Transient Artifact Reduction Algorithm (TARA) using Sparse Optimization and FilteringIvan W. Selesnick1, Harry L. Graber2, Yin Ding1, Tong Zhang1, and Randall L. Barbour2

(1) Dept. Electrical and Computer Engineering, NYU School of Engineering, Brooklyn, NY 11201, USA(2) Department of Pathology, SUNY Downstate Medical Center, Brooklyn, NY 11203, USA

Abstract

Weaddress suppression of artifacts in NIRS time-seriesimaging. We report a fast algorithm, combining sparseoptimization and filtering, that jointly estimates twoexplicitly modeled artifact types: transient disruptionsand step discontinuities.

Introduction

This work addresses the a�enuation of artifacts aris-ing in biomedical time series, such as those acquiredusing near infrared spectroscopic (NIRS) imagingdevices [1]. We model the measured time series,y(t), as

y(t) = f (t) + x1(t) + x2(t) + w(t) t ∈ R, (1)

• f (t) is a low-pass signal, i.e., HPF{f} ≈ 0.• x1(t) is a ‘Type 1’ artifact signal, intended to

model spikes. We model a Type 1 artifact signalas being sparse and having a sparse derivative. Itadheres to a baseline value of zero.

• x2(t) is a ‘Type 2’ artifact signal, intended tomodel additive step discontinuities. We model aType 2 artifact signal as having a sparse deriva-tive. It is composed of (approximate) step discon-tinuities.

•w(t) is white Gaussian noise.•We devise the ‘Transient Artifact Reduction Algo-rithm’ (TARA) to estimate both artifacts types simul-taneously, so they can be subtracted from the rawdata. TARA has high computational e�iciency andlow memory requirements.

Problem Formulation

We address the problem in the discrete-time se�ing.We propose the optimization problem

{x̂1, x̂2} = arg minx1,x2

{12‖H(y − x1 − x2)‖22

+ λ0∑n

φ0([x1]n) + λ1∑n

φ1([Dx1]n)

+ λ2∑n

φ2([Dx2]n)}, λi > 0. (2)

H denotes the high-pass filter suppressing the low-pass signal f . D is discrete-time first-order di�er-ence operator, given by [Dx]n = [x]n+1 − [x]n. Thelow-pass signal is estimated as

f̂ = L(y − x̂1 − x̂2) (3)

where L denotes the low-pass filter L = I−H.The functions φi are chosen to promote sparsity, e.g.,

φ(u) =2

a√3

(tan−1

(1 + 2a|u|√

3

)− π

6

), a > 0.

The high-pass filter, H, is implemented as

H = BA−1, (4)

where A and B are banded matrices.

Example 1

A special case of TARA is for Type 1 artifacts only(x2 is absent from (2)). We use a simulated signal(Fig. 1(a)) consisting of additive step-transients.With (λ0, λ1) = (λ∗0, 0), x̂ deviates infrequently fromthe baseline value of zero (Fig. 2(a)). With (λ0, λ1) =(0, λ∗1), it is approximately piecewise constant butdoes not adhere to a baseline of zero (Fig. 2(b)). With

(λ0, λ1) = (θλ∗0, (1− θ)λ∗1), 0 6 θ 6 1, (5)

with θ tuned to 0.3, it is reasonably sparse and has asparse derivative (Fig. 2(c)). The interpolation givenby (5) provides a trade-o�.

0 50 100 150 200

−2

0

2

4Data(a)

0 50 100 150 200

−2

0

2

4LPF/CSD with L1 penalty(b)

0 50 100 150 200

−2

0

2

4LPF/CSD with atan penalty

Time (samples)

(c)

Figure 1: (a) Simulated data. Processing with the`1-norm penalty (b) and the arctangent penalty (c).

