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), 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 non-causal recursive filters for finite-length data formulated in terms of banded matrices, makes the algorithms computationally efficient and effective. The computational 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.
Simultaneous Low-Pass Filtering and Total Variation Denoising.
I. W. Selesnick, H. L. Graber, D. S. Pfeil, and R. L. Barbour.
IEEE Trans. on Signal Processing, 62(5):1109-1124, March 2014.
Download preprint: LPFTVD_2014_Feb04.pdf
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Download Matlab software: LPFTVD_software.zip
Ivan W. Selesnick (1), Harry L. Graber (2), Douglas S. Pfeil (2), and
Randall L. Barbour (2)
(1) Polytechnic Institute of New York University, Brooklyn, NY 11201
(2) SUNY Downstate Medical Center, Brooklyn, NY 11203
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.
