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EFFICIENT INVERSION OF THE CONE BEAM TRANSFORM FOR A GENERAL CLASS OF CURVES
Title
EFFICIENT INVERSION OF THE CONE BEAM TRANSFORM FOR A GENERAL CLASS OF CURVES.
Creator
Kapralov, Mikhail, Katsevich, Alexander, University of Central Florida
Abstract / Description
We extend an efficient cone beam transform inversion formula, proposed earlier for helices, to a general class of curves. The conditions that describe the class are very natural. Curves C are smooth, without self-intersections, have positive curvature and torsion, do not bend too much in a certain sense, and do not admit lines which are tangent to C at one point and intersect C at another point. A domain U is found where reconstruction is possible with a filtered backprojection type algorithm...
Show moreWe extend an efficient cone beam transform inversion formula, proposed earlier for helices, to a general class of curves. The conditions that describe the class are very natural. Curves C are smooth, without self-intersections, have positive curvature and torsion, do not bend too much in a certain sense, and do not admit lines which are tangent to C at one point and intersect C at another point. A domain U is found where reconstruction is possible with a filtered backprojection type algorithm. Results of numerical experiments demonstrate very good image quality. The algorithm developed is useful for image reconstruction in computerized tomography.
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Date Issued
2007
Identifier
CFE0001579, ucf:47120
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0001579
EFFICIENT CONE BEAM RECONSTRUCTION FOR THE DISTORTED CIRCLE AND LINE TRAJECTORY
Title
EFFICIENT CONE BEAM RECONSTRUCTION FOR THE DISTORTED CIRCLE AND LINE TRAJECTORY.
Creator
Konate, Souleymane, Katsevich, Alexander, University of Central Florida
Abstract / Description
We propose an exact filtered backprojection algorithm for inversion of the cone beam data in the case when the trajectory is composed of a distorted circle and a line segment. The length of the scan is determined by the region of interest , and it is independent of the size of the object. With few geometric restrictions on the curve, we show that we have an exact reconstruction. Numerical experiments demonstrate good image quality.
Date Issued
2009
Identifier
CFE0002530, ucf:47669
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0002530
On the range of the Attenuated Radon Transform in strictly convex sets.
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 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.
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Date Issued
2014
Identifier
CFE0005408, ucf:50437
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005408
Inversion of the Broken Ray Transform
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 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.
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Date Issued
2014
Identifier
CFE0005514, ucf:50324
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005514
Curvelets and the Radon Transform
Title
Curvelets and the Radon Transform.
Creator
Dickerson, Jill, Katsevich, Alexander, Tamasan, Alexandru, Moore, Brian, University of Central Florida
Abstract / Description
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.
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Date Issued
2013
Identifier
CFE0004674, ucf:49852
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004674
Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of Riemann-Hilbert problem
Title
Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of Riemann-Hilbert problem.
Creator
Blackstone, Elliot, Tovbis, Alexander, Katsevich, Alexander, Tamasan, Alexandru, Pang, Sean, University of Central Florida
Abstract / Description
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 $\hat{K}$.
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Date Issued
2019
Identifier
CFE0007602, ucf:52527
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007602
Signal processing with Fourier analysis, novel algorithms and applications
Title
Signal processing with Fourier analysis, novel algorithms and applications.
Creator
Syed, Alam, Foroosh, Hassan, Sun, Qiyu, Bagci, Ulas, Rahnavard, Nazanin, Atia, George, Katsevich, Alexander, University of Central Florida
Abstract / Description
Fourier analysis is the study of the way general functions may be represented or approximatedby sums of simpler trigonometric functions, also analogously known as sinusoidal modeling. Theoriginal idea of Fourier had a profound impact on mathematical analysis, physics, and engineeringbecause it diagonalizes time-invariant convolution operators. In the past signal processing was atopic that stayed almost exclusively in electrical engineering, where only the experts could cancelnoise, compress...
Show moreFourier analysis is the study of the way general functions may be represented or approximatedby sums of simpler trigonometric functions, also analogously known as sinusoidal modeling. Theoriginal idea of Fourier had a profound impact on mathematical analysis, physics, and engineeringbecause it diagonalizes time-invariant convolution operators. In the past signal processing was atopic that stayed almost exclusively in electrical engineering, where only the experts could cancelnoise, compress and reconstruct signals. Nowadays it is almost ubiquitous, as everyone now dealswith modern digital signals.Medical imaging, wireless communications and power systems of the future will experience moredata processing conditions and wider range of applications requirements than the systems of today.Such systems will require more powerful, efficient and flexible signal processing algorithms thatare well designed to handle such needs. No matter how advanced our hardware technology becomeswe will still need intelligent and efficient algorithms to address the growing demands in signalprocessing. In this thesis, we investigate novel techniques to solve a suite of four fundamentalproblems in signal processing that have a wide range of applications. The relevant equations, literatureof signal processing applications, analysis and final numerical algorithms/methods to solvethem using Fourier analysis are discussed for different applications in the electrical engineering /computer science. The first four chapters cover the following topics of central importance in thefield of signal processing: Fast Phasor Estimation using Adaptive Signal Processing (Chapter 2) Frequency Estimation from Nonuniform Samples (Chapter 3) 2D Polar and 3D Spherical Polar Nonuniform Discrete Fourier Transform (Chapter 4)iv Robust 3D registration using Spherical Polar Discrete Fourier Transform and Spherical Harmonics(Chapter 5)Even though each of these four methods discussed may seem completely disparate, the underlyingmotivation for more efficient processing by exploiting the Fourier domain signal structureremains the same. The main contribution of this thesis is the innovation in the analysis, synthesis, discretization of certain well-known problems like phasor estimation, frequency estimation, computations of a particular non-uniform Fourier transform and signal registration on the transformed domain. We conduct propositions and evaluations of certain applications relevant algorithms suchas, frequency estimation algorithm using non-uniform sampling, polar and spherical polar Fourier transform. The techniques proposed are also useful in the field of computer vision and medical imaging. From a practical perspective, the proposed algorithms are shown to improve the existing solutions in the respective fields where they are applied/evaluated. The formulation and final proposition is shown to have a variety of benefits. Future work with potentials in medical imaging, directional wavelets, volume rendering, video/3D object classifications, high dimensional registration are also discussed in the final chapter. Finally, in the spirit of reproducible research, we release the implementation of these algorithms to the public using Github.
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Date Issued
2017
Identifier
CFE0006803, ucf:51775
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006803