Current Search: Nashed, M (x)


Title

On the range of the Attenuated Radon Transform in strictly convex sets.

Creator

Sadiq, Kamran, Tamasan, Alexandru, Nashed, M, Katsevich, Alexander, Dogariu, Aristide, University of Central Florida

Abstract / Description

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 Aanalytic 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 Aanalytic 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 Aanalytic 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 Xray 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.
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Date Issued

2014

Identifier

CFE0005408, ucf:50437

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0005408


Title

Inversion of the Broken Ray Transform.

Creator

Krylov, Roman, Katsevich, Alexander, Tamasan, Alexandru, Nashed, M, Zeldovich, Boris, University of Central Florida

Abstract / Description

The broken ray transform (BRT) is an integral of a functionalong a union of two rays with a common vertex.Consider an Xray 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 Xray 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 Xrays 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 Xray 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 pinhole collimated detectors.We obtain a range condition for the case of three curved or pinhole 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 energydependent 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.
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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 1Laplacian.

Creator

Veras, Johann, Tamasan, Alexandru, Mohapatra, Ram, Nashed, M, Dogariu, Aristide, University of Central Florida

Abstract / Description

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 1Laplacian 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 1Laplacian. 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 nonunique. We characterize the nonuniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.
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Date Issued

2014

Identifier

CFE0005437, ucf:50388

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0005437


Title

Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments.

Creator

Adams, Christine, Nashed, M, Mohapatra, Ram, Kaup, David, University of Central Florida

Abstract / Description

The central theme of this thesis deals with problems related to the question, (")Can one hear the shape of a drum?(") first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years...
Show moreThe central theme of this thesis deals with problems related to the question, (")Can one hear the shape of a drum?(") first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as (")Can one hear the shape of a graph associated with the Schr(&)#246;dinger operator?("), (")Can you hear the shape of your throat?("), (")Can you feel the shape of a manifold with Brownian motion?("), (")Can one hear the crack in a beam?("), (")Can one hear into the sun?("), etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac's question in general is (")no.(") However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem wellposed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of (")can one hear(") problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.
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Date Issued

2013

Identifier

CFE0004643, ucf:49886

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0004643


Title

Robust, Scalable, and Provable Approaches to High Dimensional Unsupervised Learning.

Creator

Rahmani, Mostafa, Atia, George, Vosoughi, Azadeh, Mikhael, Wasfy, Nashed, M, Pensky, Marianna, University of Central Florida

Abstract / Description

This doctoral thesis focuses on three popular unsupervised learning problems: subspace clustering, robust PCA, and column sampling. For the subspace clustering problem, a new transformative idea is presented. The proposed approach, termed Innovation Pursuit, is a new geometrical solution to the subspace clustering problem whereby subspaces are identified based on their relative novelties. A detailed mathematical analysis is provided establishing sufficient conditions for the proposed method...
Show moreThis doctoral thesis focuses on three popular unsupervised learning problems: subspace clustering, robust PCA, and column sampling. For the subspace clustering problem, a new transformative idea is presented. The proposed approach, termed Innovation Pursuit, is a new geometrical solution to the subspace clustering problem whereby subspaces are identified based on their relative novelties. A detailed mathematical analysis is provided establishing sufficient conditions for the proposed method to correctly cluster the data points. The numerical simulations with both real and synthetic data demonstrate that Innovation Pursuit notably outperforms the stateoftheart subspace clustering algorithms. For the robust PCA problem, we focus on both the outlier detection and the matrix decomposition problems. For the outlier detection problem, we present a new algorithm, termed Coherence Pursuit, in addition to two scalable randomized frameworks for the implementation of outlier detection algorithms. The Coherence Pursuit method is the first provable and noniterative robust PCA method which is provably robust to both unstructured and structured outliers. Coherence Pursuit is remarkably simple and it notably outperforms the existing methods in dealing with structured outliers. In the proposed randomized designs, we leverage the low dimensional structure of the low rank component to apply the robust PCA algorithm to a random sketch of the data as opposed to the full scale data. Importantly, it is analytically shown that the presented randomized designs can make the computation or sample complexity of the low rank matrix recovery algorithm independent of the size of the data. At the end, we focus on the column sampling problem. A new sampling tool, dubbed Spatial Random Sampling, is presented which performs the random sampling in the spatial domain. The most compelling feature of Spatial Random Sampling is that it is the first unsupervised column sampling method which preserves the spatial distribution of the data.
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Date Issued

2018

Identifier

CFE0007083, ucf:52010

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0007083


Title

Weighted LowRank Approximation of Matrices:Some Analytical and Numerical Aspects.

