The cost and environmental damage of reclaims is a large problem within thepaper industry. With certain types of paper, so called crepe marks on the paper’ssurface is a common issue, leading to printing defects and consequentlyreclaims. This thesis compares four different image analysis methods for evaluatingcrepe marks and predicting printing results. The methods evaluated consistsof one established methods, two adaptations of established methods andone novel method. All methods were evaluated on the same data, topographicheight images of paper samples from 4 paper rolls of similar type but differingin roughness. The method based on 1D Fourier analysis and the method basedon fully convolutional networks performs best, depending on if speed or detailedcharacteristics is a priority.

abstract: Phase contrast magnetic resonance angiography (PCMRA) is a non-invasive imaging modality that is capable of producing quantitative vascular flow velocity information. The encoding of velocity information can significantly increase the imaging acquisition and reconstruction durations associated with this technique. The purpose of this work is to provide mechanisms for reducing the scan time of a 3D phase contrast exam, so that hemodynamic velocity data may be acquired robustly and with a high sensitivity. The methods developed in this work focus on the reduction of scan duration and reconstruction computation of a neurovascular PCMRA exam. The reductions in scan duration are made through a combination of advances in imaging and velocity encoding methods. The imaging improvements are explored using rapid 3D imaging techniques such as spiral projection imaging (SPI), Fermat looped orthogonally encoded trajectories (FLORET), stack of spirals and stack of cones trajectories. Scan durations are also shortened through the use and development of a novel parallel imaging technique called Pretty Easy Parallel Imaging (PEPI). Improvements in the computational efficiency of PEPI and in general MRI reconstruction are made in the area of sample density estimation and correction of 3D trajectories. A new method of velocity encoding is demonstrated to provide more efficient signal to noise ratio (SNR) gains than current state of the art methods. The proposed velocity encoding achieves improved SNR through the use of high gradient moments and by resolving phase aliasing through the use measurement geometry and non-linear constraints. / Dissertation/Thesis / Ph.D. Bioengineering 2011

Introdução: As doenças cerebrovasculares são consideradas atualmente a segunda principal causa de óbito no mundo e a terceira causa de anos de vida perdidos por mortalidade1,2. A aterosclerose de grandes artérias é uma das principais condições subjacentes associadas ao AVC isquêmico (AVCi) e correlaciona-se com maiores taxas de recorrência. A detecção e a estratificação da estenose extracraniana podem ser feitas através do Doppler, da angiotomografia computadorizada (ATC), da angiorressonância magnética (ARM) e da angiografia intra-arterial digital (AIA), sendo esta última considerada o método padrão-ouro. Objetivo: Avaliar a acurácia diagnóstica e a concordância entre os métodos de ATC, ARM com contraste (AM CC) e AIA na análise da estenose extracraniana. Determinar a frequência e caracterizar a estenose extracraniana em população brasileira sintomática. Metodologia: Foram avaliados retrospectivamente os exames AIA, ATC e ARM, de pacientes adultos sintomáticos admitidos na Unidade de Emergência do Hospital das Clínicas da Faculdade de Medicina de Ribeirão Preto (UE HCFMRP) no período 03/2014-03/2015, para detecção e estratificação da estenose extracraniana. Foram coletados ainda dados demográficos, clínicos e laboratoriais desses pacientes. Resultados: Observou-se alta acurácia da ATC e da ARM na avaliação de segmentos carotídeos, com sensibilidade de 100% e especificidade variando entre 78% e 100% para estenoses >=50%. Para segmentos vertebrais, observou-se menor acurácia dos métodos, sobretudo da ARM, com valores de sensibilidade e especificidade variando entre 60%-85% e 33-62%, respectivamente, para estenoses >=50%. A prevalência de estenose extracraniana >=50% em nossa população foi de 38,6% e de 23,4% para estenoses >=70%. O segmento vascular mais acometido foi a origem das artérias vertebrais. Os fatores de risco associados a estenose extracraniana foram: sexo masculino, idade avançada, tabagismo e elevada pressão arterial sistólica na admissão. Conclusão: A ATC e ARM representam métodos adequados para investigação de estenoses extracranianas, sobretudo carotídeas. Ambos os métodos, sobretudo a ARM, têm acurácia limitada na avaliação dos óstios vertebrais. / Introduction: Cerebrovascular diseases are considered the second cause of death worldwide and the third cause of years of life lost (YLL)1,2. Large arteries atherosclerosis is one of the main conditions associated with stroke and correlates with higher recurrence rates. Imaging methods such as Doppler, computed tomography angiography (CTA), magnetic resonance angiography (MRA) and intra-arterial angiography (IAA) are available for extra cranial vascular study. Intra-arterial angiography remains the gold standard. Objectives: Evaluate diagnostic accuracy and concordance between CTA, MRA (contrast enhanced) and IAA for extracranial stenosis analysis. Determine frequency and distribution of extra cranial stenosis in a brazilian symptomatic population. Methods: IAA, CTA and MRA exams from adult symptomatic patients admitted between 03/2014 and 03/3015 at the Emergency Unit of Clinics Hospital of Ribeirão Preto\'s School of Medicine were retrospectively reviewed for detection and grading of extra cranial stenosis. Clinical and laboratorial information from patients was obtained. Results: Both CTA and MRA showed high sensitivity (100%) and specificity (78%-100%) in the evaluation of carotid vessels for stenosis >=50%. However, lower sensitivity (60-85%) and specificity (33-62%) were obtained for vertebral segments, Extra cranial stenosis prevalence was 38,6% for stenosis >=50% and 23,4% for stenosis >=70% among our patients. Vertebral artery origins were the most affected segments. Risk factors identified for extra cranial stenosis were: male gender, age, smoking and high systolic blood pressure on admission. Conclusions: CTA and MRA are considered adequate methods for extra cranial stenosis, specially for carotid segments. Both methods have lower accuracy in the investigation of vertebral artery origins.

L'analyse des structures curvilignes en 3 dimensions est un problème difficile en analyse d'images. En effet, ces structures sont fines, facilement corrompues par le bruit et présentent une géométrie complexe. Depuis plusieurs années, de nombreuses méthodes spécialement dédiées au traitement d'images contenant des structures curvilignes ont vu le jour. Ces méthodes concernent diverses applications en science des matériaux, télédétection ou encore en imagerie médicale. Malgré cela, l'analyse des structures curvilignes demeure une tâche complexe.Dans cette présentation nous parlerons de la caractérisation des structures curvilignes pour l'analyse d'images. Nous présenterons en premier lieu une nouvelle méthode appelée RORPO, à partir de laquelle deux caractéristiques peuvent être calculées. La première est une caractéristique d'intensité, qui préserve l'intensité des structures curvilignes tout en réduisant celle des autres structures. La deuxième est une caractéristique de direction, qui fournit en chaque point d'une image, la direction d'une structure curviligne potentielle.RORPO, contrairement à la plupart des méthodes de la littérature, est une méthode non locale, non linéaire et mieux adaptées à l'anisotropie intrinsèque des structures curvilignes. Cette méthode repose sur une notion récente de Morphologie Mathématique: les opérateurs par chemins.RORPO peut directement servir au filtrage d'images contenant des structures curvilignes, afin de spécifiquement les préserver, mais aussi de réduire le bruit. Mais les deux caractéristiques de RORPO peuvent aussi être utilisées comme information a priori sur les structure curvilignes, afin d'être intégrées dans une méthode plus complexe d'analyse d'image.Dans un deuxième temps, nous présenterons ainsi un terme de régularisation destiné à la segmentation variationnelle, utilisant les deux caractéristiques de RORPO.L'information apportée par ces deux caractéristiques permet de régulariser les structures curvilignes seulement dans la direction de leur axe principal. De cette manière, ces structures sont mieux préservées, et certaines structures curvilignes déconnectées par le bruit peuvent aussi être reconnectées.Des résultats en 2D et 3D de ces méthodes seront enfin présentées sur des images de vaisseaux sanguins provenant de diverses modalités / The analysis of curvilinear structures in 3D images is a complex and challenging task. Curvilinear structures are thin, easily corrupted by noise and present a complex geometry. Despite the numerous applications in material sciences, remote sensing and medical imaging and the large number of dedicated methods developed the last few years, the detection of such structures remains a difficult problem.In this thesis, we work on the characterization of curvilinear structures. We first propose a new framework called RORPO, to characterize such structures through two features: an intensity feature which preserves the intensity of curvilinear structures while decreasing the intensity of other structures, and a directional feature providing at each point, the direction of the curvilinear structure.RORPO, unlike classic other state of the art methods, is non-local and non-linear, which are desirable properties adapted to the intrinsic anisotropy of curvilinear structures. RORPO is based on recent advances in mathematical morphology: the path operators.We provide a full description of the structural and algorithmic details of RORPO, and we also conduct a quantitative comparative study of our features with three popular curvilinear structure analysis filters: the Frangi Vesselness, the Optimally Oriented Flux, and the Hybrid Diffusion with Continuous Switch.Besides