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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) for 1-dimensional problems which uses edge information to recover the Fourier transform at its integer coefficients, thereby enabling the use…

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.
ContributorsGutierrez, Alexander Jay (Author) / Platte, Rodrigo (Thesis director) / Gelb, Anne (Committee member) / Viswanathan, Adityavikram (Committee member) / Barrett, The Honors College (Contributor) / School of International Letters and Cultures (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2013-05
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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). Edge detection can be important as a goal in and of itself in the identification of tissue boundaries such as…

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.
ContributorsMartinez, Adam (Author) / Gelb, Anne (Thesis director) / Cochran, Douglas (Committee member) / Platte, Rodrigo (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2013-05
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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 these applications, such as magnetic resonance imaging (MRI), data are sampled in the Fourier domain. When the data are sampled…

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.
ContributorsMoore, Rachael (Author) / Gelb, Anne (Thesis director) / Davis, Jacueline (Committee member) / Barrett, The Honors College (Contributor)
Created2015-05