Autism spectrum disorder (ASD) is a neurodevelopmental disorder emerging in the first few years of life. Currently, three characteristics are required for a diagnosis of ASD, impaired social interactions, impaired verbal communication and restricted and repetitive patterns of behaviour or interests. This last category can optionally include hyper- or hypo-reactivity to sensory input. Individuals with autism can also display superior performance on visual tasks where it may help to ignore global detail, behaviour sometimes described as ‘not seeing the forest for the trees’. At present, the exact mechanisms underlying the perceptual differences between autistic and neurotypical groups remain unknown, but they may reflect an imbalance in the contributions that bottom-up and top-down processing make in perceptual processing. Visual perception is thought to rely on interactions between the ‘bottom-up’ flow of ambiguous information from the retina and the ‘top-down’ flow of disambiguating information from higher cortical areas, via cortical circuits that have been shaped by a lifetime’s experience. These interactions lead to the activation of internal representations (of objects) which are necessary for the successful navigation of our environment. In order to investigate these perceptual differences, we employed three well-known experimental paradigms with a group of thirteen autistic participants and their matched controls. We investigated visual integration (involving bottom-up and top-down interactions) across low and intermediate stage neural mechanisms. A dim line (target) is easier to detect when flanked by two brighter collinear lines (flankers), an effect known as collinear facilitation, and we used two variations of this task to investigate low-level visual integration. In the first, we varied the orientation of the collinear flankers and found reduced integration for an autistic compared to a neurotypical group, a finding that conflicts with previous research. In a second collinear facilitation experiment with neurotypical participants, in which the target could be presented before, during or after flanker presentation, we were able to isolate facilitation that we believe was due to feedforward and feedback processing. However, in a subsequent study in which we compared autistic and neurotypical performance on this task, we found no significant difference. Moving onto intermediate level visual integration, we used a contour integration task consisting of open (lines) and closed (square) contours and found reduced integration for the autistic compared to the neurotypical groups when integrating closed contours. In our final study, we looked at global motion integration, and made use of a translating diamond. This is a bistable stimulus in which four lines can be perceived as independent line fragments moving vertically, or as a single integrated shape - a diamond moving horizontally. In this experiment, the autistic group showed an unexpected bias to perceiving the stimuli in its integrated form as a diamond. Perceptual processing of shapes based on squares or diamonds reflects visual integration at a global level, and so the differences we have found in shape processing between our experimental groups (reduced integration for the square and increased integration for the diamond in autism) are more likely to be the result of differences in top-down processing.

This thesis investigates the numerical treatment of nonlinear eigenvalue problems. These problems are defined by the condition $T(lambda) v = boldsymbol{0}$, with $T: C to C^{n times n}$, where we seek to compute the scalar-vector pairs, $lambda in C$ and nonzero $ v in C^{n}$. The first contribution of this work connects recent contour integration methods to the theory and practice of system identification. This observation leads us to explore rational interpolation for system realization, producing a Loewner matrix contour integration technique. The second development of this work studies the application of rational interpolation to the function $T(z)^{-1}$, where we use the poles of this interpolant to approximate the eigenvalues of $T$. We then expand this idea to several iterative methods, where at each step the approximate eigenvalues are taken as new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton's method for a particular scalar function. / Master of Science / This thesis investigates the numerical treatment of nonlinear eigenvalue problems. The solutions to these problems often reveal characteristics of an underlying physical system. One popular methodology for handling these problems uses contour integrals to compute a set of the solutions. The first contribution of this work connects these contour integration methods to the theory and practice of system identification. This leads us to explore other techniques for system identification, resulting in a new method. Another common methodology approximates the nonlinear problem directly. The second development of this work studies the application of rational interpolation for this purpose. We then use this idea to form several iterative methods, where at each step the approximate solutions are taken to be new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton’s method for a particular scalar function.

Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative method for the numerical solution of PDEs. Effective methods for the numerical inversion are based on the approximation of the Bromwich integral. In this thesis, a numerical study is undertaken to compare the efficiency of the Laplace inversion method with more conventional time integrator methods. Particularly, we consider the method-of-lines based on MATLAB’s ODE15s and the Crank-Nicolson method. Our studies include an introductory chapter on the Laplace inversion method. Then we proceed with spectral methods for the space discretization where we introduce the interpolation polynomial and the concept of a differentiation matrix to approximate derivatives of a function. Next, formulas of the numerical differentiation formulas (NDFs) implemented in ODE15s, as well as the well-known second order Crank-Nicolson method, are derived. In the Laplace method, to compute the Bromwich integral, we use the trapezoidal rule over a hyperbolic contour. Enhancement to the computational efficiency of these methods include the LU as well as the Hessenberg decompositions. In order to compare the three methods, we consider two criteria: The number of linear system solves per unit of accuracy and the CPU time per unit of accuracy. The numerical results demonstrate that the new method, i.e., the Laplace inversion method, is accurate to an exponential order of convergence compared to the linear convergence rate of the ODE15s and the Crank-Nicolson methods. This exponential convergence leads to high accuracy with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the Crank-Nicolson method as the results show. Finally, we apply with satisfactory results the inversion method to the axial dispersion model and the heat equation in two dimensions. / AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die benadering van die Bromwich integraal. In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer op MATLAB se ODE15s en die Crank-Nicolson metode. Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode. Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie, waar ons die interpolasie polinoom invoer sowel as die konsep van ’n differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word. Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste van al hierdie metodes word verbeter met die LU sowel as die Hessenberg ontbindings. Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode, akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson metode. Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.

Visual acuity is reduced when optotypes are viewed in the presence of surrounding contours. This reduction in acuity is known as the crowding effect and is thought to be caused by a varying combination of contour interaction, gaze instability and attention. Traditional studies have used single optotypes surrounded by flanking bars to investigate crowding. Such targets may not realistically replicate the crowding effect inherent in clinical vision charts. The aim of this thesis was to systematically investigate the effect of crowding on visual thresholds in subjects with normal vision and in subjects with amblyopia, using specially designed charts. In the 1st and 2nd experiment, contour interaction was assessed using a high (80 %) and low contrast (5.8%) Sheridan Gardiner repeat letter (SGRL) chart in subjects with normal vision. The effect of contour interaction was investigated by varying the inter-letter separation in the SGRL chart. Significant contour interaction was obtained at the abutting condition for both the contrast conditions. In the 3rd experiment the same protocol was repeated but in amblyopes. Significant contour interaction was obtained at 0.2 letter separation and the abutting condition for both the contrast conditions. The effect of contour interaction appears to be less for low contrast than for high contrast letters in normal, non-amblyopic and amblyopic eyes. Finally, in the 4th experiment a Sheridan Gardiner Complex Interaction (SGCI) chart that requires imposed gaze fixations was constructed to measure visual acuity in normal’s and amblyopes. The effect of any gaze instability on crowding was investigated by comparing SGRL thresholds to SGCI thresholds. The SGCI thresholds were higher than the SGRL thresholds at all the separations measured, suggesting an important effect of gaze instability on crowding. In conclusion, this research has shown that gaze instability is an important component of the crowding effect for letter chart acuity measurements. Visual acuity especially when screening for amblyopia should be measured using a whole optotype chart that requires optotype to optotype fixation.

<p>Human vision has an incredible ability to translate light reaching the retinae into a coherent, three-dimensional representation of the outside world in a fraction of a second. Much research has been devoted to understanding how local orientation information is integrated to form global contours and shapes -– a crucial step in visual processing. This dissertation describes experiments examining how contour and shape perception are affected in healthy aging.</p> <p>Chapter 2 examined contour grouping at low contrast and in the absence of distracters. Unlike younger subjects, older subjects did not benefit from co-alignment of local orientations with the contour’s outline, suggesting that grouping by orientation co-alignment is impaired in older age in low contrast. Chapters 3 and 4 examined the effects of aging on the ability to detect and discriminate high-contrast contours embedded in a dense field of distracters, as real life situations often require detecting objects among clutter, such as a snake hiding among tall grass. Results showed that older adults require significantly more time to discriminate contours in clutter, especially for less salient contours. Moreover, increasing the relative density of background clutter had a greater detrimental effect on older, compared to younger, subjects. However, aging did not seem to affect the ability to group contours across a range of spatial distances, or the sensitivity of contour integration to orientation misalignment. Lastly, Chapter 5 examined the influence of local orientation information on the perception of a contour's shape. Results revealed that older and younger subjects perceived the shape of a sampled contour in the same way, even when the contour's orientation and position information were in conflict. These findings indicated that the integration of orientation and position information in shape perception does not change with age.</p> / Doctor of Philosophy (PhD)