the straightforward filtering application, both RORPO features are designed to be used as prior information to characterize curvilinear structures. We propose a regularization term for variational segmentation which embed these features. Classic regularization terms are not adapted to curvilinear structures and usually lead to the loss of most of the low-contrasted ones. We propose to only regularize curvilinear structures along their main axis thanks to both RORPO features. This directional regularization better preserves curvilinear structures but also reconnect parts of these structures which may have been disconnected by noise.We present results of the segmentation of retinal images with the Chan et al. model either with the classic total variation or our directional regularization term. This confirm that our regularization term is better suited for images with curvilinear structures

Cette thèse constitue à la fois un apport de nature clinique et technologique dans l’approche diagnostique du glaucome. Plus précisément, nous nous proposons d’étudier une nouvelle façon de détecter le glaucome par la mesure de l’asymétrie du rapport de la surface de l’anneau neurorétinien et de la surface de la papille ou du disque optique ou rim to disc area asymmetry ratio (RADAAR). Pour atteindre cet objectif, nous avons recours à une base de données composée d’une population subdivisée en 4 différents groupes de diagnostic (normal, glaucome possible, glaucome probable et glaucome définitif). Les mesures du RADAAR sont calculées de différentes façons à partir des paramètres stéréométriques de la tête du nerf optique des sujets, produits par la microscopie confocale à balayage laser (Heidelberg Retina Tomograph (HRT) (Heidelberg Engineering, Germany)). Nous procédons à une analyse de données grâce au logiciel SPSS où nous mettons en exergue la distribution du RADAAR dans les différentes populations, sa validité et son utilité dans le dépistage du glaucome. Nous enrôlons donc 523 sujets dans cette étude avec 82 sujets atteints de glaucome définitif. La moyenne d’âge est de 62 ans. Il y a plus de femmes que d’hommes et plus de Caucasiens que d’Africains Caribéens. Nous trouvons que la distribution de la mesure du RADAAR est différente des sujets d’un groupe de diagnostic à l’autre. En termes de performance, la sensibilité de la mesure du RADAAR est très basse c'est-à-dire que sa capacité de détecter la maladie est basse. En revanche la mesure du RADAAR est plus spécifique c'est-à-dire que sa capacité d’identifier les sujets exempts de la maladie est plus grande. Elle tendrait à être aussi plus performante chez les Africains Caribéens que chez les Caucasiens. De même, elle serait plus sensible chez les hommes que chez les femmes. La mesure du RADAAR est utile si on l’associe à une autre méthode de diagnostic comme l’analyse de Régression de Moorfields (MRA) incluse dans le logiciel du HRT3 spécialement lors de la détection du glaucome dans la population à haut risque. En définitive, nous déterminons que la mesure du RADAAR se veut un outil d’aide au diagnostic. Elle est particulièrement intéressante dans le contexte de dépistage de glaucome. / This thesis describes a new clinical and technological approach to the diagnosis of glaucoma. Specifically, we intend to study a new way to detect glaucoma by measuring rim area to disc area asymmetry ratio (RADAAR). For this purpose, we use a database consisting of a population divided into 4 different diagnostic groups (normal, possible glaucoma, probable glaucoma and definitive glaucoma). The RADAAR measurements are calculated in different ways based on the stereometric parameters of the optic nerve head of subjects, produced by confocal scanning laser ophthalmoscopy (Heidelberg Retina Tomograph (HRT) (Heidelberg Engineering, Germany)). We conduct an analysis of data with SPSS or we put forward the RADAAR distribution in different populations, its validity in detecting open angle glaucoma and its usefulness in screening for glaucoma. We therefore enroll 523 subjects in this study with about 82 subjects with definitive glaucoma. The average age is 62 years. There are more females than males and more Caucasians than Africans Caraibeans. We find that the distribution of RADAAR measures is different in each diagnosis group. In terms of performance, the sensitivity of the RADAAR measurement is very low. So, its ability to detect the disease is low. However the RADAAR measure is much more specific so, its ability to identify subjects free of the disease is high. RADAAR measure would also be much more effective in African Caribbean’s than in Caucasians. Similarly, it would be much more sensitive in males than in females. The RADAAR measurement is useful if it is combined with another method of diagnosis like the Moorfields regression analysis (MRA) included in the HRT3 software especially in case of the detection of glaucoma in populations at high risk. Ultimately, we determine that the RADAAR is an interesting tool for the diagnosis of glaucoma particularly in the context of screening for glaucoma.