0 50 100 150 200

−2

0

2

θ = 1.0, λ

0 = 1.42, λ

1 = 0 [Atan penalty](a)

True

0 50 100 150 200

−2

0

2

θ = 0.0, λ

0 = 0, λ

1 = 1.59 [Atan penalty](b)

True

0 50 100 150 200

−2

0

2

Time (samples)

θ = 0.30, λ

0 = 0.43, λ

1 = 1.11 [Atan penalty](c)

True

Figure 2: Estimated transient signals, x̂, obtainedwith various (λ0, λ1). (a) θ = 1. (b) θ = 0. (c) θ = 0.3.

•The result x1(t) shown in Fig. 2(c) is used to gen-erate the signal f (t) + x1(t) shown in Fig. 1(c). Itcan be seen that the arctangent penalty φ be�er pre-serves the full height of transients compared to theL1 norm penalty shown in Fig. 1(b).

Example 2

This example shows TARA, for Type 1 artifacts, asapplied to a near infrared spectroscopic (NIRS) timeseries. The data exhibits artifacts due to eye blinks.Signals corresponding to (λ∗0, 0) and (0, λ∗1) areshown in Figs. 3(b, c). With (λ0, λ1) set using (5) withθ = 0.05, we obtain an apparently accurate estimateof the transient artifacts (Fig. 3(d)). The correctedtime series is obtained by subtracting the estimatedartifact signal x̂ from the original data (Fig. 3(e)).The algorithm run time was about 80 milliseconds.

900 1000 1200 1400 1600 1800−20

0

20

40

60Data(a)

900 1000 1200 1400 1600 1800−20

0

20

40

60LPF/CSD [λ

0 = 1.669, λ

1 = 0](b)

900 1000 1200 1400 1600 1800−20

0

20

40

60LPF/CSD [λ

0 = 0, λ

1 = 1.370](c)

900 1000 1200 1400 1600 1800−20

0

20

40

60LPF/CSD [λ

0 = 0.083, λ

1 = 1.301] (θ = 0.050)(d)

900 1000 1200 1400 1600 1800−20

0

20

40

60

Time (samples)

Corrected data (atan penalty)(e)

Figure 3: Reduction of transient artifacts in NIRStime-series data. (a) Raw data. (b, c, d) Artifact es-timation with (λ∗0, 0), (0, λ

∗1), and (5). (e) Corrected

data.

Example 3

•TARA is applied to a NIRS time series acquired us-ing a pair of optodes on the back of a subject’s head.The data exhibits a motion-induced abrupt shi� ofthe baseline, at time index 470. Other motion arti-facts also are visible.•The Type 1 and Type 2 artifact signals as estimatedby TARA, are sparse and approximately piecewiseconstant, as intended. Note that the corrected timeseries has both low-frequency and high-frequencyspectral content.

0 200 400 600 800 1000 1200 1400 1600 1800

−20

−10

0

0

10

20

0

10

0

10

20

0

10

20

Raw data

Type 1 artifact

Type 2 artifact

Total artifact

Corrected data

Time (n)

Figure 4: Artifact reduction with TARA as applied toa NIRS time series.

Wavelet artifact estimation. Wavelet methodscompare favorably to other methods for the cor-rection of motion artifacts in single-channel NIRStime series [2, 3, 4]. In comparison with TARA, thewavelet method does not correct additive step dis-continuities as well. The wavelet-estimated artifactsignal smooths the additive step discontinuity.

1200 1300 1400 1500 1600 1700

Time (n)

Raw data

Artifact (wavelet)

Artifact (TARA)

Corrected (wavelet)

Corrected (TARA)

Figure 5: Artifact estimation and correction usingwavelets and TARA.

The artifact in the interval 1370-1420 is estimatedby TARA with distinct pre- and post-artifact base-line values; whereas the wavelet-estimated artifactsignal exhibits a small change. In addition, TARAfinds an abrupt change at time index 1530; whilethe wavelet method exhibits only a small bi-phasic(zero-mean) pulse at that instant. TARA is bet-ter able to estimate abrupt step-changes than thewavelet method because it is explicitly based on atwo-component model.

