Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / Contains bibliography. / ENGLISH ABSTRACT: See full text for summary. / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Subdivision is an important and e cient tool for rendering smooth curves and surfaces in computer graphics, by repeatedly applying a subdivision (re ning) scheme to a given set of points. In the literature, attention has been mostly restricted to developing binary subdivision schemes. The primary emphasis of this thesis is on ternary subdivision, and in particular on the interpolatory case. We will derive a symmetric ternary interpolatory subdivision scheme for the rendering of curves, satisfying analogous properties to the Dubuc-Deslauriers binary scheme. Explicit construction methods, as well as a corresponding convergence analysis, will be presented. Graphical illustrations of the results will also be provided. / AFRIKAANSE OPSOMMING: Subdivisie bied 'n belangrike en doeltre ende metode om gladde krommes en oppervlakke in rekenaargra ka te genereer. Hierdie metode behels dat 'n subdivisieskema (of verfyningskema) herhaaldelik toegepas word op 'n gegewe versameling punte. In die literatuur word daar hoofsaaklik gefokus op die ont- wikkeling van bin^ere subdivisieskemas. In hierdie tesis word die klem gel^e op tern^ere subdivisieskemas, en in die besonder op interpolerende skemas. Ons sal 'n simmetriese tern^ere interpolerende subdivisieskema, wat analo e eienskappe as di e van die Dubuc-Deslauriers bin^ere skema bevredig, ontwikkel, om krom- mes te lewer. Eksplisiete konstruksiemetodes en ooreenkomstige konvergensie- analise, asook gra ese illustrasies van die resultate, sal getoon word.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / We study refinable functions where the dilation factor is not always assumed to be 2. In our investigation, the role of convolutions and refinable step functions is emphasized as a framework for understanding various previously published results. Of particular importance is a class of polynomial factors, which was first introduced for dilation factor 2 by Berg and Plonka and which we generalise to general integer dilation factors. We obtain results on the existence of refinable functions corresponding to certain reduced masks which generalise similar results for dilation factor 2, where our proofs do not rely on Fourier methods as those in the existing literature do. We also consider subdivision for general integer dilation factors. In this regard, we extend previous results of De Villiers on refinable function existence and subdivision convergence in the case of positive masks from dilation factor 2 to general integer dilation factors. We also obtain results on the preservation of subdivision convergence, as well as on the convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial factors are added to the mask symbol. We obtain sufficient conditions for the occurrence of polynomial sections in refinable functions and construct families of related refinable functions. We also obtain results on the regularity of a refinable function in terms of the mask symbol factorisation. In this regard, we obtain much more general sufficient conditions than those previously published, while for dilation factor 2, we obtain a characterisation of refinable functions with a given number of continuous derivatives. We also study the phenomenon of subsequence convergence in subdivision, which explains some of the behaviour that we observed in non-convergent subdivision processes during numerical experimentation. Here we are able to establish different sets of sufficient conditions for this to occur, with some results similar to standard subdivision convergence, e.g. that the limit function is refinable. These results provide generalisations of the corresponding results for subdivision, since subsequence convergence is a generalisation of subdivision convergence. The nature of this phenomenon is such that the standard subdivision algorithm can be extended in a trivial manner to allow it to work in instances where it previously failed. Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can be used to calculate the exact values of the refinable function at rational points. Various examples with accompanying figures are given throughout the text to illustrate our results.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: See full text / AFRIKAANSE OPSOMMING: Sien volteks

Engineering design often requires the optimisation of multiple objectives, and becomes significantly more difficult and time consuming when the response surfaces are multimodal, rather than unimodal. A surrogate model, also known as a metamodel, can be used to replace expensive computer simulations, accelerating single and multi-objective optimisation and the exploration of new design concepts. The main research focus of this work is to investigate the use of a neural network surrogate model to improve optimisation of multimodal surfaces. Several significant contributions derive from evaluating the Cascade Correlation neural network as the basis of a surrogate model. The contributions to the neural network community ultimately outnumber those to the optimisation community. The effects of training this surrogate on multimodal test functions are explored. The Cascade Correlation neural network is shown to map poorly such response surfaces. A hypothesis for this weakness is formulated and tested. A new subdivision technique is created that addresses this problem; however, this new technique requires excessively large datasets upon which to train. The primary conclusion of this work is that Cascade Correlation neural networks form an unreliable basis for a surrogate model, despite successes reported in the literature. A further contribution of this work is the enhancement of an open source optimisation toolkit, achieved by the first integration of a truly multi-objective optimisation algorithm.

