Simultaneous low-pass filtering and total variation denoising

Ivan Selesnick, Harry L. Graber, Douglas S. Pfeil, Randall L. Barbour

Research output: Contribution to journalArticle

Abstract

This paper seeks to combine linear time-invariant (LTI) filtering and sparsity-based denoising in a principled way in order to effectively filter (denoise) a wider class of signals. LTI filtering is most suitable for signals restricted to a known frequency band, while sparsity-based denoising is suitable for signals admitting a sparse representation with respect to a known transform. However, some signals cannot be accurately categorized as either band-limited or sparse. This paper addresses the problem of filtering noisy data for the particular case where the underlying signal comprises a low-frequency component and a sparse or sparse-derivative component. A convex optimization approach is presented and two algorithms derived: one based on majorization-minimization (MM), and the other based on the alternating direction method of multipliers (ADMM). It is shown that a particular choice of discrete-time filter, namely zero-phase noncausal recursive filters for finite-length data formulated in terms of banded matrices, makes the algorithms effective and computationally efficient. The efficiency stems from the use of fast algorithms for solving banded systems of linear equations. The method is illustrated using data from a physiological-measurement technique (i.e., near infrared spectroscopic time series imaging) that in many cases yields data that is well-approximated as the sum of low-frequency, sparse or sparse-derivative, and noise components.

Original languageEnglish (US)
Article number6705694
Pages (from-to)1109-1124
Number of pages16
JournalIEEE Transactions on Signal Processing
Volume62
Issue number5
DOIs
StatePublished - Mar 1 2014

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Derivatives
Convex optimization
Linear equations
Frequency bands
Time series
Infrared radiation
Imaging techniques

Keywords

  • Butterworth filter
  • low-pass filter
  • sparse signal
  • sparsity
  • Total variation denoising
  • zero-phase filter

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Signal Processing

Cite this

Simultaneous low-pass filtering and total variation denoising. / Selesnick, Ivan; Graber, Harry L.; Pfeil, Douglas S.; Barbour, Randall L.

In: IEEE Transactions on Signal Processing, Vol. 62, No. 5, 6705694, 01.03.2014, p. 1109-1124.

Research output: Contribution to journalArticle

Selesnick, Ivan ; Graber, Harry L. ; Pfeil, Douglas S. ; Barbour, Randall L. / Simultaneous low-pass filtering and total variation denoising. In: IEEE Transactions on Signal Processing. 2014 ; Vol. 62, No. 5. pp. 1109-1124.
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