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. 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.
KeywordsDigital signal representations Noncanonical dual frame Quantization Sigma-Delta quantization Sobolev duals
The authors thank Sinan Güntürk, Mark Lammers, and Thao Nguyen for valuable discussions and collaborations on frame theory and quantization.
A. Powell was supported in part by NSF DMS Grant 0811086 and also gratefully acknowledges the hospitality and support of the Academia Sinica Institute of Mathematics (Taipei, Taiwan).
R. Saab was supported by a Banting Postdoctoral Fellowship, administered by the Natural Science and Engineering Research Council of Canada.
Ö. Yılmaz was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). He was also supported in part by the NSERC CRD Grant DNOISE II (375142-08). Finally, Yılmaz acknowledges the Pacific Institute for the Mathematical Sciences (PIMS) for supporting a CRG in Applied and Computational Harmonic Analysis.
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