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Signal Processing

Signal Processing 120 (2016) 236–248

http://d0165-16

n CorrE-m

selesi@n

journal homepage: www.elsevier.com/locate/sigpro

Sparsity-based correction of exponential artifacts

Yin Ding n, Ivan W. SelesnickSchool of Engineering, New York University, 6 Metrotech Center, Brooklyn, NY 11201, USA

a r t i c l e i n f o

Article history:Received 26 June 2015Received in revised form17 September 2015Accepted 18 September 2015Available online 30 September 2015

Keywords:SparsitySignal decompositionArtifact removal

x.doi.org/10.1016/j.sigpro.2015.09.01784/& 2015 Elsevier B.V. All rights reserved.

esponding author.ail addresses: yd372@nyu.edu (Y. Ding),yu.edu (I.W. Selesnick).

a b s t r a c t

This paper describes an exponential transient excision algorithm (ETEA). In biomedicaltime series analysis, e.g., in vivo neural recording and electrocorticography (ECoG), somemeasurement artifacts take the form of piecewise exponential transients. The proposedmethod is formulated as an unconstrained convex optimization problem, regularized bysmoothed ℓ1-norm penalty function, which can be solved by majorization–minimization(MM) method. With a slight modification of the regularizer, ETEA can also suppressmore irregular piecewise smooth artifacts, especially, ocular artifacts (OA) in electro-encephalography (EEG) data. Examples of synthetic signal, EEG data, and ECoG data arepresented to illustrate the proposed algorithms.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

This work is motivated by the problem of suppressingvarious types of artifacts in recordings of neural activity. In arecent study [25], typical artifacts in in vivo neural recordingsare classified into four types (Type 0–3, see Section 2.2 andFigure 3 in [25]). This classification covers many artifacts inthe scope of human brain activity recordings, e.g., electro-encephalography (EEG) and electrocorticography (ECoG). Inthis paper, we consider the suppression of Type 0 and Type1 artifacts. For the purpose of flexibility and generality, weredefine them in terms of morphological characteristics:

� Type 0: a smooth protuberance that can be modeled asx̂ðtÞ ¼ te�αt , when tZt0.

� Type 1: an abrupt jump followed by an exponentialdecay that can be modeled as x̂ðtÞ ¼ e�αt , when tZt0.

Fig. 1 shows examples of the two types of artifacts.We donot consider the other two types in this work because ourprevious works have addressed efficient algorithms to

remove such artifacts. For instance, low-pass filtering/totalvariation denoising (LPF=TVD) [61] suppresses Type 2 arti-facts (Fig. 1(c)), and lowpass filtering/compound sparsedenoising (LPF/CSD) [60,61] can remove sparse and blockyspikes (Type 3 shown in Fig. 1(d)).

The approach proposed in this paper is based onan optimization problem intended to capture the prim-ary morphological characteristics of the artifacts usingsparsity-inducing regularization. To formulate the pro-blem, we model the observed time series as

yðtÞ ¼ f ðtÞþxðtÞþwðtÞ; ð1Þ

where f is a lowpass signal, x is a piecewise smooth tran-sient signal (i.e., Type 0 or Type 1 artifacts), and w is sta-tionary white Gaussian noise. More specifically, f isassumed to be restricted to a certain range of low fre-quencies. In other words, Hðf Þ � 0, where H is a high-passfilter. Note that in the signal model (1), conventional LTIfiltering is not suitable to estimate either f or x from y,because component x, as a piecewise smooth signal com-prised of transients, is not band limited.

In order to estimate the components, we combine LTIfiltering with sparsity-based techniques. We formulate anoptimization problem for both decomposition and denois-ing. A computationally efficient algorithm is derived to

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

Type 0

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

Type 1

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

Type 2

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

Type 3

Fig. 1. Examples of (a) Type 0 artifact, and (b) Type 1 artifact, and (c) Type 2 artifact, and (d) Type 3 artifact.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 237

solve the optimization problem, based on the theory ofmajorization–minimization (MM) [15,24,36].

In addition, this paper specifies how to generate asmoothed penalty function and its majorizer from a non-smooth one, in order to overcome a numerical issue thatarises when the penalty function is not differentiable.

1.1. Related works

Some recent works recover signals with transients byvarious algorithms. In [19], a slowly varying signal ismodeled as a local polynomial and an optimization pro-blem using Tikhonov regularization is formulated to cap-ture it. In [47], the slowly varying trend is modeled as ahigher-order sparse-derivative signal (e.g., the third-orderderivative is sparse).

