Parâmetro de regularização em problemas inversos: estudo numérico com a transformada de Radon

In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov´s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered

Extração de linhas de imagens de sensoriamento remoto.

Essa dissertação tem o objetivo de verificar a contribuição de diferentes abordagens para extração de linhas, à classificação de imagens multiespectrais, com o possível uso na discriminação e mapeamento de classes de cobertura da terra. Nesse contexto, é efetuada a comparação entre diferentes técnicas de extração de características para extração de linhas de transmissão em áreas rurais, a saber, técnicas de realce utilizando variação de contraste e filtragem morfológica, bem como detecção de bordas utilizando filtro Canny e detector SUSAN, citando como técnica de extração de linhas a Transformada de Hough e Transformada de Radon, utilizando diferentes algoritmos, em imagens aéreas e de sensoriamento remoto. O processo de análise de imagens, com diferentes abordagens leva a resultados variados em diferentes tipos de coberturas do solo. Tais resultados foram avaliados e comparados produzindo tabelas de eficiência para cada procedimento. Estas tabelas direcionam a diferentes encaminhamentos, que vão variar de abordagem dependendo do objetivo final da extração das Linhas de Transmissão.

Recuperação de sinais esparsos. Investigação numérica sobre a quantidade de medidas necessárias para recuperar um sinal esparso

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Identificação de ondas de Rossby a partir de diagramas de Hovmoller da concentração de clorofila entre as latitudes de 25o S e 45o S do Oceano Atlântico

Ondas de Rossby são ondas planetárias oceânicas de ocorrência quase global com comprimentos de onda da ordem de centenas a milhares de quilômetros, períodos da ordem de meses a anos e amplitudes superficiais da ordem de centímetros, causando anomalia da altura da superfície do mar (AASM). Transportando grandes quantidades de energia e influenciando diretamente a circulação de larga escala e o clima global, sua propagação de fase acontece sempre para oeste e sua velocidade é predominantemente uma função da latitude. É sabido que relações entre AASM e concentração de clorofila (CC) podem ser encontradas em todas as bacias oceânicas, evidenciando uma clara influência de fenômenos físicos em fenômenos biológicos. O presente trabalho visa a identificação de ondas de Rossby através da visualização de dados de CC, que, quando comparados a de AASM, apresentaram comprimento, período e velocidade de fase conforme previsto teoricamente para ondas de Rossby baroclínicas do primeiro modo vertical. A região de estudo, entre as latitudes de 25o S e 45o S, é caracterizada pela presença da Corrente do Atlântico Sul, da convergência subtropical e da faixa de rotacional nulo do vento e é, portanto, sede de importantes fenômenos oceanográficos de larga escala. Ambos os conjuntos de dados foram obtidos por sensoriamento remoto e submetidos a uma cadeia de filtros bidimensionais de resposta impulsiva finita (FIR2D) cujo funcionamento de baseia no princípio de convolução entre a série temporal e uma matriz-filtro. Os sinais filtrados foram decompostos em componentes propagantes e não-propagantes e entre os propagantes foi feita a separação das componentes em diversas bandas espectrais definidas por seus períodos centrais. Diagramas de Hovmoller foram confeccionados e as velocidades de fase foram obtidas através da transformada de Radon. A conclusão do trabalho evidenciou uma clara influência das ondas de Rossby sobre a distribuição de clorofila ao longo da região estudada.

Identificação de ondas de Rossby a partir de diagramas de Hovmoller da concentração de clorofila entre as latitudes de 25o S e 45o S do Oceano Atlântico

Ondas de Rossby são ondas planetárias oceânicas de ocorrência quase global com comprimentos de onda da ordem de centenas a milhares de quilômetros, períodos da ordem de meses a anos e amplitudes superficiais da ordem de centímetros, causando anomalia da altura da superfície do mar (AASM). Transportando grandes quantidades de energia e influenciando diretamente a circulação de larga escala e o clima global, sua propagação de fase acontece sempre para oeste e sua velocidade é predominantemente uma função da latitude. É sabido que relações entre AASM e concentração de clorofila (CC) podem ser encontradas em todas as bacias oceânicas, evidenciando uma clara influência de fenômenos físicos em fenômenos biológicos. O presente trabalho visa a identificação de ondas de Rossby através da visualização de dados de CC, que, quando comparados a de AASM, apresentaram comprimento, período e velocidade de fase conforme previsto teoricamente para ondas de Rossby baroclínicas do primeiro modo vertical. A região de estudo, entre as latitudes de 25o S e 45o S, é caracterizada pela presença da Corrente do Atlântico Sul, da convergência subtropical e da faixa de rotacional nulo do vento e é, portanto, sede de importantes fenômenos oceanográficos de larga escala. Ambos os conjuntos de dados foram obtidos por sensoriamento remoto e submetidos a uma cadeia de filtros bidimensionais de resposta impulsiva finita (FIR2D) cujo funcionamento de baseia no princípio de convolução entre a série temporal e uma matriz-filtro. Os sinais filtrados foram decompostos em componentes propagantes e não-propagantes e entre os propagantes foi feita a separação das componentes em diversas bandas espectrais definidas por seus períodos centrais. Diagramas de Hovmoller foram confeccionados e as velocidades de fase foram obtidas através da transformada de Radon. A conclusão do trabalho evidenciou uma clara influência das ondas de Rossby sobre a distribuição de clorofila ao longo da região estudada

