An Augmented Lagrangian Model for Signal Segmentation

• Published in: Mediterranean journal of mathematics Vol. 19; no. 3
• Main Authors: Moll, Salvador, Pallardó, Vicent
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-06-2022; Springer Nature B.V
• Subjects: Algorithms; Energy methods; Functionals; Mathematics; Mathematics and Statistics; Segmentation; 49J10; Segmentation; signal processing; 35Q68; 35J92; minimizers; 35G60; total variation; one dimensional
• ISSN: 1660-5446; 1660-5454
• DOI: 10.1007/s00009-022-02029-y

In this paper, we provide a new insight to the two-phase signal segmentation problem. We propose an augmented Lagrangian variational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.