Instead of estimating the slowly varying component viaregularization, the LPF=TVD method [61] estimates alowpass component by LTI filtering and a piecewise con-stant component by optimization. In this case, an optimi-zation problem is formulated to estimate the piecewiseconstant component. The approach proposed here uses asimilar technique to recover the lowpass component, butin contrast to LPF=TVD, it is more general—the regular-ization is more flexible with a tunable parameter, so thatLPF=TVD can be considered as a special case.

Another algorithm related to the approach taken in thispaper is the transient artifact reduction algorithm (TARA)[60] which is utilized to suppress additive piecewise con-stant artifacts and spikes (similar to a hybrid of Type 2 andType 3 artifacts). The approaches proposed in this work

target different types of artifacts (Type 0 and Type 1) andapplied in different applications.

The irregularity of Type 0 transients leads to a morecomplicated artifact removal problem, where the artifactare irregular fluctuations. A typical example in EEG is ocularartifacts (OA) caused by the blink and/or movement of eyes.To suppress OA, there are approaches based on empiricalmode decomposition (EMD) [41–43,69], and on indepen-dent component analysis (ICA) methods [1,12,20,26,37,48].The concept of spatial-frequency in acoustic analysis is alsoused to remove OA from multichannel signals [44,64]. Inthis work, we present a new method to suppress ocularartifacts by proposing a specific model and using sparseoptimization.

This paper adopts a regularizer inspired by the gen-eralized 1-D total variation [27], wherein the derivativeoperator in conventional total variation regularizer isgeneralized to a recursive filter. The regularizers adoptedin ETEA and second-order ETEA coincide with first-orderand second-order cases of generalized 1-D total variation,respectively. Some differences to the problem discussed in[27] are as follows. Firstly, the signal model (1) allows alowpass baseline as a component, hence, ETEA can be seenas a combination of conventional LTI filtering and gen-eralized 1-D total variation. Secondly, we consider a for-mulation in terms of banded matrices, for computationalefficiency. Thirdly, we give optimality conditions of theproposed problems, and use these conditions as a guide toset the regularization parameters.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248238

2. Preliminaries

2.1. Notation

We use bold uppercase letters for matrices, e.g., A andB, and bold lowercase letters for vectors, e.g., x and y. Weuse column vectors for one-dimensional series. Forexample, a vector xARN is written as

x¼ xð0Þ; xð1Þ;…; xðN�1Þ½ �T ð2Þ

where ½ � �T denotes matrix transpose. The ℓ1-norm andsquared ℓ2-norm of x are defined as

JxJ1≔XnjxðnÞj; JxJ22≔

Xnx2ðnÞ: ð3Þ

The inverse transpose of a matrix is denoted as A�T. Thefirst-order difference operator D of size ðN�1Þ � N is

D≔

�1 1�1 1

⋱ ⋱�1 1

26664

37775: ð4Þ

We use f ðx; aÞ to denote a function of x determined byparameter a, to distinguish it from a function with twoordered arguments, e.g., f ðx; yÞ.

2.2. LTI filter with matrix formulation

A discrete-time filter can be described as

Xk

ak yðn�kÞ ¼Xk

bk xðn�kÞ; ð5Þ

with transfer function HðzÞ ¼ BðzÞ=AðzÞ. For a finite lengthsignal, (5) can be rewritten as

Ay¼ Bx; ð6Þ

where A;BARN�N are both banded matrices. When H is azero-phase filter with a�k ¼ ak and b�k ¼ bk, then A and Bare both symmetric Toeplitz matrices

A¼

a1 a0a0 a1 a0

⋱ ⋱ ⋱a0 a1 a0

a0 a1

26666664

37777775; ð7aÞ

Table 1Sparsity-promoting penalty functions (a40).

Penalty ϕðuÞ ϕϵðuÞ

abs juj ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p

log 1a

log 1þa ujÞ��� 1

alog 1þa

p�atan 2

affiffiffi3

p tan �1 1þ2ajujffiffi3

p� �

� π6

� �2

affiffiffi3

p tan �1

B¼

b1 b0b0 b1 b0

⋱ ⋱ ⋱b0 b1 b0

b0 b1

26666664

37777775: ð7bÞ

In this case, disregarding a few samples on both edges, ycan be re-written as

y¼ BA�1x¼Hx: ð8ÞA detailed description of such zero-phase filters is given inSection V of Ref. [61], where two parameters, d and fc,determine the order of the filter and the cut-off frequency,respectively. In this paper, B is a square matrix, in contrastto [61]. The simplest case (d¼1) is given by (7).

