1) Polynomial Smoothing of Time Series with Additive Step Discontinuities

I. W. Selesnick, S. Arnold, and V. R. Dantham, 'Polynomial Smoothing of Time Series With Additive Step Discontinuities', IEEE Trans. Signal Processing, 60(12):6305-6318, Dec. 2012.

Abstract: 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 L1-norm cost function. A computationally efficient algorithm is presented based on variable splitting and the alternating direction method of multipliers (ADMM). 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.

Download paper: PSTV2012.pdf

Matlab software

Download Matlab software: PATV_toolbox (zip file)

Download PATV toolbox guide: PATV_guide.pdf

2) Simultaneous Polynomial Approximation and Total Variation Denoising

This paper presents an improved algorithm for the PATV problem discussed in the above paper.

I. W. Selesnick, 'Simultaneous Polynomial Approximation and Total Variation Denoising', IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP). May, 2013.

Abstract: This paper addresses the problem of smoothing data with additive step discontinuities. The problem formulation is based on least square polynomial approximation and total variation denoising. In earlier work, an ADMM algorithm was proposed to minimize a suitably defined sparsity-promoting cost function. In this paper, an algorithm is derived using the majorization-minimization (MM) optimization procedure. The new algorithm converges faster and, unlike the ADMM algorithm, has no parameters that need to be set. The proposed algorithm is formulated so as to utilize fast solvers for banded systems for high computational efficiency. This paper also gives optimality conditions so that the optimality of a result produced by the numerical algorithm can be readily validated.

Download paper: PATVMM_2013_ICASSP.pdf

Presentation slides: PATVMM_2013_slides.pdf

Matlab software

Download Matlab software: PATVMM_software.zip (zip file)

3) Example

Local-polynomial approximation + total variation (LoPATV) filtering.

Example of PATV


Ivan Selesnick
Electrical and Computer Engineering
NYU Polytechnic School of Engineering
New York University
Brooklyn, New York


This material is based upon work supported by the National Science Foundation under Grant No. 1018020.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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