This dissertation addresses the problem of adaptive image filtering. Although the topic has a long history in the image processing community, researchers continuously present novel methods to obtain ever better image restoration results. With an expanding market for individuals who wish to share their everyday life on social media, imaging techniques such as compact cameras and smart phones are important factors. Naturally, every producer of imaging equipment desires to exploit cheap camera components while supplying high quality images. One step in this pipeline is to use sophisticated imaging software including, e.g., noise reduction to reduce manufacturing costs, while maintaining image quality. This thesis is based on traditional formulations such as isotropic and tensor-based anisotropic diffusion for image denoising. The difference from main-stream denoising methods is that this thesis explores the effects of introducing contextual information as prior knowledge for image denoising into the filtering schemes. To achieve this, the adaptive filtering theory is formulated from an energy minimization standpoint. The core contributions of this work is the introduction of a novel tensor-based functional which unifies and generalises standard diffusion methods. Additionally, the explicit Euler-Lagrange equation is derived which, if solved, yield the stationary point for the minimization problem. Several aspects of the functional are presented in detail which include, but are not limited to, tensor symmetry constraints and convexity. Also, the classical problem of finding a variational formulation to a given tensor-based partial differential equation is studied. The presented framework is applied in problem formulation that includes non-linear domain transformation, e.g., visualization of medical images. Additionally, the framework is also used to exploit locally estimated probability density functions or the channel representation to drive the filtering process. Furthermore, one of the first truly tensor-based formulations of total variation is presented. The key to the formulation is the gradient energy tensor, which does not require spatial regularization of its tensor components. It is shown empirically in several computer vision applications, such as corner detection and optical flow, that the gradient energy tensor is a viable replacement for the commonly used structure tensor. Moreover, the gradient energy tensor is used in the traditional tensor-based anisotropic diffusion scheme. This approach results in significant improvements in computational speed when the scheme is implemented on a graphical processing unit compared to using the commonly used structure tensor. / VIDI / NACIP / GARNICS / EMC^2

In this thesis we study Sobolev gradient flows for Perona-Malik style energy functionals and generalizations thereof. We begin with first order isotropic flows which are shown to be regularizations of the heat equation. We show that these flows are well-posed in the forward and reverse directions which yields an effective linear sharpening algorithm. We furthermore establish a number of maximum principles for the forward flow and show that edges are preserved for a finite period of time. We then go on to study isotropic Sobolev gradient flows with respect to higher order Sobolev metrics. As the Sobolev order is increased, we observe an increasing reluctance to destroy fine details and texture. We then consider Sobolev gradient flows for non-linear anisotropic diffusion functionals of arbitrary order. We establish existence, uniqueness and continuous dependence on initial data for a broad class of such equations. The well-posedness of these new anisotropic gradient flows opens the door to a wide variety of sharpening and diffusion techniques which were previously impossible under L2 gradient descent. We show how one can easily use this framework to design an anisotropic sharpening algorithm which can sharpen image features while suppressing noise. We compare our sharpening algorithm to the well-known shock filter and show that Sobolev sharpening produces natural looking images without the "staircasing" artifacts that plague the shock filter. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2010-08-25 10:44:12.23