2.3. Majorization–minimization

Majorization–minimization (MM) [15,35,57] is a pro-cedure to replace a difficult optimization problem by asequence of simpler ones [15,24,36,57]. Suppose a mini-mization problem has an objective function J:RN-R, thenthe majorizer Gðu; vÞ:RN � RN-R satisfies

Gðu;vÞ ¼ JðuÞ; when u¼ v; ð9aÞ

Gðu;vÞZ JðuÞ; when uav: ð9bÞThen MM iteratively solves

uðkþ1Þ ¼ arg minu

Gðu;uðkÞÞ ð10Þ

until convergence, where k is iteration index. The detailedderivation and proof of convergence of MM are given in[35].

3. Majorization of smoothed penalty function

When using sparse-regularized optimization to recovera signal, the ℓ1-norm is widely utilized. To further enhancesparsity, some methods use iteratively re-weighting pro-cedures [4,34,45,63], or a non-convex pseudo ℓp-normð0opo1Þ, or a mixed-norm in the regularizer [5,68,66,58,33]. Other non-convex penalty functions can also beused. In [46] a logarithm penalty function is discussed(‘log’ function in Table 1), and in [59], an arctangentfunction is used (‘atan’ function Table 1). However, eachof these functions is non-differentiable at zero, whereϕ0ð0� Þaϕ0ð0þ Þ. A method, to avoid the discontinuity inthe derivative, is to smooth the function.

ψðuÞ ¼ u=ϕ0ϵðuÞ

ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

pffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

� ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p1þa

ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p� �1þ2a

ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

pffiffiffi3

p !

� π6

! ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p1þa

ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

pþa2ðu2þϵÞ

� �

Fig. 2. (a) Convex non-smooth penalty function and corresponding smoothed penalty function. (b) Non-convex and smoothed penalty functions.(c) Majorizer (gray) of the smoothed penalty function in (a). (d) Majorizer (gray) of the smoothed non-convex penalty function in (b). For all the figures weset ϵ¼ 0:05; v¼ 0:5; a¼ 2.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 239

Consider the smoothed function ϕϵ:R-Rþ

ϕϵðuÞ≔ϕðρðu; ϵÞÞ; ð11Þ

where function ρ:R-Rþ is defined as

ρðu; ϵÞ≔ffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p; ϵ40: ð12Þ

Becauseffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

pis always greater than zero, ϕϵ has a

continuous first-order derivative

ϕ0ϵðuÞ ¼

uffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

p ϕ0ðffiffiffiffiffiffiffiffiffiffiffiffiu2þϵ

pÞ: ð13Þ

Table 1 gives the smoothed penalty functions ϕϵ corre-sponding to ϕ in the first column, and Fig. 2(a) and(b) illustrates the penalty function and its smoothed ver-sion. The parameter ϵ controls the similarity of ϕϵ to non-smooth function ϕ. When ϵ-0, ϕϵðuÞ � ϕðuÞ. In practice,we set ϵ to a very small number (e.g., 10�10).

3.1. Majorization of smoothed penalty function

We assume the non-smooth function ϕ satisfies thefollowing conditions:

1. ϕðuÞ is continuous on R.2. ϕðuÞ is twice continuously differentiable on R⧹f0g.3. ϕðuÞ is even symmetric: ϕðuÞ ¼ ϕð�uÞ.4. ϕðuÞ is increasing and concave on Rþ .

Under such assumptions, we find a quadratic functiong:R� R-R,

gðu; v;ϕÞ ¼ ϕ0ðvÞ2v

u2þϕðvÞ�v2ϕ0ðvÞ; ð14Þ

majorizing ϕ when va0 [10, Lemma 1]. However, ϕ mightnot be differentiable at zero, e.g., all functions ϕ in Table 1.To avoid this issue, we use (11) to ensure that the objectivefunction is continuous and differentiable. Then the MMmethod can be applied without the occurrence of num-erical issues (e.g., divide by zero). The following proposi-tion indicates a suitable majorizer of the smoothed penaltyfunctions.

Proposition 1. If function ϕ satisfies the listed four condi-tions in the previous paragraph and g in (14) is a majorizer ofϕ, then

gϵðu; v;ϕϵÞ ¼ gðρðu; ϵÞ; ρðv; ϵÞ;ϕÞ; ð15Þis majorizer of ϕϵðuÞ ¼ ϕðρðu; ϵÞÞ for u; vAR, and if writingexplicitly, then gϵ in (15) is

gϵðu; v;ϕϵÞ ¼u2

2ψðvÞþϕϵðvÞ�v2ϕ0ϵðvÞ; ð16Þ

where ψðvÞ ¼ v=ϕ0ϵðvÞa0.

