Simultaneous Low-Pass Filtering and Total Variation Denoising

Abstract

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.

Transient Artifact Reduction Algorithm (TARA) Based on Sparse Optimization

“…The derivation of TARA in Section III for model (1) builds on the algorithm developed here. In previous work [34], an iterative algorithm based on the 'alternating direction method of multipliers' (ADMM) was derived for the 'LPF/CSD' problem. Here, we revisit the problem and present several improvements in comparison with [34].…”

“…We considered model (2) in previous work [34], where we used it to formulate what we called the 'LPF/CSD' problem. In this paper we present an improved algorithm for the LPF/CSD problem and provide an approach for setting the parameters.…”

“…Although the suppression of Type 1 and Type 2 artifacts was addressed in our previous work [34], it was assumed that the measured time series is affected by the presence of either Type 1 or Type 2 artifacts, but not both. Two algorithms, one for each artifact type, are described in [34].…”

“…Two algorithms, one for each artifact type, are described in [34]. The respective algorithms were illustrated on time series acquired using a NIRS system [23], which frequently have artifacts of both types [15].…”

Sparse Signal Estimation by Maximally Sparse Convex Optimization

“…Specific motivating applications include nano-particle detection for bio-sensing and near infrared spectroscopic time series imaging [61], [62]. This paper explores the use of non-convex penalty functions φ n , under the constraint that the total cost function F is convex and therefore reliably minimized.…”