Tumormodelle, in denen die maligne Transformation durch definierte Onkogene experimentell ausgelöst und unterhalten wird, bieten vielfältige Möglichkeiten, die komplexen Mechanismen der Tumorentstehung und Therapieresistenz zu untersuchen und neue Ansätze für Diagnostik und Therapie auszuarbeiten. KRAS-Onkogen-„getriebene“ Transformationsmodelle spiegeln neben anderen tumorspezifischen Veränderungen insbesondere die charakteristischen Änderungen des Transkriptoms wider. In der vorliegenden Arbeit wird ein Modell für Ovarialtumore auf Grundlage von Rose Zellen („Rat ovarian surface epithelium“) verwendet, um die Rolle von Transkriptionsfaktoren, welche durch die KRAS-vermittelte Signaltransduktion hoch reguliert werden, zu untersuchen. Die KRAS-transfomierten Derivate der normalen Rose Zellen zeigen die typischen Merkmale von ankerunabhängigen und invasiven Tumorzellen. Aufgrund der hohen Komplexität sind die Interaktionen zwischen der zytoplasmatischen Signaltransduktion und dem durch sie regulierten Transkriptionsfaktornetzwerk noch weitgehend unverstanden. Die Transkriptionsfaktoren Fosl1, Hmga2, Klf6, JunB, Otx1, Gfi1 und RelA wurden systematisch mittels RNA-Interferenz in KRAS-transformierten Rose Zellen transient ausgeschaltet. Danach wurden Proliferation, Morphologie (epithelial-mesenchymale Transition, EMT) und Ankerunabhängigkeit der Zellen bestimmt. Alle untersuchten Transkriptionsfaktoren beeinflussten die KRAS-induzierten morphologischen Veränderungen teilweise, belegt durch die Abnahme der EMT-Merkmale nach siRNA-vermittelter Ausschaltung. Der Knock-down der Transkriptionsfaktoren Otx1, Gfi1 und RelA hemmte die Proliferation, während Fosl1, Hmga2, Klf6 und JunB die generelle Proliferationsfähigkeit nicht beeinflussten, jedoch spezifisch die ankerunabhängige Proliferation blockierten. Diesen Faktoren kommt daher eine spezifische Funktion in der neoplastischen Transformation zu, da die Ankerunabhängigkeit sehr gut mit der Tumorigenität korreliert ist. Um die Beteiligung der Transkriptionsfaktoren an der Deregulation von Zielgenen zu erfassen, wurden Genexpressionsmuster aller Zellen, in denen jeweils ein Faktor durch siRNA ausgeschaltet war, mittels Microarray-Analyse identifiziert. Auf dieser Grundlage wurde ein Netzwerk-Modell der regulatorischen Interaktionen zwischen den Transkriptionsfaktoren berechnet. Die Existenz der beiden funktionellen Gruppen wurde im Modell bestätigt. Darüber hinaus zeigte sich eine gegenseitige Abhängigkeit des transkriptionellen Netzwerks und der zytoplasmatischen Signaltransduktion, gemessen mittels Proteinanalyse der mitogenabhängigen Signalkinasen (MAPK). Diese wird als kompensatorische Regulation interpretiert, welche trotz Pertubation, experimentell durch siRNA, das effiziente Überleben der transformierten Zellen sicherstellt. Die vorliegende Studie schafft somit die Voraussetzung und Motivation, das reduzierte Netzwerk aus sieben Komponenten auf alle differentiell exprimierten Transkriptionsfaktoren zu erweitern. Möglicherweise behindern solche Regulationskreise in der klinischen Situation die effektive Wirkung zielgerichteter Therapien. / Tumor models, in which malignant transformation was experimentally triggered and maintained through defined oncogenes, offer manifold opportunities to determine the complex mechanisms of tumor progression and resistance to therapies, and to develop new strategies for diagnosis and therapy. Particularly, KRAS oncogene driven models of transformation reflect the characteristic alterations of the transcriptome, among other tumor specific changes. In the present work a model for ovarian cancer based on Rose („Rat ovarian surface epithelium“) cells has been used to evaluate the role of transcription factors, which are up-regulated through KRAS dependent signaling. The KRAS transformed derivates of normal ROSE cells exhibit typical characteristics of anchorage-independent and invasive tumor cells. Due to the high complexity of cellular networks, the interactions between cytoplasmic signalling and their regulated transcription factors are not well understood. The transcription factors Fosl1, Hmga2, Klf6, JunB, Otx1, Gfi1 and RelA were systematically eliminated by transient RNA interference in KRAS transformed ROSE cells. The proliferation, morphology (epithelial-mesenchymal transition, EMT) and anchorage-independence of the cells were determined. All of the selected transcription factors had partial effect on the