The proof of Proposition 1 is given in Appendix A. Fig. 2(c)and (d) gives examples of using function (16) to majorize thesmoothed penalty functions in Fig. 2(a) and (b), respectively.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248240

4. Exponential transient excision algorithm

To formulate the problem of exponential transientexcision, we define R as

R≔

�r 1�r 1

⋱ ⋱�r 1

26664

37775: ð17Þ

The first-order derivative operator D is a special case ofR with r¼1. In this paper we restrict 0oro1 to distin-guish them. As a filter, R can be seen to be a first-orderfilter attenuating low frequencies, with a transfer function

RðzÞ≔1�rz�1: ð18Þ

Driving the system RðzÞ with the step exponential

xðnÞ ¼0; non0;

rn�n0 ; nZn0;

(ð19Þ

produces an impulse δðn�n0Þ as an output. If input signal xis a step exponential with rate r, i.e., a Type 1 artifact, then

v¼ Rx ð20Þwill be sparse.

Note that r should be known beforehand. In practice,we can estimate it from the time-constant of a Type1 artifact. Using observation data y, if we can measure thetime N0 over which a Type 1 transient decays to half itsinitial height, then r can be found by solving rN0 ¼ 0:5. Weignore the influence of the slowly varying baseline, as weassume the exponential decays much faster. If the tran-sients have an approximately equal decay rate, then wecan use an average measured form multiple transients.

4.1. Problem definition

In signal model (1), if an estimate x̂ is known, then wecan estimate f by

f̂ ¼ ðy� x̂Þ�Hðy� x̂Þ; ð21Þ

where H¼ BA�1 is a highpass filter, where the A and Bmatrices are both banded with a structure as (7). Corre-spondingly, noise w is y�ðx̂þ f̂ Þ ¼Hðy� x̂Þ, which indi-cates a suitable data fidelity term JHðy�xÞJ22. Moreover,when the component x is modeled as (19), we can usev¼ Rx as its sparse representation. Hence, we propose toformulate the estimation of x from noisy data y as theunconstrained optimization problem:

x� ¼ arg minx

P1ðxÞ ¼ JHðy�xÞJ22þλXnϕϵð½Rx�nÞ

( ); ð22Þ

where λ40 is the regularization parameter related tonoise level.

4.2. Algorithm derivation

Using the majorizer of smoothed penalty function (16),and recalling that H¼ BA�1 illustrated in Section 2.2, we

majorize the objective function (22),

Gðx; z; P1Þ ¼ JHðy�xÞJ22þλXngϵð½Rx�n; ½Rz�n;ϕϵÞ

¼ JBA�1ðy�xÞJ22þxTRT ΛðRzÞ½ �RxþC ð23Þwhere C does not depend on x, and ΛðRzÞ½ �ARðN�1Þ�ðN�1Þ isa diagonal matrix

ΛðRzÞ½ �n;n ¼λ

2ψð½Rz�nÞð24Þ

where ψðuÞ≔u=ϕ0ϵðuÞ, which is listed in Table 1. Then the

minimizer of (23) is given by

x¼ A�TBTBA�1þRT ΛðRzÞ½ �Rh i�1

A�TBTBA�1y: ð25Þ

Note that calculating (25) directly requires the solution toa large dense system of equations. However, since A isinvertable, we can factor A out and write (25) as

xðkþ1Þ ¼A BTBþATRT ΛðRxðkÞÞh i

RAh i�1

BTBA�1y; ð26Þ

where the MM procedure (10) is adopted with z¼ xðkÞ.Note that the matrix to be inverted in (26) is banded, sothat the solution can be computed efficiently by fastbanded system solvers (e.g., [28,29]). Moreover, in (26), allmatrix multiplications are between banded ones. Table 2summarizes the algorithm to solve (22).

4.3. Optimality condition and parameter selection

For problem (22), it is difficult to derive optimalityconditions directly. Here, we consider the optimality con-dition indirectly via an equivalent problem, similar to thediscussion of optimality condition for total variation pro-blems in [2,17] and for LPF=TVD in [61].

To facilitate our derivation, we firstly denote the opti-mal solution of (22) by x�. We then define v� ¼ Rx� whereR is given by (17). We define GARN�ðN�1Þ,

G≔

01 0r 1 0r2 r 1 0⋮ ⋱ ⋱

rN�3 ⋯ ⋯ r 1 0rN�2 ⋯ ⋯ r2 r 1

2666666666664

3777777777775

ð27Þ

which satisfies

RG¼ I: ð28ÞThe operator G acts as an inverse filter of R. Using theabove assumptions, we derive the following propositionregarding optimality conditions for (22).

