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(1 - 13 of 13)
- Title
- WEIGHTED L^P-STABILITY FOR LOCALIZED INFINITE MATRICES.
- Creator
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Shi, Qiling, Sun, Qiyu, University of Central Florida
- Abstract / Description
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This dissertation originates from a classical result that the l^p-stability of the convolution operator associated with a summable sequence are equivalent to each other for different p . This dissertation is motivated by the recent result by C. E. Shin and Q. Sun (Journal ofFunctional Analysis, 256(2009), 2417–2439), where the l^p-stability of infinite matrices in the Gohberg-Baskakov-Sjostrand class are proved to be equivalent to each other for different p. In the dissertation, for an...
Show moreThis dissertation originates from a classical result that the l^p-stability of the convolution operator associated with a summable sequence are equivalent to each other for different p . This dissertation is motivated by the recent result by C. E. Shin and Q. Sun (Journal ofFunctional Analysis, 256(2009), 2417–2439), where the l^p-stability of infinite matrices in the Gohberg-Baskakov-Sjostrand class are proved to be equivalent to each other for different p. In the dissertation, for an infinite matrix having certain off-diagonal decay, its weighted l^p-stability for different p are proved to be equivalent to each other and hence a result by Shin and Sun is generalized.
Show less - Date Issued
- 2009
- Identifier
- CFE0002685, ucf:48238
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0002685
- Title
- Frames and Phase Retrieval.
- Creator
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Juste, Ted, Han, Deguang, Sun, Qiyu, Dutkay, Dorin, Wang, Dingbao, University of Central Florida
- Abstract / Description
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Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring aframe...
Show morePhase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d ? 1 and d(d + 1)/2, inclusive. The complex Hilbert space case remains open.In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalizationof the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ? M such that |supp(x)| = k.Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices.Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recov-ering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.
Show less - Date Issued
- 2019
- Identifier
- CFE0007660, ucf:52503
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0007660
- Title
- Applications of Compressive Sensing To Surveillance Problems.
- Creator
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Huff, Christopher, Mohapatra, Ram, Sun, Qiyu, Han, Deguang, University of Central Florida
- Abstract / Description
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In many surveillance scenarios, one concern that arises is how to construct an imager that is capable of capturing the scene with high fidelity. This could be problematic for two reasons: first, the optics and electronics in the camera may have difficulty in dealing with so much information; secondly, bandwidth constraints, may pose difficulty in transmitting information from the imager to the user efficiently for reconstruction or realization. In this thesis, we will discuss a mathematical...
Show moreIn many surveillance scenarios, one concern that arises is how to construct an imager that is capable of capturing the scene with high fidelity. This could be problematic for two reasons: first, the optics and electronics in the camera may have difficulty in dealing with so much information; secondly, bandwidth constraints, may pose difficulty in transmitting information from the imager to the user efficiently for reconstruction or realization. In this thesis, we will discuss a mathematical framework that is capable of skirting the two aforementioned issues. This framework is rooted in a technique commonly referred to as compressive sensing. We will explore two of the seminal works in compressive sensing and will present the key theorems and definitions from these two papers. We will then survey three different surveillance scenarios and their respective compressive sensing solutions. The original contribution of this thesis is the development of a distributed compressive sensing model.
Show less - Date Issued
- 2012
- Identifier
- CFE0004317, ucf:49473
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004317
- Title
- Tiling the Integers.
- Creator
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Li, Shasha, Dutkay, Dorin, Han, Deguang, Sun, Qiyu, University of Central Florida
- Abstract / Description
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A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on ?nite Abelian groups show that Fuglede conjecture is false inhigh dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the ?nite groups ZN. Based on the Coven- Meyerowitz conjecture, it has...
Show moreA set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on ?nite Abelian groups show that Fuglede conjecture is false inhigh dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the ?nite groups ZN. Based on the Coven- Meyerowitz conjecture, it has been proved that if any spectral set in Z satis?es the the Coven-Meyerowitz properties, then everyspectral set in R is a tile. We will present some of the main results related to integer tiles and give a self-contained description of the theory with detailed proofs.