KRAS induced morphologic changes, documented by reduction of EMT-properties after siRNA treatment. The knock-down of the transcription factors Otx1, Gfi1 and RelA blocked proliferation in general, whereas Fosl1, Hmga2, Klf6 and JunB had no influence on proliferation but specifically blocked the anchorage-independence. Thus, these factors exhibited essential functions in the process of neoplastic transformation, because the anchorage-independence correlates very well with tumorigenicity. In order to elucidate the involvement of the transcription factors in the genetic deregulation of their target genes, microarray based gene expression profiles were determined from all cells in which one factor was eliminated by siRNA. Based on these data, a network model of regulatory interactions among these transcription factors was calculated. The existence of both functional groups was confirmed by the model. Furthermore, an interdependence of the transcriptional networks and cytoplasmatic signaling was observed by protein analysis of the mitogen dependent signal kinases (MAPK). This was interpreted as compensatory regulation, which in spite of experimental perturbation by siRNA, permitted efficient survival of the transformed cells. Thus, the present work provides the basis and motivation to extend the reduced network composed of seven components to all regulated transcription factors. Potentially, such regulatory networks diminish the efficacy of targeted therapies in clinical situations.

Cette thèse constitue à la fois un apport de nature clinique et technologique dans l’approche diagnostique du glaucome. Plus précisément, nous nous proposons d’étudier une nouvelle façon de détecter le glaucome par la mesure de l’asymétrie du rapport de la surface de l’anneau neurorétinien et de la surface de la papille ou du disque optique ou rim to disc area asymmetry ratio (RADAAR). Pour atteindre cet objectif, nous avons recours à une base de données composée d’une population subdivisée en 4 différents groupes de diagnostic (normal, glaucome possible, glaucome probable et glaucome définitif). Les mesures du RADAAR sont calculées de différentes façons à partir des paramètres stéréométriques de la tête du nerf optique des sujets, produits par la microscopie confocale à balayage laser (Heidelberg Retina Tomograph (HRT) (Heidelberg Engineering, Germany)). Nous procédons à une analyse de données grâce au logiciel SPSS où nous mettons en exergue la distribution du RADAAR dans les différentes populations, sa validité et son utilité dans le dépistage du glaucome. Nous enrôlons donc 523 sujets dans cette étude avec 82 sujets atteints de glaucome définitif. La moyenne d’âge est de 62 ans. Il y a plus de femmes que d’hommes et plus de Caucasiens que d’Africains Caribéens. Nous trouvons que la distribution de la mesure du RADAAR est différente des sujets d’un groupe de diagnostic à l’autre. En termes de performance, la sensibilité de la mesure du RADAAR est très basse c'est-à-dire que sa capacité de détecter la maladie est basse. En revanche la mesure du RADAAR est plus spécifique c'est-à-dire que sa capacité d’identifier les sujets exempts de la maladie est plus grande. Elle tendrait à être aussi plus performante chez les Africains Caribéens que chez les Caucasiens. De même, elle serait plus sensible chez les hommes que chez les femmes. La mesure du RADAAR est utile si on l’associe à une autre méthode de diagnostic comme l’analyse de Régression de Moorfields (MRA) incluse dans le logiciel du HRT3 spécialement lors de la détection du glaucome dans la population à haut risque. En définitive, nous déterminons que la mesure du RADAAR se veut un outil d’aide au diagnostic. Elle est particulièrement intéressante dans le contexte de dépistage de glaucome. / This thesis describes a new clinical and technological approach to the diagnosis of glaucoma. Specifically, we intend to study a new way to detect glaucoma by measuring rim area to disc area asymmetry ratio (RADAAR). For this purpose, we use a database consisting of a population divided into 4 different diagnostic groups (normal, possible glaucoma, probable glaucoma and definitive glaucoma). The RADAAR measurements are calculated in different ways based on the stereometric parameters of the optic nerve head of subjects, produced by confocal scanning laser ophthalmoscopy (Heidelberg Retina Tomograph (HRT) (Heidelberg Engineering, Germany)). We conduct an analysis of data with SPSS