Tube-Based Taut String Algorithms for Total Variation Regularization

• Main Authors: Artyom Makovetskii, Sergei Voronin, Vitaly Kober, Aleksei Voronin
• Format: Article
• Language: English
• Published: MDPI AG 2020-07-01
• Series: Mathematics
• Subjects: inverse problem; signal restoration; total variation; noise filtering; non-smooth optimization
• Online Access: https://www.mdpi.com/2227-7390/8/7/1141

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.