Proposition 2. If x� is an optimal solution to (22), thenv� ¼ Rx� is an optimal solution to problem

v� ¼ arg minv

Q ðvÞ; ð29Þ

where

Q ðvÞ ¼ JHðy0�GvÞJ22þλXnϕϵð½v�nÞ: ð30Þ

and y0 ¼ y�ðx��GRx�Þ.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 241

Proof. We define x0ARN as

x0≔Gv� ¼GRx� ð31Þwhere x� is the optimal solution to (22). Note that x0 andx� share an identical v� ¼ Rx0 ¼ Rx�. Moreover, we definethe difference between x� and x0 as

d≔x��x0: ð32ÞIt follows that Rd¼ Rx��Rx0 ¼ 0. As a consequence, x0 isthe optimal solution to problem

x0 ¼ arg minx

P1ðuÞ ð33aÞ

s:t: x¼ u�d: ð33bÞNote x0ð0Þ ¼ 0 must be satisfied in (33)) because of thedefinition of G. Thus, we can write an equivalent problemto (15) using (31),

fx0; v�g ¼ arg minx;v

P1ðuÞ ð34aÞ

s:t: x¼ u�d ð34bÞ

x¼Gv ð34cÞwhere fx0; v�g must be the optimal solution. In problem(34)?, u is uniquely determined by linear functions ofv and d, so problem (34)? can be simplified by substitutingthe variables,

fx0; v�g ¼ arg minx;v

Q ðvÞ ð35aÞ

s:t: x¼Gv ð35bÞwhere Q ðvÞ ¼ P1ðuÞ, with u¼ Gvþd, and can be writtenexplicitly,

Q ðvÞ ¼ JHðy�Gv�dÞJ22þλXnϕϵð½RGvþRd�nÞ: ð36Þ

Because RG¼ I and Rd¼ 0, we simplify (36) as

Q ðvÞ ¼ JHðy0�GvÞJ22þλXnϕϵð½v�nÞ ð37Þ

where y0 ¼ y�d.In this case, the equality constraint in (35) is redundant.

We can solve the unconstrained problem (29) first, andthen compute x0 by (31). This implies that v� is the opti-mal solution to (29) and satisfies ∇Q ðv�Þ ¼ 0.□

Using Proposition 2, instead of problem (22), we alter-natively consider the optimality condition of problem (29)where as long as x� is an optimal solution of (22), v� is anoptimal solution of (29). Moreover, we can rewrite theoptimality condition of (29) as

pðnÞ ¼ λϕ0ϵð½v��nÞ ¼ λϕ0

ϵð½Rx��nÞ; ð38Þwhere p¼ 2GTHTHðy0�x0Þ, which can be rewritten as

p¼ 2GTHTHðy�x�Þ: ð39ÞNote that, writing p as (39) does not require the solution toproblem (29) or to compute d and y0. Therefore, althoughwe derived an indirect way to verify the optimality con-dition for (22), the final procedure is direct.

Setting parameter λ: Equation (38) can be used as aguide to set the regularization parameter λ. Suppose theobservation data is composed of Gaussian noise only, then

a proper value of λ should make the solution of (22)almost identically zero. Thus, its sparse representation v�,given by (20), should be all zero as well. In this case, wecan calculate a vector q, which depends only on noise,

q¼ 2GTHTHw: ð40ÞWhen p is calculated by (39), we find a constraint that�λopðnÞoλ, by the property of ϕϵ. Furthermore, sincex� � 0 is expected when the observation is pure noise (i.e.,y¼w), the values of q� p can be considered bounded:

qðnÞ A ½�λ; þλ� when jvðnÞjrβϵ for nAZN ð41Þwith high probability. The value βϵ here is related to ϵ, andsince ϵ is extremely small, it can be simply assumed to be0. As a consequence, the value of λ needs to satisfy

λZmax j2½GTHTHw�nj; n A ZN

n o: ð42Þ

Further, if the statistical property of the noise can beexploited, e.g., wðnÞ N ð0; σ2wÞ, λ can be set statistically,such that

λ¼ 2:5σw J2h1 J2 ð43Þwhere σw is the standard deviation of the noise and h1 isthe impulse response corresponding to system GTHTH.Note that, GTHTH is a filter with a transfer functionP1ðzÞ ¼H2ðzÞ=RðzÞ. Hence, although we give the matrix G toderive the approach to set λ, it is not required to computeh1 because the filter P1ðzÞ can be implemented directlyusing H(z) in (8), and R(z) in (18).