or we put forward the RADAAR distribution in different populations, its validity in detecting open angle glaucoma and its usefulness in screening for glaucoma. We therefore enroll 523 subjects in this study with about 82 subjects with definitive glaucoma. The average age is 62 years. There are more females than males and more Caucasians than Africans Caraibeans. We find that the distribution of RADAAR measures is different in each diagnosis group. In terms of performance, the sensitivity of the RADAAR measurement is very low. So, its ability to detect the disease is low. However the RADAAR measure is much more specific so, its ability to identify subjects free of the disease is high. RADAAR measure would also be much more effective in African Caribbean’s than in Caucasians. Similarly, it would be much more sensitive in males than in females. The RADAAR measurement is useful if it is combined with another method of diagnosis like the Moorfields regression analysis (MRA) included in the HRT3 software especially in case of the detection of glaucoma in populations at high risk. Ultimately, we determine that the RADAAR is an interesting tool for the diagnosis of glaucoma particularly in the context of screening for glaucoma.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Multi-resolution analysis (MRA) has become a very popular eld of mathematical study in the past two decades, being not only an area rich in applications but one that remains lled with open problems. Building on the foundation of re nability of functions, MRA seeks to lter through levels of ever-increasing detail components in data sets { a concept enticing to an age where development of digital equipment (to name but one example) needs to capture more and more information and then store this information in di erent levels of detail. Except for designing digital objects such as animation movies, one of the most recent popular research areas in which MRA is applied, is inpainting, where \lost" data (in example, a photograph) is repaired by using boundary values of the data set and \smudging" these values into the empty entries. Two main branches of application in MRA are subdivision and wavelet analysis. The former uses re nable functions to develop algorithms with which digital curves are created from a nite set of initial points as input, the resulting curves (or drawings) of which possess certain levels of smoothness (or, mathematically speaking, continuous derivatives). Wavelets on the other hand, yield lters with which certain levels of detail components (or noise) can be edited out of a data set. One of the greatest advantages when using wavelets, is that the detail data is never lost, and the user can re-insert it to the original data set by merely applying the wavelet algorithm in reverse. This opens up a wonderful application for wavelets, namely that an existent data set can be edited by inserting detail components into it that were never there, by also using such a wavelet algorithm. In the recent book by Chui and De Villiers (see [2]), algorithms for both subdivision and wavelet applications were developed without using Fourier analysis as foundation, as have been done by researchers in earlier years and which have left such algorithms unaccessible to end users such as computer programmers. The fundamental result of Chapter 9 on wavelets of [2] was that feasibility of wavelet decomposition is equivalent to the solvability of a certain set of identities consisting of Laurent polynomials, referred to as Bezout identities, and it was shown how such a system of identities can be solved in a systematic way. The work in [2] was done in the univariate case only, and it will be the purpose of this thesis to develop similar results in the bivariate case, where such a generalization is entirely non-trivial. After introducing MRA in Chapter 1, as well as discussing the re nability of functions and introducing box splines as prototype examples of functions that are re nable in the bivariate setting, our fundamental result will also be that wavelet decomposition is equivalent to solving a set of Bezout identities; this will be shown rigorously in Chapter 2. In Chapter 3, we give a set of Laurent polynomials of shortest possible length satisfying the system of Bezout identities in Chapter 2, for the particular case of the Courant hat function, which will have been introduced as a linear box spline in Chapter 1. In Chapter 4, we