Description
- Abstract:
- This dissertation is divided into two separate parts. In Part I, we develop methods to obtain information on a system when the distributions of some variables are known exactly, others are known only approximately, and perhaps others are not modeled as random variables at all. The main tool used is the duality between risk-sensitive integrals and relative entropy, and we obtain explicit bounds on standard performance measures (variances, exceedance probabilities) over families of distributions whose distance from a nominal distribution is measured by relative entropy. The evaluation of the risk-sensitive expectations is based on polynomial chaos expansions, which help keep the computational aspects tractable. In Part II, we propose a new algorithm to reconstruct sparse gradient images from Fourier and edge measurements. In many imaging applications, such as functional Magnetic Resonance Imaging (fMRI), full, uniformly-sampled Cartesian Fourier measurements are acquired to reconstruct an image. In order to reduce scan time and increase temporal resolution for fMRI studies, one would like to accurately reconstruct these images from the smallest possible set of Fourier measurements. Compressed Sensing (CS) has given rise to techniques that can provide exact and stable recovery of sparse images from a relatively small set of Fourier measurements. In particular, if the images are sparse with respect to their gradient, e.g., piece-wise constant, total-variation minimization techniques can be used to recover those images from a highly incomplete set of Fourier measurements. Our algorithm aims to further reduce the number of Fourier measurements required for exact or stable recovery by utilizing prior edge information from a high resolution reference image. This reference image, or more precisely, the fully sampled Fourier measurements of this reference image, is obtained prior to an fMRI study in order to provide edge information for the region of interest. By combining this edge information with CS techniques for sparse gradient images, numerical experiments show that we can further reduce the number of Fourier measurements required for exact or stable recovery by an additional factor of 1.6 to 3 , compared with CS techniques alone, without edge information.
- Notes:
- Thesis (Ph.D. -- Brown University (2012)
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Citation
Chowdhary, Kamaljit,
"Distinguishing and Integrating Aleatoric and Epistemic Variation in Uncertainty Quantification/ Sparse Gradient Image Reconstruction from Fourier and Edge Measurements"
(2012).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0MK6B6M
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....