Modelado y Mitigación de Fenómenos de Multitrayecto en Imágenes Radar mediante Técnicas basadas en Inversión Temporal

Las técnicas SAR (Synthetic Aperture Radar, radar de apertura sintética) e ISAR (Inverse SAR, SAR inverso) son sistemas radar coherentes de alta resolución, capaces de proporcionar un mapa de la sección radar del blanco en el dominio espacial de distancia y acimut. El objetivo de ambas técnicas radica en conseguir una resolución acimutal más fina generando una apertura sintética a partir del movimiento relativo entre radar y blanco. Los radares imagen complementan la labor de los sistemas ópticos e infrarrojos convencionales, especialmente en condiciones meteorológicas adversas. Los sistemas SAR e ISAR convencionales se diseñan para iluminar blancos en situaciones de línea de vista entre sensor y blanco. Por este motivo, presentan un menor rendimiento en escenarios complejos, como por ejemplo en bosques o entornos urbanos, donde los retornos multitrayecto se superponen a los ecos directos procedentes de los blancos. Se conocen como "imágenes fantasma", puesto que enmascaran a los verdaderos blancos y dan lugar a una calidad visual pobre, complicando en gran medida la detección del blanco. El problema de la mitigación del multitrayecto en imágenes radar adquiere una relevancia teórica y práctica. En esta Tesis Doctoral, se hace uso del concepto de inversión temporal (Time Reversal, TR) para mejorar la calidad visual de las imágenes SAR e ISAR eliminando las "imágenes fantasma" originadas por la propagación multitrayecto (algoritmos TR-SAR y TR-ISAR, respectivamente). No obstante, previamente a la aplicación de estas innovadoras técnicas de mitigación del multi-trayecto, es necesario resolver el problema geométrico asociado al multitrayecto. Centrando la atención en la mejora de las prestaciones de TR-ISAR, se implementan una serie de técnicas de procesado de señal avanzadas antes y después de la etapa basada en inversión temporal (el eje central de esta Tesis). Las primeras (técnicas de pre-procesado) están relacionadas con el multilook averaging, las transformadas tiempo-frecuencia y la transformada de Radon, mientras que las segundas (técnicas de post-procesado) se componen de un conjunto de algoritmos de superresolución. En pocas palabras, todas ellas pueden verse como un valor añadido al concepto de TR, en lugar de ser consideradas como técnicas independientes. En resumen, la utilización del algoritmo diseñado basado en inversión temporal, junto con algunas de las técnicas de procesado de señal propuestas, no deben obviarse si se desean obtener imágenes ISAR de gran calidad en escenarios con mucho multitrayecto. De hecho, las imágenes resultantes pueden ser útiles para posteriores esquemas de reconocimiento automático de blancos (Automatic Target Recognition, ATR). Como prueba de concepto, se hace uso tanto de datos simulados como experimentales obtenidos a partir de radares de alta resolución con el fin de verificar los métodos propuestos.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

The role of vegetation in enhancing radon concentration and ion production in the atmosphere

The role of ions in the production of atmospheric particles has gained wide interest due to their profound impact on climate. Away from anthropogenic sources, molecules are ionized by alpha radiation from radon exhaled from the ground and cosmic gamma radiation from space. These molecular ions quickly form into ‘cluster ions’, typically smaller than about 1.5 nm. Using our measurements and the published literature, we present evidence to show that cluster ion concentrations in forest areas are consistently higher than outside. Since alpha radiation cannot penetrate more than a few centimetres of soil, radon present deep in the ground cannot directly contribute to the measured cluster ion concentrations. We propose an additional mechanism whereby radon, which is water soluble, is brought up by trees and plants through the uptake of groundwater and released into the atmosphere by transpiration. We estimate that, in a forest comprising eucalyptus trees spaced 4m apart, approximately 28% of the radon in the air may be released by transpiration. Considering that 24% of the earth’s land area is still covered in forests; these findings have potentially important implications for atmospheric aerosol formation and climate.

