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Spectrally Uniform Frames and Spectrally Optimal Dual Frames
Title
Spectrally Uniform Frames and Spectrally Optimal Dual Frames.
Creator
Pehlivan, Saliha, Han, Deguang, Mohapatra, Ram, Sun, Qiyu, Tatari, Mehmet, University of Central Florida
Abstract / Description
Frames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in...
Show moreFrames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof one-erasure optimal Parseval frames, the connection between two-erasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in the reconstruction process of the signal vector, then spectral radiusmeasurement for the error operators is more appropriate then the operator norm measurement.We obtain a complete characterization of spectrally one-uniform frames (i.e., one-erasure optimalframes with respect to the spectral radius measurement) in terms of the redundancy distributionof the frame. Our characterization relies on the connection between spectrally optimal frames andthe linear connectivity property of the frame. We prove that the linear connectivity property isequivalent to the intersection dependence property, and is also closely related to the well-knownconcept of k-independent set. For spectrally two-uniform frames, it is necessary that the framemust be linearly connected. We conjecture that it is also necessary that a two-uniform frame mustbe n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is thenumber of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establishseveral necessary and sufficient conditions for the existence of an alternate dual frame to make the iterated reconstruction to work.
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Date Issued
2013
Identifier
CFE0005111, ucf:50747
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005111
OPTIMAL DUAL FRAMES FOR ERASURES AND DISCRETE GABOR FRAMES
Title
OPTIMAL DUAL FRAMES FOR ERASURES AND DISCRETE GABOR FRAMES.
Creator
Lopez, Jerry, Han, Deguang, University of Central Florida
Abstract / Description
Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work...
Show moreSince their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in $\mathbb^n$, but very little is known about the $\ell^2(\mathbb)$ case or the $\ell^2(\mathbb^d)$ case. We establish some basic Gabor frame theory for $\ell^2(\mathbb)$ and then generalize to the $\ell^2(\mathbb^d)$ case.
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Date Issued
2009
Identifier
CFE0002614, ucf:48274
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0002614