector spaces, is proved for these spaces.Show more - ItemDental implant recognition(Stellenbosch : Stellenbosch University, 2023-09) Kohlakala, Aviwe; Coetzer, Johannes; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division.
Show more ENGLISH ABSTRACT: Deep learning-based frameworks have recently been steadily outperforming existing state-of-the-art systems in a number of computer vision applications, but these models require a large number of training samples in order to effectively train the model parameters. Within the medical field the limited availability of training data is one of the main challenges faced when using deep learning to create practical clinical applications in medical imaging. In this dissertation a novel algorithm for generating artificial training samples from triangulated three-dimensional (3D) surface models within the context of dental implant recognition is proposed. The proposed algorithm is based on the calculation of two-dimensional (2D) parallel projections from a number of different angles of 3D volumetric representations of computer-aided design (CAD) surface models. A fully convolutional network (FCN) is subsequently trained on the artificially generated X-ray images for the purpose of automatically identifying the connection type associated with a specific dental implant in an actual X-ray image. An ensemble of image processing and deep learning-based techniques capable of distinguishing between pixels that belong to an implant from those belonging to the background in an actual X-ray image is developed. Normalisation and preprocessing techniques are subsequently applied to the segmented dental implants within the questioned actual X-ray image. The normalised dental implants are presented to the trained FCN for classification purposes. Experiments are conducted on two data sets that contain the simulated and actual X-ray images in order to gauge the proficiency of the proposed systems. Given the fact that the novel systems proposed in this study utilise an ensemble of techniques that has not been employed for the purpose of dental implant classification/recognition on any previous occasion, the results achieved in this study are encouraging and constitute a significant contribution to the current state of the art, especially in scenarios where the proposed systems are combined with existing systems.Show more - ItemHand vein-based biometric authentication using neural networks(Stellenbosch : Stellenbosch University, 2024-03) Beukes, Emile; Coetzer, Johannes; Stellenbosch University. Faculty of Science. Dept. of Applied Mathematics.
Show more ENGLISH ABSTRACT: The feasibility of employing convolutional neural networks for the purpose of authenticating an individual based on a near infra-red image of his/her dorsal hand vein pattern is inves- tigated in this study. The proficiency of different architectural designs associated with sim- ilarity measure networks (SMNs), in particular two-channel SMNs and Siamese SMNs, are compared. Four different combinations of neural network layers are investigated for each of the aforementioned SMNs. Three different levels of preprocessing are applied to the hand vein images in order to investigate the relevance of information surrounding the actual hand veins on the proficiency of the networks. The proficiency of the proposed systems is gauged within the context of two real-world scenarios, namely the individual dependent scenario (IDS) and the individual independent scenario (IIS). A tailor-made network is trained for each client during enrolment in mere minutes within the context of the IDS, while a single net- work is trained in a once-off fashion prior to the enrolment of any clients within the context of the IIS. Two publicly available hand vein databases namely the Bosphorus and Wilches databases are investigated within the context of this study. An artificially generated hand vein database, namely the GenVeins database, is developed in this study for the purpose of acquiring a set of training individuals that is large enough so as to be representative of the entire population. The motivation behind the creation of the GenVeins database constitutes the fact that experimental results indicate that system proficiency is severely impaired when training on an insufficient number of different individuals within the context of the IIS. The systems proposed in this study are therefore considered implementation-ready in the sense that they are either trained in a (1) tailor-made fashion for each client enrolled into the sys- tem in real time or in a (2) once-off fashion on a set of fictitious individuals that is sufficiently representative of the entire population. The proposed systems do therefore not merely serve as so-called proofs-of-concept (POCs) in which a system is trained and tested on the same set of individuals. These POCs are clearly not feasible within the context of any real world scenario.Show more - ItemMathematical pore-scale modelling of kinematic and geometric properties of fibrous porous media(Stellenbosch : Stellenbosch University, 2023-12) Maré, Esmari; Fidder, Sonia; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division.