4.4. Example: synthetic signal

To illustrate ETEA, the algorithm is tested on a simu-lated signal and compared to other methods. Fig. 3 showsthe test signal and its components: a lowpass signal f, atransient component x, composed of three exponentialdecays, and white Gaussian noise with σw ¼ 0:20. Thenoisy observation is shown in Fig. 3 in gray.

Denoising: The result of conventional lowpass filteringis shown in Fig. 4(a). The result is smooth, but the stepedges of each pulse are diminished. Moreover, if comparedto the true low-pass baseline, the result is also distorted.The result of using LPF=TVD [61] is illustrated in Fig. 4(b).Although the abrupt jumps have been preserved, seriousstaircase effects appear. LPF=TVD uses the first-orderderivative operator D in regularization. The derivative ofan exponential is another exponential, therefore the deri-vative is unlikely to have a sufficiently sparse representa-tion. A compensative method is to widen the bandwidth ofthe lowpass filter, however, this leads to a noisy result. Theresult presented in Fig. 4(b) is selected by tuning allnecessary parameters of LPF=TVD to optimize RMSE (root-mean-square error) value.

The result of the proposed ETEA method is shown inFig. 4(c). The step edges and decay behavior are wellpreserved (RMSE¼0.057). We use a smoothed ℓ1-normpenalty function. In this example, we set ϵ¼ 10�10,r¼0.94, filter order parameter d¼1, cut-off frequencyf c ¼ 0:013 cycle=sample, and regularization parameter iscalculated by (43) based on the noise principle.

Time (sample)0 50 100 150 200 250 300 350 400

-0.50

0.51

1.5

-0.50

0.5

-0.50

0.51

1.52

2.5 Simulated data ( = 0.20)

Low-pass

Exponential

Clean Noisy

Fig. 3. Example 1: Simulated data comprising a lowpass component andexponential transients.

Time (samples)

-1

0

1

2

3Lowpass filter

Time (samples)

-1

0

1

2

3

LPF/TVD (RMSE = 0.073)

Time (sample)

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400

0 50 100 150 200 250 300 350 400-1

0

1

2

3ETEA (RMSE = 0.057)

Fig. 4. Example 1: The denoising results of (a) lowpass filtering, (b) LPF/TVD, (c) ETEA (proposed method).

0 50 100 150 200 250 300 350 400

−0.50

0.51

1.5

−0.50

0.5

−0.50

0.51

1.52

2.5 Artifact−free Wavelet (RMSE = 0.072)

Time (sample)

Denoising

Low−pass subband

Other subbands

0 50 100 150 200 250 300 350 400

−0.50

0.51

1.5

−0.50

0.5

−0.50

0.51

1.52

2.5 ETEA (log) (RMSE = 0.043)

Time (sample)

x+f

f

x

Fig. 5. Example 1: Comparison of denoising and decomposition results using(a) Wavelet-based method and, (b) ETEA with non-convex regularization.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248242

Decomposition: Wavelet-based methods for artifact cor-rection have been described in [1,25,40,56]. In Fig. 5(a), weillustrate denoising and decomposition results obtainedusing the stationary (un-decimated) wavelet transform [11]with Haar wavelet using hard-threshold determined by theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 log N

pσw thresholding scheme. The denoised result is

further enhanced by the artifact-free denoising method[13,14] which uses total variation minimization to over-come pseudo-Gibbs oscillations. Although the denoisingoutput is relatively smooth and captures the discontinuities,the decomposed components are both distorted. Comparedto the true components in Fig. 3, the lowpass subbanddeviates from true lowpass component (especially at aboutn¼100). The transient component, reconstructed fromother subbands after denoising, has to compensate for theerror. Hence, the estimated transient component does nothave a zero baseline (see bottom of Fig. 5(a)).

Non-convex penalty functions can be used to improvethe result of ETEA. Here we use the smoothed non-convexlogarithm penalty function in Table 1 (a¼2). The filteringresult is shown in Fig. 5(b), where discontinuities are moreaccurately preserved, and the RMSE is reduced to 0.043compared with the result in Fig. 4(c). We also illustratethe decomposed f and x components in Fig. 5(b). ETEArecovers both of the components accurately. The algorithmrun-time is not affected by the choice of penalty function.In this example, 50 iterations of the algorithm takes about

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 243

25 ms on a MacBook Pro 2012 with a 2.7 GHz CPU,implemented in Matlab 8.4.

