An augmented Lagrangian model for signal segmentation

TL;DR

This paper introduces an augmented Lagrangian variational model for 1D signal segmentation, extending Chan-Vese's approach, and reveals properties of minimizers that facilitate a simple algorithm for segmentation.

Contribution

It proposes a novel augmented Lagrangian model for signal segmentation and proves key properties of its minimizers, enabling straightforward computation.

Findings

Minimizers are binary functions coinciding with Chan-Vese's minimizers.

Jump set of minimizers is a subset of the original signal's jump set.

All jump points of minimizers belong to the same level set of the signal.

Abstract

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. By 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 byproduct 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. Secondly, 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.