Run times

0 1000 2000 3000 4000 50000

50

100

150

200

250

300

350

Signal length

Run tim

e (

mill

iseconds)

Algorithm run time (50 iterations)

TARA

LPF/CSD

TARA is fast. Run times measured using a 2013 Mac-Book Pro (2.5 GHz Intel Core i5) running MatlabR2011a.

Preservation of hemodynamic response

•Physiological time-series data (e.g., NIRS, EEG) areo�en acquired in multichannel form. We applyTARA to multichannel data (Fig. 6, black) using thesame (shape) parameters for all channels. To avoidcumulative baseline dri�, each corrected time serieshas been filtered with a zero-phase second-order re-cursive dc-notch filter.•TARA e�ectively reduces transient artifacts in mostchannels, without introducing substantial distor-tion.•In the course of suppressing artifacts, biological in-formation of interest should not be distorted or at-tenuated. For NIRS, any hemodynamic response(HR) waveforms present should be preserved. Totest TARA in this regard, we add a simulated HR toeach channel of the considered multichannel data.Signals with the simulated HR is shown in gray inFig. 6. The gray-colored artifact signals, obtainedfrom the HR-added data, are nearly indistinguish-able from the original artifact signals.=⇒ TARA accurately preserves the HR in the cor-rected data.•To compare TARA with wavelets, we apply waveletsto the same data, with and without the added HR.

The wavelet-estimated artifact signals (Fig. 7) ex-hibit a noticeable portion of the HR in about half thechannels. Consequently, the HR is more a�enuatedand distorted in the wavelet-corrected data than inthe TARA-corrected data.

0 200 400 600 800 1000 1200 1400 1600 1800

Raw data

0 200 400 600 800 1000 1200 1400 1600 1800

Estimated artifact signals (TARA)

0 200 400 600 800 1000 1200 1400 1600 1800

Corrected data (TARA)

Time (n)

Figure 6: TARA applied to multichannel data.

0 200 400 600 800 1000 1200 1400 1600 1800

Estimated artifact signals (wavelet)

0 200 400 600 800 1000 1200 1400 1600 1800

Corrected data (wavelet)

Time (n)

Figure 7:Wavelet transient artifact reduction of mul-tichannel data.

•In quantitative terms, we measure the root-mean-square deviation over the HR interval (700-880) be-tween the HR and non-HR artifact signals. The valueis 0.18 for TARA and 0.53 for wavelets. By this mea-sure, the wavelet method is a�ected by the HR 2.9times more than TARA, and so TARA be�er pre-serves the HR.

References

[1] R. Al abdi, H. L. Graber, Y. Xu, and R. L. Barbour. Optome-chanical imaging system for breast cancer detection. J. Opt.Soc. Am. A, 28(12):2473–2493, December 2011.

[2] S. Brigadoi et al. Motion artifacts in functional near-infrared spectroscopy: A comparison of motion correctiontechniques applied to real cognitive data. NeuroImage,85:181–191, 2014.

[3] M. K. Islam, A. Rastegarnia, A. T. Nguyen, and Z. Yang. Arti-fact characterization and removal for in vivo neural record-ing. J. Neuroscience Methods, 226:110–123, 2014.

[4] B. Molavi and G. A. Dumont. Wavelet-based motion ar-tifact removal for functional near-infrared spectroscopy.Physio. Meas., 33(2):259, 2012.

[5] I. W. Selesnick, H. L. Graber, D. S. Pfeil, and R. L. Barbour.Simultaneous low-pass filtering and total variation denois-ing. IEEE Trans. Signal Process., 62(5):1109–1124, March2014.

This research was supported by the NSF under Grant No. CCF-1018020, the NIH under Grant Nos. R42NS050007, R44NS049734, and R21NS067278, and by DARPA project N66001-10-C-2008.