Show less - Date Issued
- 2014
- Identifier
- CFE0005199, ucf:50642
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005199
- Title
- Fast Compressed Automatic Target Recognition for a Compressive Infrared Imager.
- Creator
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Millikan, Brian, Foroosh, Hassan, Rahnavard, Nazanin, Muise, Robert, Atia, George, Mahalanobis, Abhijit, Sun, Qiyu, University of Central Florida
- Abstract / Description
-
Many military systems utilize infrared sensors which allow an operator to see targets at night. Several of these are either mid-wave or long-wave high resolution infrared sensors, which are expensive to manufacture. But compressive sensing, which has primarily been demonstrated in medical applications, can be used to minimize the number of measurements needed to represent a high-resolution image. Using these techniques, a relatively low cost mid-wave infrared sensor can be realized which has...
Show moreMany military systems utilize infrared sensors which allow an operator to see targets at night. Several of these are either mid-wave or long-wave high resolution infrared sensors, which are expensive to manufacture. But compressive sensing, which has primarily been demonstrated in medical applications, can be used to minimize the number of measurements needed to represent a high-resolution image. Using these techniques, a relatively low cost mid-wave infrared sensor can be realized which has a high effective resolution. In traditional military infrared sensing applications, like targeting systems, automatic targeting recognition algorithms are employed to locate and identify targets of interest to reduce the burden on the operator. The resolution of the sensor can increase the accuracy and operational range of a targeting system. When using a compressive sensing infrared sensor, traditional decompression techniques can be applied to form a spatial-domain infrared image, but most are iterative and not ideal for real-time environments. A more efficient method is to adapt the target recognition algorithms to operate directly on the compressed samples. In this work, we will present a target recognition algorithm which utilizes a compressed target detection method to identify potential target areas and then a specialized target recognition technique that operates directly on the same compressed samples. We will demonstrate our method on the U.S. Army Night Vision and Electronic Sensors Directorate ATR Algorithm Development Image Database which has been made available by the Sensing Information Analysis Center.
Show less - Date Issued
- 2018
- Identifier
- CFE0007408, ucf:52739
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0007408
- Title
- Spectrally Uniform Frames and Spectrally Optimal Dual Frames.
- Creator
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Pehlivan, Saliha, Han, Deguang, Mohapatra, Ram, Sun, Qiyu, Tatari, Mehmet, University of Central Florida
- Abstract / Description
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Frames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in...
Show moreFrames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in the reconstruction process of the signal vector, then spectral radiusmeasurement for the error operators is more appropriate then the operator norm measurement.We obtain a complete characterization of spectrally one-uniform frames (i.e., one-erasure optimalframes with respect to the spectral radius measurement) in terms of the redundancy distributionof the frame. Our characterization relies on the connection between spectrally optimal frames andthe linear connectivity property of the frame. We prove that the linear connectivity property isequivalent to the intersection dependence property, and is also closely related to the well-knownconcept of k-independent set. For spectrally two-uniform frames, it is necessary that the framemust be linearly connected. We conjecture that it is also necessary that a two-uniform frame mustbe n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is thenumber of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establishseveral necessary and sufficient conditions for the existence of an alternate dual frame to make the iterated reconstruction to work.
Show less - Date Issued
- 2013
- Identifier
- CFE0005111, ucf:50747
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005111
- Title
- Weighted Low-Rank Approximation of Matrices:Some Analytical and Numerical Aspects.
- Creator
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Dutta, Aritra, Li, Xin, Sun, Qiyu, Mohapatra, Ram, Nashed, M, Shah, Mubarak, University of Central Florida
- Abstract / Description
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This dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank 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 state-of-the-art unweighted and weighted low-rank approximation algorithms...