Creator

Dutta, Aritra, Li, Xin, Sun, Qiyu, Mohapatra, Ram, Nashed, M, Shah, Mubarak, University of Central Florida

Abstract / Description

This dissertation addresses some analytical and numerical aspects of a problem of weighted lowrank approximation of matrices. We propose and solve two different versions of weighted lowrank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing stateoftheart unweighted and weighted lowrank approximation algorithms...
Show moreThis dissertation addresses some analytical and numerical aspects of a problem of weighted lowrank approximation of matrices. We propose and solve two different versions of weighted lowrank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing stateoftheart unweighted and weighted lowrank approximation algorithms.Classical principal component analysis (PCA) is constrained to have equal weighting on the elements of the matrix, which might lead to a degraded design in some problems. To address this fundamental flaw in PCA, Golub, Hoffman, and Stewart proposed and solved a problem of constrained lowrank approximation of matrices: For a given matrix $A = (A_1\;A_2)$, find a low rank matrix $X = (A_1\;X_2)$ such that ${\rm rank}(X)$ is less than $r$, a prescribed bound, and $\AX\$ is small.~Motivated by the above formulation, we propose a weighted lowrank approximation problem that generalizes the constrained lowrank approximation problem of Golub, Hoffman and Stewart.~We study a general framework obtained by pointwise multiplication with the weight matrix and consider the following problem:~For a given matrix $A\in\mathbb{R}^{m\times n}$ solve:\begin{eqnarray*}\label{weighted problem}\min_{\substack{X}}\\left(AX\right)\odot W\_F^2~{\rm subject~to~}{\rm rank}(X)\le r,\end{eqnarray*}where $\odot$ denotes the pointwise multiplication and $\\cdot\_F$ is the Frobenius norm of matrices.In the first part, we study a special version of the above general weighted lowrank approximation problem.~Instead of using pointwise multiplication with the weight matrix, we use the regular matrix multiplication and replace the rank constraint by its convex surrogate, the nuclear norm, and consider the following problem:\begin{eqnarray*}\label{weighted problem 1}\hat{X} (&)=(&) \arg \min_X \{\frac{1}{2}\(AX)W\_F^2 +\tau\X\_\ast\},\end{eqnarray*}where $\\cdot\_*$ denotes the nuclear norm of $X$.~Considering its resemblance with the classic singular value thresholding problem we call it the weighted singular value thresholding~(WSVT)~problem.~As expected,~the WSVT problem has no closed form analytical solution in general,~and a numerical procedure is needed to solve it.~We introduce auxiliary variables and apply simple and fast alternating direction method to solve WSVT numerically.~Moreover, we present a convergence analysis of the algorithm and propose a mechanism for estimating the weight from the data.~We demonstrate the performance of WSVT on two computer vision applications:~background estimation from video sequences~and facial shadow removal.~In both cases,~WSVT shows superior performance to all other models traditionally used. In the second part, we study the general framework of the proposed problem.~For the special case of weight, we study the limiting behavior of the solution to our problem,~both analytically and numerically.~In the limiting case of weights,~as $(W_1)_{ij}\to\infty, W_2=\mathbbm{1}$, a matrix of 1,~we show the solutions to our weighted problem converge, and the limit is the solution to the constrained lowrank approximation problem of Golub et. al. Additionally, by asymptotic analysis of the solution to our problem,~we propose a rate of convergence.~By doing this, we make explicit connections between a vast genre of weighted and unweighted lowrank approximation problems.~In addition to these, we devise a novel and efficient numerical algorithm based on the alternating direction method for the special case of weight and present a detailed convergence analysis.~Our approach improves substantially over the existing weighted lowrank approximation algorithms proposed in the literature.~Finally, we explore the use of our algorithm to realworld problems in a variety of domains, such as computer vision and machine learning. Finally, for a special family of weights, we demonstrate an interesting property of the solution to the general weighted lowrank approximation problem. Additionally, we devise two accelerated algorithms by using this property and present their effectiveness compared to the algorithm proposed in Chapter 4.
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Date Issued

2016

Identifier

CFE0006833, ucf:51789

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0006833


Title

Calibration of Option Pricing in Reproducing Kernel Hilbert Space.

Creator

Ge, Lei, Nashed, M, Yong, Jiongmin, Qi, Yuanwei, Sun, Qiyu, Caputo, Michael, University of Central Florida

Abstract / Description

A parameter used in the BlackScholes equation, volatility, is a measure for variation of the price of a financial instrument over time. Determining volatility is a fundamental issue in the valuation of financial instruments. This gives rise to an inverse problem known as the calibration problem for option pricing. This problem is shown to be illposed. We propose a regularization method and reformulate our calibration problem as a problem of finding the local volatility in a reproducing...
Show moreA parameter used in the BlackScholes equation, volatility, is a measure for variation of the price of a financial instrument over time. Determining volatility is a fundamental issue in the valuation of financial instruments. This gives rise to an inverse problem known as the calibration problem for option pricing. This problem is shown to be illposed. We propose a regularization method and reformulate our calibration problem as a problem of finding the local volatility in a reproducing kernel Hilbert space. We defined a new volatility function which allows us to embrace both the financial and time factors of the options. We discuss the existence of the minimizer by using regu larized reproducing kernel method and show that the regularizer resolves the numerical instability of the calibration problem. Finally, we apply our studied method to data sets of index options by simulation tests and discuss the empirical results obtained.
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Date Issued

2015

Identifier

CFE0005617, ucf:50211

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0005617