This thesis describes extensions to 2D and 3D model-based segmentation algorithms for the analysis of ultrasound images and sequences. Starting from a common 2D+t "track-to-last" algorithm, it is shown that the typical method of searching for boundary candidates perpendicular to the model contour is unnecessary if, for each boundary candidate, its corresponding position on the model contour is optimised jointly with the model contour geometry. With this observation, two 2D+t segmentation algorithms, which accurately recover boundary displacements and are capable of segmenting arbitrarily long sequences, are formulated and validated. Generalising to 3D, subdivision surfaces are shown to be natural choices for continuous model surfaces, and the algorithms necessary for joint optimisation of the correspondences and model surface geometry are described. Three applications of 3D model-based segmentation for ultrasound image analysis are subsequently presented and assessed: skull segmentation for fetal brain image analysis; face segmentation for shape analysis, and single-frame left ventricle (LV) segmentation from echocardiography images for volume measurement. A framework to perform model-based segmentation of multiple 3D sequences - while jointly optimising an underlying linear basis shape model - is subsequently presented for the challenging application of right ventricle (RV) segmentation from 3D+t echocardiography sequences. Finally, an algorithm to automatically select boundary candidates independent of a model surface estimate is described and presented for the task of LV segmentation. Although motivated by challenges in ultrasound image analysis, the conceptual contributions of this thesis are general and applicable to model-based segmentation problems in many domains. Moreover, the components are modular, enabling straightforward construction of application-specific formulations for new clinical problems as they arise in the future.

O objetivo deste trabalho foi comparar a atratividade do sorriso entre os diferentes protocolos de tratamento da Classe II subdivisao (extracao de 1, 3 e 4 pre-molares) e verificar a influencia do tamanho do corredor bucal e corredor posterior na estetica do sorriso. A amostra consistiu de fotografias frontais do sorriso posado de 23, 25 e 20 individuos tratados com os protocolos de extracoes de 1, 3 e 4 pre-molares, respectivamente. Em cada fotografia, o corredor bucal e o corredor posterior foram medidos proporcionalmente a largura do sorriso. Por intermedio de um site na internet, as 68 fotografias foram avaliadas quanto a estetica por 46 leigos e 70 ortodontistas, utilizando-se de uma escala de 10 pontos. A atratividade do sorriso entre os 3 protocolos da Classe II subdivisao e os dois grupos de avaliadores foram comparados por meio da Analise de Variancia a dois criterios, enquanto que a influencia da idade e do genero do avaliador na avaliacao da atratividade foi verificada pela Analise de Covariancia. A influencia do tamanho do corredor bucal e do corredor posterior na atratividade do sorriso foi avaliada pelo teste de Correlação de Pearson. Para a comparacao do tamanho dos corredores bucais e corredores posteriores, entre os 3 grupos, foi aplicada a Analise de Variancia. Os resultados demonstraram que nao ha diferenca na atratividade do sorriso entre os pacientes tratados com extracoes de 1, 3 e 4 pre-molares, tanto para leigos quanto para ortodontistas, e o genero e a idade dos avaliadores nao influenciaram a avaliacao da estetica. Os tamanhos dos corredores bucais e dos posteriores nao tiveram correlacao com a atratividade do sorriso. Alem disso, os tamanhos dos corredores bucais e posteriores, entre os grupos, foram estatisticamente semelhantes. / The objective of this study was to compare smile attractiveness between different treatment protocols of Class II subdivision malocclusion (1, 3 and 4-premolar extraction protocols) and verify the esthetic influence of buccal and posterior corridors on the smile attractiveness. The sample consisted of posed smile photographs obtained from 68 subjects, divided into 3 groups according to the treatment extraction protocol. Group 1, treated with 1 premolar extraction protocol included 20 subjects. Group 2, treated with 4 premolars extraction included 25 subjects. The 23 patients in group 3 were treated with 3 premolars extraction. The buccal and posterior corridors of each picture were measured in proportion to the smile width. To rate the posed smile photographs, panels of 70 orthodontists and 46 laypeople used a 10-point scale. Smile attractiveness and differences between the 3 groups and between the 2 panels of raters were subjected to Two-way ANOVA. To determine the relationship between raters gender and age and the esthetic scores Analysis of Covariance was used. Pearson correlation coefficients were calculated to determine the relationship of buccal and posterior corridors size to the esthetic score. The results showed that there were no significant differences in smile attractivenesss scores between the 3 groups and between orthodontists and laypeople. Also raters gender and age did not influence the esthetic score. The buccal and posterior corridors size did not influence the esthetic score. Besides that, buccal and posterior corridors size did not differ between the groups.

