Matching Items (30)

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Computations on Spherical Domains

Description

The main goal of this project is to study approximations of functions on circular and spherical domains using the cubed sphere discretization. On each subdomain, the function is approximated by

The main goal of this project is to study approximations of functions on circular and spherical domains using the cubed sphere discretization. On each subdomain, the function is approximated by windowed Fourier expansions. Of particular interest is the dependence of accuracy on the different choices of windows and the size of the overlapping regions. We use Matlab to manipulate each of the variables involved in these computations as well as the overall error, thus enabling us to decide which specific values produce the most accurate results. This work is motivated by problems arising in atmospheric research.

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Created

Date Created
  • 2018-05

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An l1 Regularization Algorithm for Reconstructing Piecewise Smooth Functions from Fourier Data Using Wavelet Projection

Description

Imaging technologies such as Magnetic Resonance Imaging (MRI) and Synthetic Aperture Radar (SAR) collect Fourier data and then process the data to form images. Because images are piecewise smooth, the

Imaging technologies such as Magnetic Resonance Imaging (MRI) and Synthetic Aperture Radar (SAR) collect Fourier data and then process the data to form images. Because images are piecewise smooth, the Fourier partial sum (i.e. direct inversion of the Fourier data) yields a poor approximation, with spurious oscillations forming at the interior edges of the image and reduced accuracy overall. This is the well known Gibbs phenomenon and many attempts have been made to rectify its effects. Previous algorithms exploited the sparsity of edges in the underlying image as a constraint with which to optimize for a solution with reduced spurious oscillations. While the sparsity enforcing algorithms are fairly effective, they are sensitive to several issues, including undersampling and noise. Because of the piecewise nature of the underlying image, we theorize that projecting the solution onto the wavelet basis would increase the overall accuracy. Thus in this investigation we develop an algorithm that continues to exploit the sparsity of edges in the underlying image while also seeking to represent the solution using the wavelet rather than Fourier basis. Our method successfully decreases the effect of the Gibbs phenomenon and provides a good approximation for the underlying image. The primary advantages of our method is its robustness to undersampling and perturbations in the optimization parameters.

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Created

Date Created
  • 2015-12

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Computations on Parameterized Surfaces with Chebfun2

Description

Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The

Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial interpolation at Chebyshev points or by trigonometric interpolation at equispaced points for periodic functions. More recently, the system has been extended to handle bivariate functions and vector fields. These two new classes of objects are called Chebfun2 and Chebfun2v, respectively. We will show that Chebfun2 and Chebfun2v, and can be used to accurately and efficiently perform various computations on parametric surfaces in two or three dimensions, including path trajectories and mean and Gaussian curvatures. More advanced surface computations such as mean curvature flows are also explored. This is also the first work to use the newly implemented trigonometric representation, namely Trigfun, for computations on surfaces.

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Created

Date Created
  • 2016-05

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FOURFUN: A new system for automatic computations using Fourier expansions

Description

Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and

Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that are accurate to close to machine precision. Common MATLAB functions have been overloaded to keep the syntax of the Fourfun class as consistent as possible with the general MATLAB syntax. We show that the system can be used to efficiently solve several differential equations. Comparisons with Chebfun, a similar system based on Chebyshev polynomial approximations, are provided.

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Created

Date Created
  • 2014-05

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Two Approaches to MRI Reconstruction: Gaussian Radial Basis Functions and Single Shot Parse

Description

Physical limitations of Magnetic Resonance Imaging (MRI) introduce different errors in the image reconstruction process. The discretization and truncation of data under discrete Fourier transform causes oscillations near jump discontinuities,

Physical limitations of Magnetic Resonance Imaging (MRI) introduce different errors in the image reconstruction process. The discretization and truncation of data under discrete Fourier transform causes oscillations near jump discontinuities, a phenomenon known as the Gibbs effect. Using Gaussian-based approximations rather than the discrete Fourier transform to reconstruct images serves to diminish the Gibbs effect slightly, especially when coupled with filtering. Additionally, a simplifying assumption is made that, during signal collection, the amount of transverse magnetization decay at a point does not depend on that point's position in space. Though this methodology significantly reduces operational run-time, it nonetheless introduces geometric error, which can be mitigated using Single-Shot (SS) Parse.

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Created

Date Created
  • 2015-05

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Edge Detection from Non-Uniform Fourier Data via a Modified Method of Convolutional Gridding

Description

The recovery of edge information in the physical domain from non-uniform Fourier data is of importance in a variety of applications, particularly in the practice of magnetic resonance imaging (MRI).