Anisotropy of articular cartilage reflects the ECM gradient architecture : Hough-Radon Transform Analysis

Tissue-specific extracellular matrix (ECM) is known to be an ideal bioscaffold to inspire the future of regenerative medicine. It holds the secret of how nature has developed such an organization of molecules into a unique functional complexity. This work exploited an innovative image processing algorithm and high resolution microscopy associated with mechanical analysis to establish a correlation between the gradient organization of cartiligous ECM and its anisotropic biomechanical response. This was hypothesized to be a reliable determinant that can elucidate how microarchitecture interrelates with biomechanical properties. Hough-Radon transform of the ECM cross-section images revealed its conformational variation from tangential interface down to subchondral region. As the orientation varied layer by layer, the anisotropic mechanical response deviated relatively. Although, results were in good agreement (Kendall's tau-b > 90%), there were evidences proposing that alignment of the fibrous network, specifically in middle zone, is not as random as it was previously thought.

Radon-222 activity flux measurement using activated charcoal canisters: Revisiting the methodology

The measurement of radon ((222)Rn) activity flux using activated charcoal canisters was examined to investigate the distribution of the adsorbed (222)Rn in the charcoal bed and the relationship between (222)Rn activity flux and exposure time. The activity flux of (222)Rn from five sources of varying strengths was measured for exposure times of one, two, three, five, seven, 10, and 14 days. The distribution of the adsorbed (222)Rn in the charcoal bed was obtained by dividing the bed into six layers and counting each layer separately after the exposure. (222)Rn activity decreased in the layers that were away from the exposed surface. Nevertheless, the results demonstrated that only a small correction might be required in the actual application of charcoal canisters for activity flux measurement, where calibration standards were often prepared by the uniform mixing of radium ((226)Ra) in the matrix. This was because the diffusion of (222)Rn in the charcoal bed and the detection efficiency as a function of the charcoal depth tended to counterbalance each other. The influence of exposure time on the measured (222)Rn activity flux was observed in two situations of the canister exposure layout: (a) canister sealed to an open bed of the material and (b) canister sealed over a jar containing the material. The measured (222)Rn activity flux decreased as the exposure time increased. The change in the former situation was significant with an exponential decrease as the exposure time increased. In the latter case, lesser reduction was noticed in the observed activity flux with respect to exposure time. This reduction might have been related to certain factors, such as absorption site saturation or the back diffusion of (222)Rn gas occurring at the canister-soil interface.

On the interaction between radon progeny and particles generated by electronic and traditional cigarettes

During their entire lives, people are exposed to the pollutants present in indoor air. Recently, Electronic Nicotine Delivery Systems, mainly known as electronic cigarettes, have been widely commercialized: they deliver particles into the lungs of the users but a “second-hand smoke” has yet to be associated to this indoor source. On the other hand, the naturally-occurring radioactive gas, i.e. radon, represents a significant risk for lung cancer, and the cumulative action of these two agents could be worse than the agents separately would. In order to deepen the interaction between radon progeny and second-hand aerosol from different types of cigarettes, a designed experimental study was carried out by generating aerosol from e-cigarette vaping as well as from second-hand traditional smoke inside a walk-in radon chamber at the National Institute of Ionizing Radiation Metrology (INMRI) of Italy. In this chamber, the radon present in air comes naturally from the floor and ambient conditions are controlled. To characterize the sidestream smoke emitted by cigarettes, condensation particle counters and scanning mobility particle sizer were used. Radon concentration in the air was measured through an Alphaguard ionization chamber, whereas the measurement of radon decay product in the air was performed with the Tracelab BWLM Plus-2S Radon daughter Monitor. It was found an increase of the Potential Alpha-Energy Concentration (PAEC) due to the radon decay products attached to aerosol for higher particle number concentrations. This varied from 7.47 ± 0.34 MeV L−1 to 12.6 ± 0.26 MeV L−1 (69%) for the e-cigarette. In the case of traditional cigarette and at the same radon concentration, the increase was from 14.1 ± 0.43 MeV L−1 to 18.6 ± 0.19 MeV L−1 (31%). The equilibrium factor increases, varying from 23.4% ± 1.11% to 29.5% ± 0.26% and from 30.9% ± 1.0% to 38.1 ± 0.88 for the e-cigarette and traditional cigarette, respectively. These growths still continue for long time after the combustion, by increasing the exposure risk.

Moments estimation in Radon space

In this paper, we show a method of obtaining general and orthogonal moments, specifically Legendre and Zernicke moments, from the Radon Transform data of a two-dimensional function. The regular or geometric moments are first evaluated directly from the projection data and the orthogonal moments are derived from these regular moments.

End point estimates for Radon transform of radial functions on non-Euclidean spaces

We prove end point estimate for Radon transform of radial functions on affine Grasamannian and real hyperbolic space. We also discuss analogs of these results on the sphere.