Polynomial smoothing of time series with additive step discontinuities

This paper addresses the problem of estimating simultaneously a local polynomial signal and an approximately piecewise constant signal from a noisy additive mixture. The approach developed in this paper synthesizes the total variation filter and least-square polynomial signal smoothing into a unified problem formulation. The method is based on formulating an $\ell- 1$-norm regularized inverse problem. A computationally efficient algorithm, based on variable splitting and the alternating direction method of multipliers (ADMM), is presented. Algorithms are derived for both unconstrained and constrained formulations. The method is illustrated on experimental data involving the detection of nano-particles with applications to real-time virus detection using a whispering-gallery mode detector.