Show more ENGLISH ABSTRACT: This study involves the mathematical modelling of permeability ( of both the Darcy and Forchheimer flow regimes) and specific surface area of fibre-type and foamlike porous media using geometric models. Several existing models for predicting these properties have been studied in the literature, with the Representative Unit Cell (RUC) model being of particular interest due to its simple rectangular geometry and good performance compared to other models and experimental data from the literature. This study includes a comparative analysis of the permeability and specific surface area prediction of different versions of the RUC model for fibrous media involving the 2D RUC models for in-plane and through plane flow, the 3D RUC model, the two-strut RUC models for in-plane and through plane flow, and the three-strut RUC model. It furthermore incorporates novel contributions such as the adaptation of the three-strut ( or foam) RUC model by adding solid material to account for the observed lump at the intersection of struts in actual metal and ceramic foams. The RUC models are also adapted analytically to take secondary effects such as compression or variable rectangular geometry into account. Additionally, the models are adapted to include changes in the predictions of the permeability due to the Klinkenberg effect, an effect that accounts for the increase in the permeability of gas flow as opposed to that of a liquid. The novelty of this study lies in the incorporation of these effects into the model predictions, which extends the range of applicability of the proposed models beyond those available in the literature. In order to ensure the user-friendliness of the analytical models provided, the predictive equations are expressed in terms of measurable macroscopic parameters. Furthermore, the models are evaluated through comparison with other models from the literature as well as available experimental and numerical data, which yield results that are satisfactory. The findings contribute positively towards industrial applications such as filtration and heat transfer processes, facilitating their effective operation by means of analytical modelling and analysis of the physical flow processes involved.Show more - ItemPath planning for wheeled mobile robots using an optimal control approach(Stellenbosch : Stellenbosch University, 2019-12) Matebese, Belinda Thembisa; Banda, Mapundi K.; Withey, Daniel; Brink, Willie; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences (Applied Mathematics).
Show more ENGLISH ABSTRACT: The capability and practical use of wheeled mobile robots in real-world applications have resulted in them being a topic of recent interest. These systems are most prevalent because of their simple design and ease to control. In many cases, they also have an ability to move around in an environment without any human intervention. A main stream of research for wheeled mobile robots is that of planning motions of the robot under nonholonomic constraints. A typical motion planning problem is to find a feasible path in the configuration space of the mobile robot that starts at the given initial state and reaches the desired goal state while satisfying robot kinematic or dynamic constraints. A variety of methods have been used to solve various aspects of the motion planning problem. Depending on the desired quality of the solution, an optimal path is often sought. In this dissertation, optimal control is employed to obtain optimal collision-free paths for two-wheeled mobile robots and manipulators mounted on wheeled mobile platforms from an initial state to a goal state while avoiding obstacles. Obstacle avoidance is mathematically modelled using the potential field technique. The optimal control problem is then solved using an indirect method approach. This approach employs Pontryagin’s minimum principle where analytical solutions for optimality conditions are derived. Solving the optimality condition leads to two sets of differential equations that have to be solved simultaneously and whose conditions are given at different times. This set of equations is known as a two-point boundary value problem (TPBVP) and can be solved using numerical techniques. An indirect method, namely Leapfrog, is then implemented to solve the TPBVP. The Leapfrog method begins with a feasible trajectory, which is divided into smaller subdivisions where the local optimal controls are solved. The locally optimal trajectories are added and following a certain scheme of updating the number of subdivisions, the algorithm ends with the generation of an optimal trajectory along with the corresponding cost. An advantage of using the Leapfrog method is that it does not depend on the provision of good initial guesses along a path. In addition, the solution provided by the method satisfies both boundary conditions at every step. Moreover, in each iteration the paths generated are feasible and their cost decreases asymptotically. To illustrate the effectiveness of the algorithm numerically, a quadratic cost with the control objective of steering the mobile robot from an initial state to a final state while avoiding obstacles is minimized. Simulations and numerical results are presented for environments with and without obstacles. A comparison is made between the Leapfrog method and the BVP4C optimization algorithm, and also the kinodynamic-RRT algorithm. The Leapfrog method shows value for continued development as a path planning method since it initializes easily, finds kinematically feasible paths without the need of post processing and where other techniques may fail. To our knowledge the work presented here is the first application of the Leapfrog method to find optimal trajectories for motion planning on a two-wheeled mobile robot and mobile manipulator.Show more - ItemPore-scale modelling of transport phenomena in homogeneous porous media(Stellenbosch : Stellenbosch University, 1999-12) Diedericks, Gerhardus Petrus Jacobus; Du Plessis, J. P.; Stellenbosch University. Faculty of Science. Dept. of Mathematics.
