Quantization and Finite Frames
Frames are a tool for providing stable and robust signal representations in a wide variety of pure and applied settings. Frame theory uses a set of frame vectors to discretely represent a signal in terms of its associated collection of frame coefficients.
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Quantization and Finite Frames Alexander M. Powell, Rayan Saab, and Özgür Yılmaz
Abstract Frames are a tool for providing stable and robust signal representations in a wide variety of pure and applied settings. Frame theory uses a set of frame vectors to discretely represent a signal in terms of its associated collection of frame coefficients. Dual frames and frame expansions allow one to reconstruct a signal from its frame coefficients—the use of redundant or overcomplete frames ensures that this process is robust against noise and other forms of data loss. Although frame expansions provide discrete signal decompositions, the frame coefficients generally take on a continuous range of values and must also undergo a lossy step to discretize their amplitudes so that they may be amenable to digital processing and storage. This analog-to-digital conversion step is known as quantization. We shall give a survey of quantization for the important practical case of finite frames and shall give particular emphasis to the class of Sigma-Delta algorithms and the role of noncanonical dual frame reconstruction. Keywords Digital signal representations · Noncanonical dual frame · Quantization · Sigma-Delta quantization · Sobolev duals
8.1 Introduction Data representation is crucial in modern signal processing applications. Among other things, one seeks signal representations that are numerically stable, robust against noise and data loss, computationally tractable, and well adapted to specific A.M. Powell () Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA e-mail: [email protected] R. Saab Department of Mathematics, Duke University, Durham, NC 27708, USA e-mail: [email protected] Ö. Yılmaz Department of Mathematics, University of British Columbia, Vancouver, BC Canada V6T 1Z2 e-mail: [email protected] P.G. Casazza, G. Kutyniok (eds.), Finite Frames, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8373-3_8, © Springer Science+Business Media New York 2013
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applied problems. Frame theory has emerged as an important tool for meeting these requirements. Frames use redundancy or overcompleteness to provide robustness and design flexibility, and the linearity of frame expansions makes them simple to use in practice. The linear representations given by frame expansions are a cornerstone of frame M N N N theory. If (ϕi )M i=1 ⊂ R is a frame for R and if (ψi )i=1 ⊂ R is any associated dual frame, then the following frame expansion holds:
∀x ∈ RN ,
x=
M x, ϕi ψi .
(8.1)
i=1
Equivalently, if Φ ∗ is the analysis operator associated to (ϕi )M i=1 and Ψ is the synM thesis operator associated to (ψi )i=1 , then ∀x ∈ RN ,
x = Ψ Φ ∗ x.
(8.2)
The frame expansion (8.1) discretely encodes x ∈ RN by the frame coefficients (x, ϕi )M i=1 . Consequently, frame expansions can be interpreted as generalized sampling formulas, where frame coefficients play the role of samples of the underlying object. Our technology nowadays is overwhelmingly digita
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