Introdução: O Stent Carotídeo (SC) surgiu como uma alternativa à cirurgia de endarterectomia para o tratamento de estenose carotídea extracraniana com o objetivo de prevenir o acidente vascular encefálico (AVE). O sucesso do SC depende de estratégias que minimizem o risco de AVE. No início do estudo não existiam estudos randomizados comparando o implante de stent carotídeo com diferentes tipos de proteção cerebral. Objetivos: Testar de forma aleatória a eficácia de dois diferentes princípios de proteção embólica no território carotídeo (Angioguard®) e Mo.Ma), utilizando a ressonância magnética ponderada em Difusão (RM-PD) para detectar novas lesões isquêmicas no encéfalo analisando número, tamanho e localização. Métodos: Sessenta pacientes submetidos ao implante do stent carotídeo (SC), foram alocados aleatoriamente para utilizar filtro distal Angioguard® (30 pacientes) e balão de oclusão proximal Mo.Ma (30 pacientes) desde julho de 2008 a 2011. Todos os pacientes realizaram RM-PD pré e 48 horas pós o SC. Os resultados foram avaliados por neurologista independente e cego ao tipo de proteção cerebral utilizada. Foram acompanhados por um período de pelo menos um ano. Os dados qualitativos foram resumidos em frequências absolutas e relativas (porcentagens) e comparados utilizando o teste quiquadrado com correção de continuidade de Yates ou o teste exato de Fisher. Os dados quantitativos foram expressos em médias e desvio-padrão, e/ou medianas e intervalos interquartis e foram comparadas utilizando o teste t de Student ou não paramétrico de Mann-Whitney. Resultados: Não houve diferença estatisticamente significativa em quanto a antecedentes clínicos ou características das lesões carotídeas entre os grupos, apenas as lesões eram mais calcificadas no grupo Angioguard® (p < 0,01). Não houve diferença entre os grupos quanto a incidência de novas lesões isquêmicas (63,3% do Angioguard® vs 66,7% do Mo.Ma, p = 0,787). Quando presentes, as lesões isquêmicas por pacientes o fizeram em número significativamente menor no grupo Mo.Ma, entre 1 e 43 lesões (mediana = 6), comparado ao grupo AngioguardÒ, entre 1 e 76 lesões (mediana = 10) com p < 0,001. Três pacientes (5%) apresentaram eventos neurológicos em até 30 dias e no seguimento de um ano, 1 paciente teve um infarto agudo do miocárdio. Conclusão: Foram observadas novas lesões isquêmicas cerebrais em mais de 60% dos pacientes que utilizaram os dois dispositivos de proteção cerebral, entretanto houve significativamente menos lesões por paciente no grupo Mo.Ma, com significância estatística p = < 0,001. A maioria das lesões foi pequena < 0,5 mm, e encontradas em território ipsilateral. Não foi observado óbito ou AVE maior no seguimento de pelo menos um ano. / Background: Carotid Stent (CAS) has emerged as an alternative to surgical carotid endarterectomy for the treatment of extracranial carotid stenosis in order to prevent stroke. The success of the CAS depends on estrategies that minimize the risk of stroke. When this study began there were no randomized trial comparing different types of cerebral protection during carotid stenting. Objectives: Randomly test the effectiveness of two different embolic protection principles in carotid artery (Angioguard®) vs Mo.Ma) using diffusion-weighted magnetic resonance imaging (DWI) to detect new ischemic lesions in the brain, analyzing the number, size and location of this new ischemic lesions between groups. Methods: Sixty patients undergoing CAS, were randomly assigned to use distal filter AngioguardÒ (30p) and proximal balloon occlusion Mo.Ma (30p) from July 2008 to July 2011. All patients underwent DWI before and 48 hours after the CAS. The results were evaluated by an independent neuroradiologist blind to the type of cerebral protection used. The patients were followed during at least year. Qualitative data were summarized as absolute and relative frequencies (percentages) and compared using chisquare test with Yates continuity correction or Fisher\'s exact test. Quantitative data were expressed as means and standard deviations, and / or medians and interquartile ranges and were compared using the Student t test or nonparametric Mann-Whitney test. Results: Demographic, clinical and lesion characteristics were not different between the two groups, there were more calcified lesion in the Angioguard® group (p < 0.001). There was no difference between groups regarding the incidence of new ischemic lesions in the Angioguard® group compared to the Mo.Ma group (63.3% vs 66.7% p = 0.787). When present, the number of ischemic cerebral lesions per pacient were in fewer number in the Mo.Ma group (1 to 43 lesions; median = 6) compared to the Angioguard® group (1 to 76 lesions; median = 10) p < 0.001 and this difference was significant. Three patients (5%) had neurological events within 30 days with complete regression of symptoms, and one patient develop an infarction during the first year of follow-up. Conclusions: There were new cerebral ischemic lesions detected by DWI in more than 60% of the patients in both groups, on the other hand there were significantly fewer lesions per patient in those allocated to Mo.Ma as compared to Angioguard® with statistical significance p = 0.001. Most lesions were small < 0.5mm, and localized in ipsilateral territory. There was no death or disabling stroke in at least one year of follow-up.