Show more ENGLISH ABSTRACT: The main purpose of this study is to develop deterministic, process-based models of incompressible Newtonian flow and electrical c01iduction in homogeneous, anisotropic porous media. The foundation of the models is provided by the volume averaging theory which is used to obtain the macroscopic balance equations for momentum transport and electrical conduction. These volume averaged equations contain, amongst others, integral terms over the fluid-solid surface area where the integrands are related to the microscopic fluxes of the transport quantities. The closure modelling is conducted by employing a pore-scale model which requires explicit assumptions regarding the mean geometric properties of the porous medium microstructure and accounts for the configuration of the fluid-solid surface area. The pore-scale model also provides an estimate of the microscopic flow paths. The average geometry of different anisotropic materials, namely two types of foamlike materials, granular porous media and fibre beds, is captured in representative unit cells which form the core of the physical pore-scale model. This particular type of closure modelling further requires a direct transformation of microscopic fluxes to the macroscopic level. It is indicated, in context of the volume averaging theory, that microscopic fluxes may be estimated by the respective macroscopic channel average fluxes. The transformation of the microscopic flux to the channel average flux is accomplished through the flux related tortuosity tensor. New definitions for the tortuosity and lineality as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. It is shown that the definitions produce results which conform with several other published results and are applicable to anisotropic media. Application of the modelling technique to Newtonian flow results in momentum transport equations valid for both the Darcy and Forchheimer flow regimes. The coefficients appearing in these equations are expressed in terms of fluid properties and measurable geometric features of the porous medium. The predictions of the anisotropic foamlike materials are validated against experimental pressure gradient measurements for flow through a high porosity, anisotropic knitted wire mesh rolled up to form a cylindrical plug. The predictions compare reasonably well with the experimental results. The modelling approach is also applied to electrical conduction in anisotropic porous media saturated with an electrically conductive fluid. A macroscopic form of Ohm's law is derived as well as deterministic expressions for the formation factor. The formation factor predictions for isotropic porous media are compared to several experimental measurements as well as to semi-empirical expressions. The predictions compare favourably to the measurements.Show more - ItemVideo Surveillance Incorporating Pan-Tilt-Zoom Cameras(Stellenbosch : Stellenbosch University, 2015-03) Holtzhausen, Petrus Jacobus; Herbst, B. M.; Crnojevic, Vladimir; Brink, Willie; Stellenbosch University. Faculty of Science. Department of Applied Mathematics
Show more ENGLISH ABSTRACT : When trespassers target businesses and homes, outdoor spaces are typically the first point of illegal entry. Camera systems can help secure these environments, but typically many cameras are needed to cover large areas. In practice most camera systems are only used to review events after they have happened. It is however possible to do much more, and we explore the paradigm of active monitoring where cameras detect trespassers and give visual verification of the alarm. This detection needs to be resilient to weather effects and other environmental noise, while running in real-time on high resolution video sequences. Our goal is to replace multiple static cameras with a single Pan-Tilt-Zoom (PTZ) camera that can monitor expansive terrain. These cameras can survey, detect and zoom in on objects of interest. We develop and implement a robust, real-time algorithm based on an interaction framework between an illumination invariant and a color based background model. We also developed and implemented a novel technique where we use optical flow motion vectors to determine the size and shape of the spatial Gaussian kernels in non parametric models. Although computationally more expensive we demonstrate these more sophisticated models can be more robust. These ideas are adapted for PTZ cameras, exploiting their pan, tilt and zoom capabilities. We apply background modeling on panorama images that inform PTZ camera movement. By this we discuss the construction of a system that secures perimeters using zooming camera analytics.Show more - ItemWriter-independent handwritten signature verification(Stellenbosch : Stellenbosch University, 2015-12) Swanepoel, Jacques Philip; Coetzer, Johannes; Stellenbosch University. Faculty of Science. Department of Mathematical Sciences (Applied Mathematics).
