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(1 - 9 of 9)
- Title
- Filtering Problems in Stochastic Tomography.
- Creator
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Gomez, Tyler, Swanson, Jason, Yong, Jiongmin, Tamasan, Alexandru, Dogariu, Aristide, University of Central Florida
- Abstract / Description
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Distinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having...
Show moreDistinguishing signal from noise has always been a major goal in probabilistic analysis of data. Such is no less the case in the field of medical imaging, where both the processes of photon emission and their rate of absorption by the body behave as random variables. We explore methods by which to extricate solid conclusions from noisy data involving an X-ray transform, long the mathematical mainstay of such tools as computed axial tomography (CAT scans). Working on the assumption of having some prior probabilities assigned to various states a body can be found in, we introduce and make rigorous an understanding of how to condition these into posterior probabilities by using the scan data.
Show less - Date Issued
- 2017
- Identifier
- CFE0006740, ucf:51839
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006740
- Title
- On the range of the Attenuated Radon Transform in strictly convex sets.
- Creator
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Sadiq, Kamran, Tamasan, Alexandru, Nashed, M, Katsevich, Alexander, Dogariu, Aristide, University of Central Florida
- Abstract / Description
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In the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert...
Show moreIn the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of A. Bukhgeim. We then uses the range characterization of the Radon transform of functions to characterize the range of the attenuated Radon transform of vector fields as they appear in the medical diagnostic techniques of Doppler tomography. As an application we determine necessary and sufficient conditions for the Doppler and X-ray data to be mistaken for each other. We also characterize the range of real symmetric second order tensor field using the range characterization of the Radon transform of zero tensor field.
Show less - Date Issued
- 2014
- Identifier
- CFE0005408, ucf:50437
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005408
- Title
- Functional Data Analysis and its application to cancer data.
- Creator
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Martinenko, Evgeny, Pensky, Marianna, Tamasan, Alexandru, Swanson, Jason, Richardson, Gary, University of Central Florida
- Abstract / Description
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The objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is...
Show moreThe objective of the current work is to develop novel procedures for the analysis of functional dataand apply them for investigation of gender disparity in survival of lung cancer patients. In particular,we use the time-dependent Cox proportional hazards model where the clinical information isincorporated via time-independent covariates, and the current age is modeled using its expansionover wavelet basis functions. We developed computer algorithms and applied them to the dataset which is derived from Florida Cancer Data depository data set (all personal information whichallows to identify patients was eliminated). We also studied the problem of estimation of a continuousmatrix-variate function of low rank. We have constructed an estimator of such functionusing its basis expansion and subsequent solution of an optimization problem with the Schattennormpenalty. We derive an oracle inequality for the constructed estimator, study its properties viasimulations and apply the procedure to analysis of Dynamic Contrast medical imaging data.
Show less - Date Issued
- 2014
- Identifier
- CFE0005377, ucf:50447
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005377
- Title
- Inversion of the Broken Ray Transform.
- Creator
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Krylov, Roman, Katsevich, Alexander, Tamasan, Alexandru, Nashed, M, Zeldovich, Boris, University of Central Florida
- Abstract / Description
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The broken ray transform (BRT) is an integral of a functionalong a union of two rays with a common vertex.Consider an X-ray beam scanning an object of interest.The ray undergoes attenuation and scatters in all directions inside the object.This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed.In our work we assume the single scattering approximation when the intensity of the raysscattered more than once is negligibly small.Among all paths that...