The recovery of edge information in the physical domain from non-uniform Fourier data is of importance in a variety of applications, particularly in the practice of magnetic resonance imaging (MRI). Edge detection can be important as a goal in and of itself in the identification of tissue boundaries such as those defining the locations of tumors. It can also be an invaluable tool in the amelioration of the negative effects of the Gibbs phenomenon on reconstructions of functions with discontinuities or images in multi-dimensions with internal edges. In this thesis we develop a novel method for recovering edges from non-uniform Fourier data by adapting the "convolutional gridding" method of function reconstruction. We analyze the behavior of the method in one dimension and then extend it to two dimensions on several examples.

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Created

Date Created
  • 2013-05

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Optimal Sampling for Function Approximation

Description

This thesis project focuses on algorithms that generate good sampling points for function approximation. In one dimension, polynomial interpolation using equispaced points is unstable, with high Oscillations near the

This thesis project focuses on algorithms that generate good sampling points for function approximation. In one dimension, polynomial interpolation using equispaced points is unstable, with high Oscillations near the endpoints of the interpolated interval. On the other hand, Chebyshev nodes provide both stable and highly accurate points for polynomial interpolation. In higher dimensions, optimal sampling points are unknown. This project addresses this problem by finding algorithms that are robust in various domains for polynomial interpolation and least-squares. To measure the quality of the nodes produced by said algorithms, the Lebesgue constant will be used. In the algorithms, a number of numerical techniques will be used, such as the Gram-Schmidt process and the pivoted-QR process. In addition, concepts such as node density and greedy algorithms will be explored.

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Created

Date Created
  • 2021-05

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Edge Informed Fourier Reconstruction from Non-Uniform Spectral Data

Description

The reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This thesis presents a new polynomial based resampling method (PRM)

The reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI). This thesis presents a new polynomial based resampling method (PRM) for 1-dimensional problems which uses edge information to recover the Fourier transform at its integer coefficients, thereby enabling the use of the inverse fast Fourier transform algorithm. By minimizing the error of the PRM approximation at the sampled Fourier modes, the PRM can also be used to improve on initial edge location estimates. Numerical examples show that using the PRM to improve on initial edge location estimates and then taking of the PRM approximation of the integer frequency Fourier coefficients is a viable way to reconstruct the underlying function in one dimension. In particular, the PRM is shown to converge more quickly and to be more robust than current resampling techniques used in MRI, and is particularly amenable to highly irregular sampling patterns.

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Created

Date Created
  • 2013-05

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Some applications of vector fitting in the solution of electromagnetic fields and interactions

Description

Vector Fitting (VF) is a recent macromodeling method that has been popularized by its use in many commercial software for extracting equivalent circuit's of simulated networks. Specifically for material measurement

Vector Fitting (VF) is a recent macromodeling method that has been popularized by its use in many commercial software for extracting equivalent circuit's of simulated networks. Specifically for material measurement applications, VF is shown to estimate either the permittivity or permeability of a multi-Debye material accurately, even when measured in the presence of noise and interferences caused by test setup imperfections. A brief history and survey of methods utilizing VF for material measurement will be introduced in this work. It is shown how VF is useful for macromodeling dielectric materials after being measured with standard transmission line and free-space methods. The sources of error in both an admittance tunnel test device and stripline resonant cavity test device are identified and VF is employed for correcting these errors. Full-wave simulations are performed to model the test setup imperfections and the sources of interference they cause are further verified in actual hardware measurements. An accurate macromodel is attained as long as the signal-to-interference-ratio (SIR) in the measurement is sufficiently high such that the Debye relaxations are observable in the data. Finally, VF is applied for macromodeling the time history of the total fields scattering from a perfectly conducting wedge. This effort is an initial test to see if a time domain theory of diffraction exists, and if the diffraction coefficients may be exactly modeled with VF. This section concludes how VF is not only useful for applications in material measurement, but for the solution of modeling fields and interactions in general.

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Created

Date Created
  • 2013

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Time-dependent models of signal transduction networks

Description

Signaling cascades transduce signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling

Signaling cascades transduce signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling cascades, it is not understood why cascades typically show three or four layers. Using singular perturbation theory, Michaelis-Menten type equations are derived for open enzymatic systems. When these equations are organized into a cascade, it is demonstrated that the output signal as a function of time becomes sigmoidal with the addition of more layers. Furthermore, it is shown that the activation time will speed up to a point, after which more layers become superfluous. It is shown that three layers create a reliable sigmoidal response progress curve from a wide variety of time-dependent signaling inputs arriving at the cell membrane, suggesting that natural selection may have favored signaling cascades as a parsimonious solution to the problem of generating switch-like behavior in a noisy environment.

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Agent

Created

Date Created
  • 2013