O objetivo deste trabalho foi comparar a atratividade do sorriso entre os diferentes protocolos de tratamento da Classe II subdivisao (extracao de 1, 3 e 4 pre-molares) e verificar a influencia do tamanho do corredor bucal e corredor posterior na estetica do sorriso. A amostra consistiu de fotografias frontais do sorriso posado de 23, 25 e 20 individuos tratados com os protocolos de extracoes de 1, 3 e 4 pre-molares, respectivamente. Em cada fotografia, o corredor bucal e o corredor posterior foram medidos proporcionalmente a largura do sorriso. Por intermedio de um site na internet, as 68 fotografias foram avaliadas quanto a estetica por 46 leigos e 70 ortodontistas, utilizando-se de uma escala de 10 pontos. A atratividade do sorriso entre os 3 protocolos da Classe II subdivisao e os dois grupos de avaliadores foram comparados por meio da Analise de Variancia a dois criterios, enquanto que a influencia da idade e do genero do avaliador na avaliacao da atratividade foi verificada pela Analise de Covariancia. A influencia do tamanho do corredor bucal e do corredor posterior na atratividade do sorriso foi avaliada pelo teste de Correlação de Pearson. Para a comparacao do tamanho dos corredores bucais e corredores posteriores, entre os 3 grupos, foi aplicada a Analise de Variancia. Os resultados demonstraram que nao ha diferenca na atratividade do sorriso entre os pacientes tratados com extracoes de 1, 3 e 4 pre-molares, tanto para leigos quanto para ortodontistas, e o genero e a idade dos avaliadores nao influenciaram a avaliacao da estetica. Os tamanhos dos corredores bucais e dos posteriores nao tiveram correlacao com a atratividade do sorriso. Alem disso, os tamanhos dos corredores bucais e posteriores, entre os grupos, foram estatisticamente semelhantes. / The objective of this study was to compare smile attractiveness between different treatment protocols of Class II subdivision malocclusion (1, 3 and 4-premolar extraction protocols) and verify the esthetic influence of buccal and posterior corridors on the smile attractiveness. The sample consisted of posed smile photographs obtained from 68 subjects, divided into 3 groups according to the treatment extraction protocol. Group 1, treated with 1 premolar extraction protocol included 20 subjects. Group 2, treated with 4 premolars extraction included 25 subjects. The 23 patients in group 3 were treated with 3 premolars extraction. The buccal and posterior corridors of each picture were measured in proportion to the smile width. To rate the posed smile photographs, panels of 70 orthodontists and 46 laypeople used a 10-point scale. Smile attractiveness and differences between the 3 groups and between the 2 panels of raters were subjected to Two-way ANOVA. To determine the relationship between raters gender and age and the esthetic scores Analysis of Covariance was used. Pearson correlation coefficients were calculated to determine the relationship of buccal and posterior corridors size to the esthetic score. The results showed that there were no significant differences in smile attractivenesss scores between the 3 groups and between orthodontists and laypeople. Also raters gender and age did not influence the esthetic score. The buccal and posterior corridors size did not influence the esthetic score. Besides that, buccal and posterior corridors size did not differ between the groups.