Show more AFRIKAANSE OPSOMMING : In hierdie verhandeling stel ons 'n nuwe strategie vir outomatiese handtekening-verifikasie voor. Die voorgestelde raamwerk gebruik 'n skrywer-onafhanklike benadering tot handtekening- modellering en is dus in staat om bevraagtekende handtekeninge, wat aan enige skrywer behoort, te bekragtig, op voorwaarde dat minstens een outentieke voorbeeld vir vergelykingsdoeleindes beskikbaar is. Ons ondersoek die tradisionele statiese geval (waarin 'n bestaande pen-op-papier handtekening vanuit 'n versyferde dokument onttrek word), asook die toenemend gewilde dinamiese geval (waarin handtekeningdata outomaties tydens ondertekening m.b.v. gespesialiseerde elektroniese hardeware bekom word). Die statiese kenmerk-onttrekkingstegniek behels die berekening van verskeie diskrete Radontransform (DRT) projeksies, terwyl dinamiese handtekeninge deur verskeie ruimtelike en temporele funksie-kenmerke in die kenmerkruimte voorgestel word. Ten einde skryweronafhanklike handtekening-ontleding te bewerkstellig, word hierdie kenmerkstelle na 'n verskil-gebaseerde voorstelling d.m.v. 'n geskikte digotomie-transformasie omgeskakel. Die klassikasietegnieke, wat vir handtekeking-modellering en -verifikasie gebruik word, sluit kwadratiese diskriminant-analise (KDA) en steunvektormasjiene (SVMe) in. Die hoofbydraes van hierdie studie sluit twee nuwe tegnieke, wat op die bou van 'n robuuste skrywer-onafhanklike handtekeningmodel gerig is, in. Die eerste, 'n dinamiese tydsverbuiging digotomie-transformasie vir statiese handtekening-voorstelling, is in staat om vir redelike intra-klas variasie te kompenseer, deur die DRT-projeksies voor vergelyking nie-lineêr te belyn. Die tweede, 'n skrywer-spesieke verskil-normaliseringstrategie, is in staat om inter-klas skeibaarheid in die verskilruimte te verbeter deur slegs streng relevante statistieke tydens die normalisering van verskil-vektore te beskou. Die normaliseringstrategie is generies van aard in die sin dat dit ewe veel van toepassing op beide statiese en dinamiese handtekening-modelkonstruksie is. Die stelsels wat in hierdie studie ontwikkel is, is spesi ek op die opsporing van hoë-kwaliteit vervalsings gerig. Stelselvaardigheid-afskatting word met behulp van 'n omvattende eksperimentele protokol bewerkstellig. Verskeie groot handtekening-datastelle is oorweeg. In beide die statiese en dinamiese gevalle vaar die voorgestelde SVM-gebaseerde stelsel beter as die voorgestelde KDA-gebaseerde stelsel. Ons toon ook aan dat die stelsels wat in hierdie studie ontwikkel is, die meeste bestaande stelsels wat op dieselfde datastelle ge evalueer is, oortref. Dit is selfs meer belangrik om daarop te let dat, wanneer hierdie stelsels met bestaande tegnieke in die literatuur vergelyk word, ons aantoon dat die gebruik van die nuwe tegnieke, soos in hierdie studie voorgestel, konsekwent tot 'n statisties beduidende verbetering in stelselvaardigheid lei.Show more