Matching Items (23)

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A Statistical Framework for Detecting Edges from Noisy Fourier Data with Multiple Concentration Factors

Description

The concentration factor edge detection method was developed to compute the locations and values of jump discontinuities in a piecewise-analytic function from its first few Fourier series coecients. The method

The concentration factor edge detection method was developed to compute the locations and values of jump discontinuities in a piecewise-analytic function from its first few Fourier series coecients. The method approximates the singular support of a piecewise smooth function using an altered Fourier conjugate partial sum. The accuracy and characteristic features of the resulting jump function approximation depends on these lters, known as concentration factors. Recent research showed that that these concentration factors could be designed using aexible iterative framework, improving upon the overall accuracy and robustness of the method, especially in the case where some Fourier data are untrustworthy or altogether missing. Hypothesis testing methods were used to determine how well the original concentration factor method could locate edges using noisy Fourier data. This thesis combines the iterative design aspect of concentration factor design and hypothesis testing by presenting a new algorithm that incorporates multiple concentration factors into one statistical test, which proves more ective at determining jump discontinuities than the previous HT methods. This thesis also examines how the quantity and location of Fourier data act the accuracy of HT methods. Numerical examples are provided.

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  • 2016-05

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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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Date Created
  • 2015-12

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Designing concentration factors to detect jump discontinuities from non-uniform Fourier data

Description

Edge detection plays a significant role in signal processing and image reconstruction applications where it is used to identify important features in the underlying signal or image. In some of

Edge detection plays a significant role in signal processing and image reconstruction applications where it is used to identify important features in the underlying signal or image. In some of these applications, such as magnetic resonance imaging (MRI), data are sampled in the Fourier domain. When the data are sampled uniformly, a variety of algorithms can be used to efficiently extract the edges of the underlying images. However, in cases where the data are sampled non-uniformly, such as in non-Cartesian MRI, standard inverse Fourier transformation techniques are no longer suitable. Methods exist for handling these types of sampling patterns, but are often ill-equipped for cases where data are highly non-uniform. This thesis further develops an existing approach to discontinuity detection, the use of concentration factors. Previous research shows that the concentration factor technique can successfully determine jump discontinuities in non-uniform data. However, as the distribution diverges further away from uniformity so does the efficacy of the identification. This thesis proposes a method for reverse-engineering concentration factors specifically tailored to non-uniform data by employing the finite Fourier frame approximation. Numerical results indicate that this design method produces concentration factors which can more precisely identify jump locations than those previously developed.

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Date Created
  • 2015-05

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Edge Detection from Spectral Phase Data

Description

The detection and characterization of transients in signals is important in many wide-ranging applications from computer vision to audio processing. Edge detection on images is typically realized using small, local,

The detection and characterization of transients in signals is important in many wide-ranging applications from computer vision to audio processing. Edge detection on images is typically realized using small, local, discrete convolution kernels, but this is not possible when samples are measured directly in the frequency domain. The concentration factor edge detection method was therefore developed to realize an edge detector directly from spectral data. This thesis explores the possibilities of detecting edges from the phase of the spectral data, that is, without the magnitude of the sampled spectral data. Prior work has demonstrated that the spectral phase contains particularly important information about underlying features in a signal. Furthermore, the concentration factor method yields some insight into the detection of edges in spectral phase data. An iterative design approach was taken to realize an edge detector using only the spectral phase data, also allowing for the design of an edge detector when phase data are intermittent or corrupted. Problem formulations showing the power of the design approach are given throughout. A post-processing scheme relying on the difference of multiple edge approximations yields a strong edge detector which is shown to be resilient under noisy, intermittent phase data. Lastly, a thresholding technique is applied to give an explicit enhanced edge detector ready to be used. Examples throughout are demonstrate both on signals and images.

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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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Date Created
  • 2014-05

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Incorporating the Sparsity of Edges into the Fourier Reconstruction of Piecewise Smooth Functions

Description

In applications such as Magnetic Resonance Imaging (MRI), data are acquired as Fourier samples. Since the underlying images are only piecewise smooth, standard recon- struction techniques will yield the Gibbs

In applications such as Magnetic Resonance Imaging (MRI), data are acquired as Fourier samples. Since the underlying images are only piecewise smooth, standard recon- struction techniques will yield the Gibbs phenomenon, which can lead to misdiagnosis. Although filtering will reduce the oscillations at jump locations, it can often have the adverse effect of blurring at these critical junctures, which can also lead to misdiagno- sis. Incorporating prior information into reconstruction methods can help reconstruct a sharper solution. For example, compressed sensing (CS) algorithms exploit the expected sparsity of some features of the image. In this thesis, we develop a method to exploit the sparsity in the edges of the underlying image. We design a convex optimization problem that exploits this sparsity to provide an approximation of the underlying image. Our method successfully reduces the Gibbs phenomenon with only minimal "blurring" at the discontinuities. In addition, we see a high rate of convergence in smooth regions.

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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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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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Date Created
  • 2013-05

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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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Date Created
  • 2013-05

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Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

Description

Detecting edges in images from a finite sampling of Fourier data is important in a variety of applications. For example, internal edge information can be used to identify tissue boundaries

Detecting edges in images from a finite sampling of Fourier data is important in a variety of applications. For example, internal edge information can be used to identify tissue boundaries of the brain in a magnetic resonance imaging (MRI) scan, which is an essential part of clinical diagnosis. Likewise, it can also be used to identify targets from synthetic aperture radar data. Edge information is also critical in determining regions of smoothness so that high resolution reconstruction algorithms, i.e. those that do not “smear over” the internal boundaries of an image, can be applied. In some applications, such as MRI, the sampling patterns may be designed to oversample the low frequency while more sparsely sampling the high frequency modes. This type of non-uniform sampling creates additional difficulties in processing the image. In particular, there is no fast reconstruction algorithm, since the FFT is not applicable. However, interpolating such highly non-uniform Fourier data to the uniform coefficients (so that the FFT can be employed) may introduce large errors in the high frequency modes, which is especially problematic for edge detection. Convolutional gridding, also referred to as the non-uniform FFT, is a forward method that uses a convolution process to obtain uniform Fourier data so that the FFT can be directly applied to recover the underlying image. Carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients. Similarly, the convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges. We demonstrate our technique in one and two dimensional examples.

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Date Created
  • 2014-12-01