4.5. Example: artifact removal of ECoG data

Conventional EEG studies and most clinical EEG appli-cations are restricted below 75 Hz [51]. Advanced mea-suring methods such as ECoG records multichannel cor-tical potentials from the micro-electrodes located insidehuman skull with a higher sampling rate. In this example,we use a 5-second ECoG signal epoch, with a sampling rateof 2713 Hz, recorded from a mesial temporal lobe epilepsy(MTLE) patient.

Fig. 6(a) shows the raw data in gray and the corrected databy ETEA in black. There are two Type 1 artifacts identified inthis 5-s epoch. One is at about t¼1.20 s, and the other is atabout t¼3.60 s. In signal model (1), suppressing component xwhich represents artifacts, the corrected data ðf þwÞ shouldpreserve the components. We show the corrected data inFig. 6(a), where the two Type 1 spikes are correctly removed.The decomposed artifact is illustrated in Fig. 6(a) as well,which adheres a zero-baseline.

Since wavelet-based methods have been successfullyapplied to suppress artifacts [1,25,40]. A comparison witha wavelet-based method is shown in Fig. 6(b). We use thestationary (un-decimated) wavelet transform [11] withHaar wavelet filter and the non-negative garrote thresholdfunction [18], as recommended in [25]. The thresholdinghas been applied to all the subbands except the lowpassband, and the artifact component is obtained by sub-tracting the corrected data from the raw data. As shown,this approach estimates transient pulses but the estimatedpulse adheres to the shape of the Haar wavelet filter: apositive-negative pulse (see the bottom square box inFig. 6(b)), but not the true Type 1 artifact in Fig. 1(b).

To more clearly illustrate the estimated results by thetwo methods, we show the details of the corrected dataand the artifact from t¼3.50 to 3.65 s in dashed line boxesin Fig. 1(a) and (b), respectively. Since the wavelet-basedmethod cannot correctly estimate Type 1 artifact in thiscase, a ‘bump’ can be observed in Fig. 6(b) to compensatethe error. Moreover, this inappropriate estimation willcause the corrected data to change before Type 1 artifactactually occurs (see Fig. 1(b)). In contrast, ETEA estimatesthe artifact as the abrupt drift with a decay, then it excisesthe artifact without influencing the data before the tran-sient occurs (see Fig. 1(a)).

In this example, for the signal with a length of 13,565samples, the proposed algorithm converges within 40iterations, and takes about 330 ms on a MacBook Pro 2012with 2.7 GHz CPU, implemented in Matlab 8.4. The costfunction history is shown in Fig. 7.

5. Higher-order ETEA

In the previous section, we have presented ETEA basedon signal model (1), suitable for Type 1 artifacts, where thecomponent x has discontinuous step exponential tran-sients. Here we illustrate another version of ETEA based on

signal model (1), where the component x models Type0 artifacts.

First, we consider the operator

R2≔

r2 �2r 1r2 �2r 1

⋱ ⋱ ⋱r2 �2r 1

26664

37775; ð44Þ

where R2ARðN�2Þ�N has three non-zero coefficients ineach row. As a special case, when r¼1, R2 is the second-order difference operator, and JR2xJ1 is the same as theregularizer used in [30] for ℓ1 detrending.

Taking R2 as a filter, the transfer function is

R2ðzÞ ¼ ð1�rz�1Þ2; ð45Þwhich has a double zero at z¼ r. The impulse response ofsystem R�1

2 ðzÞ is ðnþ1Þrn, when nZ0. It has the sameshape as Fig. 1(a), which is a suitable model for the Type0 artifacts in [25]. Therefore, R2x is sparse when x iscomposed of such piecewise smooth transients. Similar to(22), we formulate the problem

x� ¼ arg minx

P2ðxÞ ¼ JHðy�xÞJ22þλXnϕϵð½R2x�nÞ

( )ð46Þ

to estimate the transient component x, so that the cor-rected data is y� x̂, and the low-pass trend f can be esti-mated by (21). Additionally, (46) can be easily solved byETEA (in Table 2) substituting R by R2.

An example of complicated and irregular artifacts isthe eye blink/movement artifacts in EEG data, which mayhave various morphologies and durations. In this case, weassume that the artifacts can be estimated by a continuouspiecewise smooth waveform, generated by applying acertain sparse impulse sequence to R�1

2 ðzÞ (45). In otherwords, we broaden our signal model so that the artifact isnot only equivalent to an isolated Type 0 transient as in[25], but also a superposition of multiple such transients,with some freedom of scaling and shifting. The problem ofestimating x can be solved by (46) in this case as well.