Show moreThe broken ray transform (BRT) is an integral of a functionalong a union of two rays with a common vertex.Consider an X-ray beam scanning an object of interest.The ray undergoes attenuation and scatters in all directions inside the object.This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed.In our work we assume the single scattering approximation when the intensity of the raysscattered more than once is negligibly small.Among all paths that the scattered rays travel inside the object we pick the one that isa union of two segments with one common scattering point.The intensity of the ray which traveled this path and exited the object can be measured by a collimated detector.The collimated detector is able to measure the intensity of X-rays from the selected direction.The logarithm of such a measurement is the broken ray transform of the attenuation coefficientplus the logarithm of the scattering coefficient at the scattering point (vertex)and a known function of the scattering angle.In this work we consider the reconstruction of X-ray attenuation coefficient distributionin a plane from the measurements on two or three collimated detector arrays.We derive an exact local reconstruction formula for three flat collimated detectorsor three curved or pin-hole collimated detectors.We obtain a range condition for the case of three curved or pin-hole detectors and provide a special caseof the range condition for three flat detectors.We generalize the reconstruction formula to four and more detectors and find anoptimal set of parameters that minimize noise in the reconstruction.We introduce a more accurate scattering model which takes into accountenergy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterativereconstruction method for the energy-dependent case.To solve the problem we assume that the radiation source is monoenergeticand the dependence of the attenuation coefficient on energy is linearon an energy interval from the minimal to the maximal scattered energy. %initial radiation energy.We find the parameters of the linear dependence of the attenuation on energy as a function of a pointin the reconstruction plane.
Show less - Date Issued
- 2014
- Identifier
- CFE0005514, ucf:50324
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005514
- Title
- Electrical Conductivity Imaging via Boundary Value Problems for the 1-Laplacian.
- Creator
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Veras, Johann, Tamasan, Alexandru, Mohapatra, Ram, Nashed, M, Dogariu, Aristide, University of Central Florida
- Abstract / Description
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We study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity...
Show moreWe study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity from the magnitude of one current density field inside. The method now known as Current Density Impedance Imaging is based on solving boundary value problems for the 1-Laplacian in an appropriate Riemann metric space. We consider two types of methods: the ones based on level sets and a variational approach, which aim to solve specific boundary value problem associated with the 1-Laplacian. We will address the Cauchy and Dirichlet problems with full and partial data, and also the Complete Electrode Model (CEM). The latter model is known to describe most accurately the voltage potential distribution in a conductive body, while taking into account the transition of current from the electrode to the body. For the CEM the problem is non-unique. We characterize the non-uniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.
Show less - Date Issued
- 2014
- Identifier
- CFE0005437, ucf:50388
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005437
- Title
- Inverse Problems in Multiple Light Scattering.
- Creator
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Broky, John, Dogariu, Aristide, Christodoulides, Demetrios, Wu, Shintson, Tamasan, Alexandru, University of Central Florida
- Abstract / Description
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The interaction between coherent waves and material systems with complex optical properties is a complicated, deterministic process. Light that scatters from such media gives rise to random fields with intricate properties. It is common perception that the randomness of these complex fields is undesired and therefore is to be removed, usually through a process of ensemble averaging. However, random fields emerging from light matter interaction contain information about the properties of the...
Show moreThe interaction between coherent waves and material systems with complex optical properties is a complicated, deterministic process. Light that scatters from such media gives rise to random fields with intricate properties. It is common perception that the randomness of these complex fields is undesired and therefore is to be removed, usually through a process of ensemble averaging. However, random fields emerging from light matter interaction contain information about the properties of the medium and a thorough analysis of the scattered light allows solving specific inverse problems. Traditional attempts to solve these kinds of inverse problems tend to rely on statistical average quantities and ignore the deterministic interaction between the optical field and the scattering structure. Thus, because ensemble averaging inherently destroys specific characteristics of random processes, one can only recover limited information about the medium. This dissertation discusses practical means that go beyond ensemble averaging to probe complex media and extract additional information about a random scattering system. The dissertation discusses cases in which media with similar average properties can be differentiated by detailed examination of fluctuations between different realizations of the random process of multiple scattering. As a different approach to this type of inverse problems, the dissertation also includes a description of how higher-order field and polarization correlations can be used to extract features of random media and complex systems from one single realization of the light-matter interaction. Examples include (i) determining the level of multiple scattering, (ii) identifying non-stationarities in random fields, and (iii) extracting underlying correlation lengths of random electromagnetic fields that result from basic interferences. The new approaches introduced and the demonstrations described in this dissertation represent practical means to extract important material properties or to discriminate between media with similar characteristics even in situations when experimental constraints limit the number of realizations of the complex light-matter interaction.
