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.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.
- 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.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.
- ItemDental implant recognition(Stellenbosch : Stellenbosch University, 2023-09) Kohlakala, Aviwe; Coetzer, Johannes; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics Division.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.
- 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.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.
- 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).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.