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Sigma-Delta Quantization for Fusion Frames and Distributed Sensor Networks

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Frames and Other Bases in Abstract and Function Spaces

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

We discuss an adaptation of Sigma-Delta (Σ Δ) quantization to the vector-valued setting of fusion frames. Stability bounds are presented for first order and higher order fusion frame Σ Δ algorithms and an analogue of Sobolev duals for fusion frames is described for the reconstruction process. Stability bounds for the first order algorithm show that stable implementations are possible with extremely coarse quantization alphabets that use less than one bit per dimension of each fusion frame subspace. Examples are given to illustrate the performance of the algorithms.

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Acknowledgements

J. Jiang was partially supported by NSF DMS 1211687. A.M. Powell was partially supported by NSF DMS 1521749 and NSF DMS 1211687. A.M. Powell gratefully acknowledges the hospitality and support of the Academia Sinica Institute of Mathematics (Taipei, Taiwan).

The authors thank John Benedetto, Xuemei Chen, Doug Hardin, Mark Lammers, Anneliese Spaeth, Nguyen Thao, and Özgür Yılmaz for valuable discussions related to the material.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Jiayi Jiang & Alexander M. Powell

Authors
  1. Jiayi Jiang
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  2. Alexander M. Powell
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Correspondence to Alexander M. Powell .

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Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

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Jiang, J., Powell, A.M. (2017). Sigma-Delta Quantization for Fusion Frames and Distributed Sensor Networks. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_6

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