Doctoral Degrees (Applied Mathematics)

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Browsing Doctoral Degrees (Applied Mathematics) by browse.metadata.advisor "Brink, Willie"

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    Analysing 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.
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    Path 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.
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    Video 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
    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.