# Simultaneous Low-Pass Filtering and Total Variation Denoising

### I.W. Selesnick *et al.* (2014)

#### Summary

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 non-causal
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.