Introdução: O Stent Carotídeo (SC) surgiu como uma alternativa à cirurgia de endarterectomia para o tratamento de estenose carotídea extracraniana com o objetivo de prevenir o acidente vascular encefálico (AVE). O sucesso do SC depende de estratégias que minimizem o risco de AVE. No início do estudo não existiam estudos randomizados comparando o implante de stent carotídeo com diferentes tipos de proteção cerebral. Objetivos: Testar de forma aleatória a eficácia de dois diferentes princípios de proteção embólica no território carotídeo (Angioguard®) e Mo.Ma), utilizando a ressonância magnética ponderada em Difusão (RM-PD) para detectar novas lesões isquêmicas no encéfalo analisando número, tamanho e localização. Métodos: Sessenta pacientes submetidos ao implante do stent carotídeo (SC), foram alocados aleatoriamente para utilizar filtro distal Angioguard® (30 pacientes) e balão de oclusão proximal Mo.Ma (30 pacientes) desde julho de 2008 a 2011. Todos os pacientes realizaram RM-PD pré e 48 horas pós o SC. Os resultados foram avaliados por neurologista independente e cego ao tipo de proteção cerebral utilizada. Foram acompanhados por um período de pelo menos um ano. Os dados qualitativos foram resumidos em frequências absolutas e relativas (porcentagens) e comparados utilizando o teste quiquadrado com correção de continuidade de Yates ou o teste exato de Fisher. Os dados quantitativos foram expressos em médias e desvio-padrão, e/ou medianas e intervalos interquartis e foram comparadas utilizando o teste t de Student ou não paramétrico de Mann-Whitney. Resultados: Não houve diferença estatisticamente significativa em quanto a antecedentes clínicos ou características das lesões carotídeas entre os grupos, apenas as lesões eram mais calcificadas no grupo Angioguard® (p < 0,01). Não houve diferença entre os grupos quanto a incidência de novas lesões isquêmicas (63,3% do Angioguard® vs 66,7% do Mo.Ma, p = 0,787). Quando presentes, as lesões isquêmicas por pacientes o fizeram em número significativamente menor no grupo Mo.Ma, entre 1 e 43 lesões (mediana = 6), comparado ao grupo AngioguardÒ, entre 1 e 76 lesões (mediana = 10) com p < 0,001. Três pacientes (5%) apresentaram eventos neurológicos em até 30 dias e no seguimento de um ano, 1 paciente teve um infarto agudo do miocárdio. Conclusão: Foram observadas novas lesões isquêmicas cerebrais em mais de 60% dos pacientes que utilizaram os dois dispositivos de proteção cerebral, entretanto houve significativamente menos lesões por paciente no grupo Mo.Ma, com significância estatística p = < 0,001. A maioria das lesões foi pequena < 0,5 mm, e encontradas em território ipsilateral. Não foi observado óbito ou AVE maior no seguimento de pelo menos um ano. / Background: Carotid Stent (CAS) has emerged as an alternative to surgical carotid endarterectomy for the treatment of extracranial carotid stenosis in order to prevent stroke. The success of the CAS depends on estrategies that minimize the risk of stroke. When this study began there were no randomized trial comparing different types of cerebral protection during carotid stenting. Objectives: Randomly test the effectiveness of two different embolic protection principles in carotid artery (Angioguard®) vs Mo.Ma) using diffusion-weighted magnetic resonance imaging (DWI) to detect new ischemic lesions in the brain, analyzing the number, size and location of this new ischemic lesions between groups. Methods: Sixty patients undergoing CAS, were randomly assigned to use distal filter AngioguardÒ (30p) and proximal balloon occlusion Mo.Ma (30p) from July 2008 to July 2011. All patients underwent DWI before and 48 hours after the CAS. The results were evaluated by an independent neuroradiologist blind to the type of cerebral protection used. The patients were followed during at least year. Qualitative data were summarized as absolute and relative frequencies (percentages) and compared using chisquare test with Yates continuity correction or Fisher\'s exact test. Quantitative data were expressed as means and standard deviations, and / or medians and interquartile ranges and were compared using the Student t test or nonparametric Mann-Whitney test. Results: Demographic, clinical and lesion characteristics were not different between the two groups, there were more calcified lesion in the Angioguard® group (p < 0.001). There was no difference between groups regarding the incidence of new ischemic lesions in the Angioguard® group compared to the Mo.Ma group (63.3% vs 66.7% p = 0.787). When present, the number of ischemic cerebral lesions per pacient were in fewer number in the Mo.Ma group (1 to 43 lesions; median = 6) compared to the Angioguard® group (1 to 76 lesions; median = 10) p < 0.001 and this difference was significant. Three patients (5%) had neurological events within 30 days with complete regression of symptoms, and one patient develop an infarction during the first year of follow-up. Conclusions: There were new cerebral ischemic lesions detected by DWI in more than 60% of the patients in both groups, on the other hand there were significantly fewer lesions per patient in those allocated to Mo.Ma as compared to Angioguard® with statistical significance p = 0.001. Most lesions were small < 0.5mm, and localized in ipsilateral territory. There was no death or disabling stroke in at least one year of follow-up.