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Numerical Linear Algebra in Signals, Systems and Control pp 1–21Cite as

The Anti-Reflective Transform and Regularization by Filtering

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Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 80))

Abstract

Filtering methods are used in signal and image restoration to reconstruct an approximation of a signal or image from degraded measurements. Filtering methods rely on computing a singular value decomposition or a spectral factorization of a large structured matrix. The structure of the matrix depends in part on imposed boundary conditions. Anti-reflective boundary conditions preserve continuity of the image and its (normal) derivative at the boundary, and have been shown to produce superior reconstructions compared to other commonly used boundary conditions, such as periodic, zero and reflective. The purpose of this paper is to analyze the eigenvector structure of matrices that enforce anti-reflective boundary conditions. In particular, a new anti-reflective transform is introduced, and an efficient approach to computing filtered solutions is proposed. Numerical tests illustrate the performance of the discussed methods.

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References

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Acknowledgements

The work of the first, second and fourth authors was partially supported by MIUR, grant numbers 2004015437 and 2006017542. The work of the third author was supported in part by the NSF under grant DMS-05-11454.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    A. Aricò

  2. Dipartimento di Fisica e Matematica, Università dell’Insubria - Sede di Como, Via Valleggio 11, 22100, Como, Italy

    M. Donatelli & S. Serra-Capizzano

  3. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    J. Nagy

Authors
  1. A. Aricò
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  2. M. Donatelli
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  3. J. Nagy
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  4. S. Serra-Capizzano
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Corresponding author

Correspondence to J. Nagy .

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Editors and Affiliations

  1. CESAME, Batiment Euler, Universite Catholique de Louvain, 4, avenue Georges Lemaitre, Louvain la Neuve, 1348, Belgium

    Paul Van Dooren

  2. Department of Electrical and Computer, Engineering, Texas A&M University, Zachry Engineering Center, College Station, 77843-3128, Texas, USA

    Shankar P. Bhattacharyya

  3. Dept. Mathematics, Chinese University Hong Kong, Shatin, New Territories, Hong Kong/PR China

    Raymond H. Chan

  4. Dept. Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, Connecticut, USA

    Vadim Olshevsky

  5. , Electrical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    Aurobinda Routray

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Aricò, A., Donatelli, M., Nagy, J., Serra-Capizzano, S. (2011). The Anti-Reflective Transform and Regularization by Filtering. In: Van Dooren, P., Bhattacharyya, S., Chan, R., Olshevsky, V., Routray, A. (eds) Numerical Linear Algebra in Signals, Systems and Control. Lecture Notes in Electrical Engineering, vol 80. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0602-6_1

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  • DOI: https://doi.org/10.1007/978-94-007-0602-6_1

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-0601-9

  • Online ISBN: 978-94-007-0602-6

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