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Description
Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain.
This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.
Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.
This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.
Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.
ContributorsScarnati, Theresa (Author) / Gelb, Anne (Thesis advisor) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Gardner, Carl (Committee member) / Sanders, Toby (Committee member) / Arizona State University (Publisher)
Created2018
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 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.
ContributorsWasserman, Gabriel Kanter (Author) / Gelb, Anne (Thesis director) / Cochran, Doug (Committee member) / Archibald, Rick (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05
ContributorsEvans, Bartlett R. (Conductor) / Schildkret, David (Conductor) / Glenn, Erica (Conductor) / Concert Choir (Performer) / Chamber Singers (Performer) / ASU Library. Music Library (Publisher)
Created2018-03-16
ContributorsOwen, Ken (Conductor) / McDevitt, Mandy L. M. (Performer) / Larson, Brook (Conductor) / Wang, Lin-Yu (Performer) / Jacobs, Todd (Performer) / Morehouse, Daniel (Performer) / Magers, Kristen (Performer) / DeGrow, Gary (Performer) / DeGrow, Richard (Performer) / Women's Chorus (Performer) / Sun Devil Singers (Performer) / ASU Library. Music Library (Publisher)
Created2004-03-24
ContributorsMetz, John (Performer) / Sowers, Richard (Performer) / Collegium Musicum (Performer) / ASU Library. Music Library (Publisher)
Created1983-01-29
ContributorsEvans, Bartlett R. (Conductor) / Glenn, Erica (Conductor) / Steiner, Kieran (Conductor) / Thompson, Jason D. (Conductor) / Arizona Statesmen (Performer) / Women's Chorus (Performer) / Concert Choir (Performer) / Gospel Choir (Conductor) / ASU Library. Music Library (Publisher)
Created2019-03-15
ContributorsKillian, George W. (Performer) / Killian, Joni (Performer) / Vocal Jazz Ensemble (Performer) / ASU Library. Music Library (Publisher)
Created1992-11-05
ContributorsButler, Robb (Conductor) / McCreary, Kimilee (Conductor) / Bakko, Nicki L. (Conductor) / Schreuder, Joel (Conductor) / Larson, Matthew (Performer) / Ortman, Mory (Performer) / Graduate Chorale I (Performer) / Graduate Chorale II (Performer) / ASU Library. Music Library (Publisher)
Created1999-12-02
ContributorsGarrett, Jennifer (Conductor) / FitzPatrick, Carole (Performer) / Aspnes, Lynne (Performer) / Campbell, Andrew (Pianist) (Performer) / Ryan, Russell (Performer) / Rockmaker, Jody (Performer) / Kocour, Mike (Performer) / McLin, Katherine (Performer) / Larson, Brook Carter (Conductor) / Women's Chorus (Performer) / Men's Chorus (Performer) / ASU Library. Music Library (Publisher)
Created2009-05-04
ContributorsLarson, Brook Carter (Conductor) / Gentry, Gregory R. (Conductor) / Garrison, Ryan D. (Conductor) / Schildkret, David (Conductor) / Men's Chorus (Performer) / Symphonic Chorale (Performer) / Women's Chorus (Performer) / Chamber Singers (Performer) / Choral Union (Performer) / ASU Library. Music Library (Publisher)
Created2007-12-03