investigate an application of our result in Chapter 3 to bivariate interpolatory subdivision. With the view to establish a general class of wavelets corresponding to the Courant hat function, we proceed in the subsequent Chapters 5 { 8 to develop a general theory for solving the Bezout identities of Chapter 2 separately, before suggesting strategies for reconciling these solution classes in order to be a simultaneous solution of the system. / AFRIKAAANSE OPSOMMING: Multi-resolusie analise (MRA) het in die afgelope twee dekades toenemende gewildheid geniet as 'n veld in wiskundige wetenskappe. Nie net is dit 'n area wat ryklik toepaslik is nie, maar dit bevat ook steeds vele oop vraagstukke. MRA bou op die grondleggings van verfynbare funksies en poog om deur vlakke van data-komponente te sorteer, of te lter, 'n konsep wat aanloklik is in 'n era waar die ontwikkeling van digitale toestelle (om maar 'n enkele voorbeeld te noem) sodanig moet wees dat meer en meer inligting vasgel^e en gestoor moet word. Behalwe vir die ontwerp van digitale voorwerpe, soos animasie- lms, word MRA ook toegepas in 'n mees vername navorsingsgebied genaamd inverwing, waar \verlore" data (soos byvoorbeeld in 'n foto) herwin word deur data te neem uit aangrensende gebiede en dit dan oor die le e data-dele te \smeer." Twee hooftakke in toepassing van MRA is subdivisie en gol e-analise. Die eerste gebruik verfynbare funksies om algoritmes te ontwikkel waarmee digitale krommes ontwerp kan word vanuit 'n eindige aantal aanvanklike gegewe punte. Die verkrygde krommes (of sketse) kan voldoen aan verlangde vlakke van gladheid (of verlangde grade van kontinue afgeleides, wiskundig gesproke). Gol es word op hul beurt gebruik om lters te bou waarmee gewensde dataof geraas-komponente verwyder kan word uit datastelle. Een van die grootste voordeel van die gebruik van gol es bo ander soortgelyke instrumente om data lters mee te bou, is dat die geraas-komponente wat uitgetrek word nooit verlore gaan nie, sodat die proses omkeerbaar is deurdat die gebruiker die sodanige geraas-komponente in die groter datastel kan terugbou deur die gol e-algoritme in trurat toe te pas. Hierdie eienskap van gol fies open 'n wonderlike toepassingsmoontlikheid daarvoor, naamlik dat 'n bestaande datastel verander kan word deur data-komponente daartoe te voeg wat nooit daarin was nie, deur so 'n gol e-algoritme te gebruik. In die onlangse boek deur Chui and De Villiers (sien [2]) is algoritmes ontwikkel vir die toepassing van subdivisie sowel as gol es, sonder om staat te maak op die grondlegging van Fourier-analise, soos wat die gebruik was in vroe ere navorsing en waardeur algoritmes wat ontwikkel is minder e ektief was vir eindgebruikers. Die fundamentele resultaat oor gol es in Hoofstuk 9 in [2], verduidelik hoe suksesvolle gol e-ontbinding ekwivalent is aan die oplosbaarheid van 'n sekere versameling van identiteite bestaande uit Laurent-polinome, bekend as Bezout-identiteite, en dit is bewys hoedat sodanige stelsels van identiteite opgelos kan word in 'n sistematiese proses. Die werk in [2] is gedoen in die eenveranderlike geval, en dit is die doelwit van hierdie tesis om soortgelyke resultate te ontwikkel in die tweeveranderlike geval, waar sodanige veralgemening absoluut nie-triviaal is. Nadat 'n inleiding tot MRA in Hoofstuk 1 aangebied word, terwyl die verfynbaarheid van funksies, met boks-latfunksies as prototipes van verfynbare funksies in die tweeveranderlike geval, bespreek word, word ons fundamentele resultaat gegee en bewys in Hoofstuk 2, naamlik dat gol e-ontbinding in die tweeveranderlike geval ook ekwivalent is aan die oplos van 'n sekere stelsel van Bezout-identiteite. In Hoofstuk 3 word 'n versameling van Laurent-polinome van korste moontlike lengte gegee as illustrasie van 'n oplossing van 'n sodanige stelsel van Bezout-identiteite in Hoofstuk 2, vir die besondere geval van die Courant hoedfunksie, wat in Hoofstuk 1 gede nieer word. In Hoofstuk 4 ondersoek ons 'n toepassing van die resultaat in Hoofstuk 3 tot tweeveranderlike interpolerende subdivisie. Met die oog op die ontwikkeling van 'n algemene klas van gol es verwant aan die Courant hoedfunksie, brei ons vervolglik in Hoofstukke 5 { 8 'n algemene teorie uit om die oplossing van die stelsel van Bezout-identiteite te ondersoek, elke identiteit apart, waarna ons moontlike strategie e voorstel vir die versoening van hierdie klasse van gelyktydige oplossings van die Bezout stelsel.

