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Tube-Based Taut String Algorithms for Total Variation Regularization

Author

Listed:
  • Artyom Makovetskii

    (Department of Mathematics, Chelyabinsk State University, Chelyabinsk 454001, Russia)

  • Sergei Voronin

    (Department of Mathematics, Chelyabinsk State University, Chelyabinsk 454001, Russia)

  • Vitaly Kober

    (Department of Mathematics, Chelyabinsk State University, Chelyabinsk 454001, Russia
    Ensenada Center for Scientific Research and Higher Education (CICESE), Ensenada 22860, Mexico
    Institute for Information Transmission Problems, Russain Academy of Sciences, Moscow 127051, Russia)

  • Aleksei Voronin

    (Department of Mathematics, Chelyabinsk State University, Chelyabinsk 454001, Russia)

Abstract

Removing noise from signals using total variation regularization is a challenging signal processing problem arising in many practical applications. The taut string method is one of the most efficient approaches for solving the 1D TV regularization problem. In this paper we propose a geometric description of the linearized taut string method. This geometric description leads to the notion of the “tube”. We propose three tube-based taut string algorithms for total variation regularization. Different weight functionals can be used in the 1D TV regularization that lead to different types of tubes. We consider uniform, vertically nonuniform, vertically and horizontally nonuniform tubes. The proposed geometric approach is used to speed-up TV regularization processing by dividing the tubes into subtubes and using parallel processing. We introduce the concept of a relatively convex tube and describe the relationship between the geometric characteristics of tubes and exact solutions to the TV regularization. The properties of exact solutions can also be used to design efficient algorithms for solving the TV regularization problem. The performance of the proposed algorithms is discussed and illustrated by computer simulation.

Suggested Citation

  • Artyom Makovetskii & Sergei Voronin & Vitaly Kober & Aleksei Voronin, 2020. "Tube-Based Taut String Algorithms for Total Variation Regularization," Mathematics, MDPI, vol. 8(7), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1141-:d:383667
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    1. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
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