Time and covariance smoothing for restoration of bivariate signals

TL;DR

This paper introduces a novel method for restoring bivariate signals by leveraging polarization properties and covariance matrices, using an efficient ADMM-based algorithm to improve reconstruction quality in noisy or incomplete data scenarios.

Contribution

It presents a new formulation and algorithm for bivariate signal reconstruction that incorporates polarization regularization via covariance matrices, addressing a complex optimization problem.

Findings

Effective reconstruction demonstrated on synthetic data

Algorithm converges efficiently with convex subproblems

Improves signal recovery in ill-posed inverse problems

Abstract

In many applications and physical phenomena, bivariate signals are polarized, i.e. they trace an elliptical trajectory over time when viewed in the 2D planes of their two components. The smooth evolution of this elliptical trajectory, called polarization ellipse, is highly informative to solve ill-posed inverse problems involving bivariate signals where the signal is collected through indirect, noisy or incomplete measurements. This work proposes a novel formulation and an efficient algorithm for reconstructing bivariate signals with polarization regularization. The proposed formulation leverages the compact representation of polarization through the instantaneous covariance matrices. To address the resulting quartic optimization problem, we propose a well-suited parameter splitting strategy which leads to an efficient iterative algorithm (alternating direction method of multipliers (ADMM)) with convex subproblems at each iteration. The performance of the proposed method is illustrated on numerical synthetic data experiments.