Show less - Date Issued
- 2012
- Identifier
- CFE0004656, ucf:49888
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004656
- Title
- Curvelets and the Radon Transform.
- Creator
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Dickerson, Jill, Katsevich, Alexander, Tamasan, Alexandru, Moore, Brian, University of Central Florida
- Abstract / Description
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Computed Tomography (CT) is the standard in medical imaging field. In this study, we look at the curvelet transform in an attempt to use it as a basis for representing a function. In doing so, we seek a way to reconstruct a function from the Radon data that may produce clearer results. Using curvelet decomposition, any known function can be represented as a sum of curvelets with corresponding coefficients. It can be shown that these corresponding coefficients can be found using the Radon data...
Show moreComputed Tomography (CT) is the standard in medical imaging field. In this study, we look at the curvelet transform in an attempt to use it as a basis for representing a function. In doing so, we seek a way to reconstruct a function from the Radon data that may produce clearer results. Using curvelet decomposition, any known function can be represented as a sum of curvelets with corresponding coefficients. It can be shown that these corresponding coefficients can be found using the Radon data, even if the function is unknown. The use of curvelets has the potential to solve partial or truncated Radon data problems. As a result, using a curvelet representation to invert radon data allows the chance of higher quality images to be produced. This paper examines this method of reconstruction for computed tomography (CT). A brief history of CT, an introduction to the theory behind the method, and implementation details will be provided.
Show less - Date Issued
- 2013
- Identifier
- CFE0004674, ucf:49852
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004674
- Title
- Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of Riemann-Hilbert problem.
- Creator
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Blackstone, Elliot, Tovbis, Alexander, Katsevich, Alexander, Tamasan, Alexandru, Pang, Sean, University of Central Florida
- Abstract / Description
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In this dissertation, we study a self-adjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a so-called ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular Riemann-Hilbert...
Show moreIn this dissertation, we study a self-adjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a so-called ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular Riemann-Hilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the spectral analysis of the finite Hilbert transform on multiple adjacent intervals. We show that $\Gamma(z;\lambda)$ also has a jump in the $\lambda$ plane which allows us to compute the jump of the resolvent of $\hat{K}$. This jump is an important step in showing that the finite Hilbert transforms has simple and purely absolutely continuous spectrum. The well known spectral theory now allows us to construct unitary operators which diagonalize the finite Hilbert transforms. Lastly, we mention some future directions which include the many interval scenario and a bispectral property of $\hat{K}$.
Show less - Date Issued
- 2019
- Identifier
- CFE0007602, ucf:52527
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0007602
- Title
- Random Transformations of Optical Fields and Applications.
- Creator
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Kohlgraf-Owens, Thomas, Dogariu, Aristide, Saleh, Bahaa, Schulzgen, Axel, Tamasan, Alexandru, University of Central Florida
- Abstract / Description
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The interaction of optical waves with material systems often results in complex, seemingly random fields. In many cases, the interaction, while complicated, is both linear and deterministic.This dissertation focuses on the possible inverse problems associated with the determination of either the excitation field or the scattering system. The scattered field can be thought of as a massive sampling and mixing of the excitation field. This dissertation will show how such complicated sampling...
Show moreThe interaction of optical waves with material systems often results in complex, seemingly random fields. In many cases, the interaction, while complicated, is both linear and deterministic.This dissertation focuses on the possible inverse problems associated with the determination of either the excitation field or the scattering system. The scattered field can be thought of as a massive sampling and mixing of the excitation field. This dissertation will show how such complicated sampling functions can be characterized and how the corresponding scattering medium can then be used as an optical device such as a lens, polarimeter, or spectrometer.Another class of inverse problems deals with extracting information about the material system from changes in the scattered field. This dissertation includes a novel technique, based on dynamic light scattering, that allows for a full polarimetric measurement of the scattered light using a reference field with controllable polarization. Another technique relates to imaging the reflectivity of a target that is being randomly illuminated. We demonstrate that a method based on the correlation between the integrated scattered intensity and the corresponding illumination intensity distribution can prove superior to standard imaging microscopy at low-light levels.
Show less - Date Issued
- 2012
- Identifier
- CFE0004786, ucf:49746
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004786