<p>Detta examensarbete har genomförts på uppdrag av Försvarets Materielverk (FMV). </p><p>Det påstås att evaluering av komplexa/sammansatta system är svårt, om inte omöjligt. Common Criteria (CC) är en ISO/IEC-standard för evaluering av produkter och system. Även Information Technology Security Evaluation Criteria (ITSEC) används för samma ändamål. Jag har undersökt vad som är möjligt och denna uppsats beskriver alternativa evalueringslösningar med hjälp av CC/ITSEC, dock primärt enligt CC; där fokus hela tiden vilar på ackreditering av system och produkter inom det svenska försvaret. </p><p>Metoden som används bygger på kvalitativ och utredande analys vilken baseras på empirisk kunskap inhämtad från nyckelaktörer på marknaden samt en teoretisk referensram vilken utgörs av nyckelstandarder och för arbetet/uppsatsen relevanta dokument. </p><p>Jag har undersökt SYSn och FTA vilka är två ytterst intressanta lösningar för två olika typer av evalueringssituationer vid informell evaluering. Båda lösningarna fokuserar på systemevalueringar men produktevalueringar stöds även de. SYSn och FTA är framtagna av UK:s myndighet CESG vid GCHQ. </p><p>SYSn är en mer pragmatisk, kostnadseffektiv, snabb (med avseende på total evalueringstid) och flexibel lösning än existerande formella CCoch ITSEC-lösningar. Samtidigt bygger SYSn på CEM vilket gör att SYSn åtnjuter många av CC:s och CEM:s fördelar med avseende på bland annat återanvändbarhet och god struktur. SYSn erbjuder motsvarande assurans som CC: s EAL (SYS2 ≈ EAL2, SYS3 ≈ EAL3, SYS4 ≈EAL4).. Spårbarhet gällande utfall, testprocedurer, använd metodik etc. bibehålls. </p><p>FTA å andra sidan är ett avsevärt snabbare evalueringsinstrument där avkall på återanvändbarhet, struktur, dokumentation och förtroende görs. FTA är en lösning vilken fokuserar på best effort och ger inga direkta garantier om assuransnivåer etc. När det är ont om tid och inget certifikat eller ett formellt utlåtande för evalueringsresultatet efterfrågas är FTA en tilltalande lösning. Evaluering genomförs i enlighet med Fast Track Assessment Methodology (FTAM).</p>

Detta examensarbete har genomförts på uppdrag av Försvarets Materielverk (FMV). Det påstås att evaluering av komplexa/sammansatta system är svårt, om inte omöjligt. Common Criteria (CC) är en ISO/IEC-standard för evaluering av produkter och system. Även Information Technology Security Evaluation Criteria (ITSEC) används för samma ändamål. Jag har undersökt vad som är möjligt och denna uppsats beskriver alternativa evalueringslösningar med hjälp av CC/ITSEC, dock primärt enligt CC; där fokus hela tiden vilar på ackreditering av system och produkter inom det svenska försvaret. Metoden som används bygger på kvalitativ och utredande analys vilken baseras på empirisk kunskap inhämtad från nyckelaktörer på marknaden samt en teoretisk referensram vilken utgörs av nyckelstandarder och för arbetet/uppsatsen relevanta dokument. Jag har undersökt SYSn och FTA vilka är två ytterst intressanta lösningar för två olika typer av evalueringssituationer vid informell evaluering. Båda lösningarna fokuserar på systemevalueringar men produktevalueringar stöds även de. SYSn och FTA är framtagna av UK:s myndighet CESG vid GCHQ. SYSn är en mer pragmatisk, kostnadseffektiv, snabb (med avseende på total evalueringstid) och flexibel lösning än existerande formella CCoch ITSEC-lösningar. Samtidigt bygger SYSn på CEM vilket gör att SYSn åtnjuter många av CC:s och CEM:s fördelar med avseende på bland annat återanvändbarhet och god struktur. SYSn erbjuder motsvarande assurans som CC: s EAL (SYS2 ≈ EAL2, SYS3 ≈ EAL3, SYS4 ≈EAL4).. Spårbarhet gällande utfall, testprocedurer, använd metodik etc. bibehålls. FTA å andra sidan är ett avsevärt snabbare evalueringsinstrument där avkall på återanvändbarhet, struktur, dokumentation och förtroende görs. FTA är en lösning vilken fokuserar på best effort och ger inga direkta garantier om assuransnivåer etc. När det är ont om tid och inget certifikat eller ett formellt utlåtande för evalueringsresultatet efterfrågas är FTA en tilltalande lösning. Evaluering genomförs i enlighet med Fast Track Assessment Methodology (FTAM).