O objetivo deste trabalho foi comparar os resultados das alterações cefalométricas dentárias, esqueléticas e tegumentares, de casos com má oclusão de Classe II, subdivisão, tratados sob dois protocolos distintos de extrações assimétricas. A amostra foi constituída por 126 teleradiografias em norma lateral de 63 pacientes, os quais foram divididos em dois grupos. O Grupo 1 foi formado por 32 pacientes com má oclusão de Classe II, subdivisão, com idade média inicial de 13,98 e final de 16,90, tratados com extração assimétrica de um pré-molar superior do lado da relação de Classe II.O Grupo 2foi composto por 31 pacientes com má oclusão de Classe II, subdivisão, com idade média inicial de 13,58 e final de 16,83, tratados com extrações assimétricas de três pré-molares, sendo dois superiores e um inferior do lado da relação de Classe I.A comparação intergrupos das alterações do tratamento foi realizada através do teste t. Os resultados demonstraram a existência de diferenças estatisticamente significantes entre as alterações proporcionadas pelos dois tratamentos em relação: ao componente dentoalveolar inferior, onde a terapêutica realizada com uma extração propiciou a vestibularização e protrusão dos incisivos inferiores, ao contrário da terapêutica com três extrações que propiciou uma lingualização e retrusão dos mesmos.Os dois grupos apresentaram uma melhora na relação maxilomandibular. O índice de assimetria dos molares superiores aumentou no grupo com uma extração, enquanto que o grupo com três extrações apresentou um aumento no índice de assimetria dos molares inferiores. Os dois grupos apresentaram semelhanças quanto às alterações do perfil tegumentar. / The purpose of this study was to cephalometrically compare the dentoskeletal and soft-tissue changes consequent to orthodontic treatment in patients with Class II subdivision malocclusion treated with asymmetric extraction protocols. The sample consisted of 126 lateral cephalograms of 63 patients, divided into two groups. Group 1 consisted of 32 patients with Class II, subdivision malocclusion, and had initial and final mean ages of 13.98 and 16.90, treated with asymmetric extraction of one maxillary premolar on the Class II side. Group 2 consisted of 31 patients with Class II, subdivision malocclusion, and had initial and final mean ages of 13.58 and 16.83, treated with asymmetric extractions of two maxillary premolars and one mandibular premolar on the Class I side. T tests were used to compare the groups at pretreatment and the treatment changes between groups. The results demonstrated that group 1 had mandibular incisor protrusion, while group 2 had lower incisor retraction. The maxillary molar asymmetry index increased in group 1, on the contrary of group 2 that had mandibular molar asymmetry index increase. Both groups had similar soft-tissue changes.

Rhinichthys osculus, the Speckled Dace, is one of the most ubiquitous fish in western North America. Within the Southern California region, the local taxon is known as the Santa Ana Speckled Dace. The purpose of this study was to characterize and identify polymorphic microsatellite markers for R. osculus in which twenty-three were identified through Illumina pair-end sequencing. Seven of these loci were then used to examine the patterns of genetic variation and population structure that occurred within and among the watersheds in the Southern California. The study also examined the regional relationships among Southern California, Central California and Owen’s River Valley. Analysis of the microsatellite data revealed highly significant moderate levels of population structure exist within the Southern California region (RST=0.160, p=0.001). This structure is best explained by watershed as well as isolation by distance (R2=.2286, p=0.010). Highly significant geographic structure also exists among the geographic regions of Southern California, Central Coast, and Owen’s River Valley regions (RST= 0.600, p-value=0.001) that are congruent with the regional differentiation elucidated by mtDNA sequence data. In both cases, the degree of population differentiation was correlated with isolation by distance. Utilizing this information we were able gain a better understanding of the evolutionary relationships among the Southern California populations of Santa Ana Speckled Dace. Within the Santa Ana Speckled Dace populations we examined four models to explain the geographic structure: watershed, mountain range, tributary, and isolation by distance. While all were significant, the tributary model exhibited the higher level of population structure (RST= 0.160, p-value=0.001) and a significant correlation was exhibited between geographic distance and population structure, suggesting isolation by distance may be playing a role. The results of the microsatellite analysis are congruent with an earlier broad scale analysis of mtDNA sequence data that suggests the Central California and the Owens Valley populations diverged from each other prior to the divergence of the Santa Ana Speckled Dace populations from the Colorado Basin populations, and that the Central Coast populations were not established as a result of a migration event from the Southern California populations, as was previously hypothesized. Primarily due to human activity, Santa Ana Speckled Dace habitat has become highly fragmented resulting in some populations becoming extirpated. We hope this study will guide the strategies for the conservation of the remaining populations of Santa Ana Speckled Dace and watershed management in Southern California.