Show moreThis dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank 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 state-of-the-art unweighted and weighted low-rank 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 low-rank 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 $\|A-X\|$ is small.~Motivated by the above formulation, we propose a weighted low-rank approximation problem that generalizes the constrained low-rank 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(A-X\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 low-rank 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}\|(A-X)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 low-rank 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 low-rank 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 low-rank approximation algorithms proposed in the literature.~Finally, we explore the use of our algorithm to real-world 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 low-rank approximation problem. Additionally, we devise two accelerated algorithms by using this property and present their effectiveness compared to the algorithm proposed in Chapter 4.
Show less - Date Issued
- 2016
- Identifier
- CFE0006833, ucf:51789
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006833
- Title
- Applied Advanced Error Control Coding for General Purpose Representation and Association Machine Systems.
- Creator
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Dai, Bowen, Wei, Lei, Lin, Mingjie, Rahnavard, Nazanin, Turgut, Damla, Sun, Qiyu, University of Central Florida
- Abstract / Description
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General-Purpose Representation and Association Machine (GPRAM) is proposed to be focusing on computations in terms of variation and flexibility, rather than precision and speed. GPRAM system has a vague representation and has no predefined tasks. With several important lessons learned from error control coding, neuroscience and human visual system, we investigate several types of error control codes, including Hamming code and Low-Density Parity Check (LDPC) codes, and extend them to...
Show moreGeneral-Purpose Representation and Association Machine (GPRAM) is proposed to be focusing on computations in terms of variation and flexibility, rather than precision and speed. GPRAM system has a vague representation and has no predefined tasks. With several important lessons learned from error control coding, neuroscience and human visual system, we investigate several types of error control codes, including Hamming code and Low-Density Parity Check (LDPC) codes, and extend them to different directions.While in error control codes, solely XOR logic gate is used to connect different nodes. Inspired by bio-systems and Turbo codes, we suggest and study non-linear codes with expanded operations, such as codes including AND and OR gates which raises the problem of prior-probabilities mismatching. Prior discussions about critical challenges in designing codes and iterative decoding for non-equiprobable symbols may pave the way for a more comprehensive understanding of bio-signal processing. The limitation of XOR operation in iterative decoding with non-equiprobable symbols is described and can be potentially resolved by applying quasi-XOR operation and intermediate transformation layer. Constructing codes for non-equiprobable symbols with the former approach cannot satisfyingly perform with regarding to error correction capability. Probabilistic messages for sum-product algorithm using XOR, AND, and OR operations with non-equiprobable symbols are further computed. The primary motivation for the constructing codes is to establish the GPRAM system rather than to conduct error control coding per se. The GPRAM system is fundamentally developed by applying various operations with substantial over-complete basis. This system is capable of continuously achieving better and simpler approximations for complex tasks.The approaches of decoding LDPC codes with non-equiprobable binary symbols are discussed due to the aforementioned prior-probabilities mismatching problem. The traditional Tanner graph should be modified because of the distinction of message passing to information bits and to parity check bits from check nodes. In other words, the message passing along two directions are identical in conventional Tanner graph, while the message along the forward direction and backward direction are different in our case. A method of optimizing signal constellation is described, which is able to maximize the channel mutual information.A simple Image Processing Unit (IPU) structure is proposed for GPRAM system, to which images are inputted. The IPU consists of a randomly constructed LDPC code, an iterative decoder, a switch, and scaling and decision device. The quality of input images has been severely deteriorated for the purpose of mimicking visual information variability (VIV) experienced in human visual systems. The IPU is capable of (a) reliably recognizing digits from images of which quality is extremely inadequate; (b) achieving similar hyper-acuity performance comparing to human visual system; and (c) significantly improving the recognition rate with applying randomly constructed LDPC code, which is not specifically optimized for the tasks.
Show less - Date Issued
- 2016
- Identifier
- CFE0006449, ucf:51413
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006449
- Title
- Modified Pal Interpolation and Sampling Bilevel Signals with Finite Rate of Innovation.
