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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
I'M BEING FRAMED: PHASE RETRIEVAL AND FRAME DILATION IN FINITE-DIMENSIONAL REAL HILBERT SPACES
Title
I'M BEING FRAMED: PHASE RETRIEVAL AND FRAME DILATION IN FINITE-DIMENSIONAL REAL HILBERT SPACES.
Creator
Greuling, Jason L, Han, Deguang, University of Central Florida
Abstract / Description
Research has shown that a frame for an n-dimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suffcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space...
Show moreResearch has shown that a frame for an n-dimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suffcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will offer phase retrieval. In this thesis, we will explore and provide what necessary and suffcient conditions must be satisfed to dilate a phase retrieval frame for an n-dimensional real Hilbert space to a phase retrieval frame for a k-dimensional real Hilbert.
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Date Issued
2018
Identifier
CFH2000319, ucf:45868
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFH2000319