- Creator
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Ramesh, Gayatri, Mohapatra, Ram, Vajravelu, Kuppalapalle, Li, Xin, Sun, Qiyu, University of Central Florida
- Abstract / Description
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Sampling and interpolation are two important topics in signal processing. Signal processing is a vast field of study that deals with analysis and operations of signals such as sounds, images, sensor data, telecommunications and so on. It also utilizes many mathematical theories such as approximation theory, analysis and wavelets. This dissertation is divided into two chapters: Modified P(&)#225;l Interpolation and Sampling Bilevel Signals with Finite Rate of Innovation. In the first chapter,...
Show moreSampling and interpolation are two important topics in signal processing. Signal processing is a vast field of study that deals with analysis and operations of signals such as sounds, images, sensor data, telecommunications and so on. It also utilizes many mathematical theories such as approximation theory, analysis and wavelets. This dissertation is divided into two chapters: Modified P(&)#225;l Interpolation and Sampling Bilevel Signals with Finite Rate of Innovation. In the first chapter, we introduce a new interpolation process, the modified P\'al interpolation, based on papers by P(&)#225;l, J(&)#243;o and Szab(&)#243;, and we establish the existence and uniqueness of interpolation polynomials of modified P(&)#225;l type.The paradigm to recover signals with finite rate of innovation from their samples is a fairly recent field of study. In the second chapter, we show that causal bilevel signals with finite rate of innovation can be stably recovered from their samples provided that the sampling period is at or above the maximal local rate of innovation, and that the sampling kernel is causal and positive on the first sampling period. Numerical simulations are presented to discuss the recovery of bilevel causal signals in the presence of noise.
Show less - Date Issued
- 2013
- Identifier
- CFE0005113, ucf:50760
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005113
- Title
- Sampling and Reconstruction of Spatial Signals.
- Creator
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Cheng, Cheng, Li, Xin, Sun, Qiyu, Yong, Jiongmin, Liu, Zhe, Xu, Mengyu, University of Central Florida
- Abstract / Description
-
Digital processing of signals f may start from sampling on a discrete set ?, f ?? f(?_n), ?_n ??.The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm...
Show moreDigital processing of signals f may start from sampling on a discrete set ?, f ?? f(?_n), ?_n ??.The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling.In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp.A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises.Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only.
Show less - Date Issued
- 2017
- Identifier
- CFE0006726, ucf:51889
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006726
- Title
- Calibration of Option Pricing in Reproducing Kernel Hilbert Space.
- Creator
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Ge, Lei, Nashed, M, Yong, Jiongmin, Qi, Yuanwei, Sun, Qiyu, Caputo, Michael, University of Central Florida
- Abstract / Description
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A parameter used in the Black-Scholes 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 ill-posed. 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 Black-Scholes 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 ill-posed. 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.
Show less - Date Issued
- 2015
- Identifier
- CFE0005617, ucf:50211
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005617
- Title
- Tiling properties of spectra of measures.
- Creator
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Haussermann, John, Dutkay, Dorin, Han, Deguang, Sun, Qiyu, Dogariu, Aristide, University of Central Florida
- Abstract / Description
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We investigate tiling properties of spectra of measures, i.e., sets ? in R with an orthogonal basis in L2 with respect to some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprising tiling properties for spectra of fractal measures, the existence of complementing sets and...
Show moreWe investigate tiling properties of spectra of measures, i.e., sets ? in R with an orthogonal basis in L2 with respect to some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprising tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the Coven-Meyerowitz property, the existence of complementing Hadamard pairs in the case ofHadamard pairs of size 2,3,4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2,3,4 or 5.
Show less - Date Issued
- 2014
- Identifier
- CFE0005182, ucf:50656
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0005182
- 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.
Show less - Date Issued
- 2017
- Identifier
- CFE0006803, ucf:51775
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006803