5.1. Example: simulated data

Fig. 8 shows the simulated data and its lowpass andtransient components. The transient component has sev-eral pulses with different heights and widths. They areobtained by feeding a sequence of impulses into systemR�12 ðzÞ. The filter is given in (5) with ak ¼ ½1; �2r; r2�,

b0 ¼ 1, and r¼0.950.Empirical mode decomposition (EMD) [22,23], a powerful

method for analyzing signals, has been successfully utilizedin different fields, including neuroscience, biometrics, speechrecognition, electrocardiogram (ECG) analysis, and fault detec-tion [8,16,21,39,54,62,65,67].

As a comparison to the proposed approach, we usethe EMD based denoising algorithm in [32], which useswavelet coefficient thresholding techniques on decom-posed IMFs. More specifically, we use clear first iterativeinterval thresholding (CIIT) with smoothly clipped abso-lute deviation (SCAD) penalty thresholding [31,32], with20 iterations, and the result is shown in Fig. 9(b). In this

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−100

0

100

200

300

100

200

300

400

Time (second)

ETEA

Artifact

Raw data Corrected

Time (second)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-100

0

100

200

300

100

200

300

400 Wavelet

Artifact

Raw data Corrected

Fig. 6. Example 2: Comparison of artifacts removal for ECoG data.(a) ETEA (proposed method), (b) wavelet-based method.

5 10 15 20 25 30 35 400.8

1

1.2

1.4

1.6

1.8

2x 105

Iterations

Cost function history

Fig. 7. Example 2: Cost function history.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248244

example, in order to perform decomposition, among theentire eight IMFs, the thresholding is performed on IMFs1–5, their summation is considered as the transients illu-strated in Fig. 9(a), and IMFs 6–8 are considered as theestimation of the lowpass component. The EMD basedmethod achieves a decent denoising performance, butdoes not accurately estimate the components. For instance,the simulated data has a smooth dip at about n¼ 700(circled in Fig. 9(a)). EMD decomposes it into higherIMFs since they are varying slowly, which degrades theestimation. We may group the lowpass and transient

components differently to avoid this problem, for instance,grouping IMF 1–6 together in order to include moreoscillations into transient component, but this causesother distortion, where the decomposed transient com-ponent contains a lowpass signal and does not adhere to abaseline of zero.

The result obtained using second-order ETEA is illu-strated in Fig. 9(b). ETEA estimates both the low-pass andtransient components well, and recovers the signal by xþ fprecisely with RMSE¼0.87 (with the smoothed ℓ1-penaltyfunction). The regularization parameter λ for problem (46)was similar to (43),

λ� 2:5σw J2h2 J2; ð47Þ

where h2 is the impulse response of system H2ðzÞ=R2ðzÞ,and R2ðzÞ is defined in (45). The decomposition is accurate.There are no compensating waveforms between the esti-mated x and f at n¼ 700, comparing to the estimation inFig. 9(a).

Through numerical experiments, we found that second-order ETEA is not very sensitive to parameter r. Fig. 9(c) shows the RMSE of denoising the data in Fig. 8(a), usingrA ½0:85;0:99�. In this test, all filter parameters (fc and d) arethe same and λ is set by (47). In most cases, the results arebetter than EMD-CIIT. In addition, r should not be too smallor very close to 1. If r has to be very small to yield a goodestimation of component x, it must fluctuate extremelyrapidly, then it must be closer to a sequence of sparse spikes(i.e., Type 3 artifacts in [25]), which differs from the signalmodel. For such a signal, other algorithms are more sui-table, e.g., LPF/CSD [60,61]. Similarly, if r has to be very closeto 1 to fit the transients, then the transients must be veryclose to a piecewise linear signal, which is also not how wemodel the signal initially. As a consequence, we suggest toset r in the range 0:90oro0:98.

5.2. Example: ocular artifacts suppression

In this example, we use ETEA with R2 to correct EEGwith eye blink/movement artifacts. Fig. 10a shows a 10-ssignal from channel Fp1, with sampling rate f s ¼ 256 Hz,downloaded from [38]. As a channel located on forehead,Fp1 is very sensitive to the motion of eyes and eyebrows,and in this example, eye movement artifacts with largeamplitudes are present through the entire signal. Applying

0 100 200 300 400 500 600 700 800 900 1000

−1.5−1

−0.50

0.51

1.5

−0.50

0.5

−1−0.5

00.5

11.5

22.5

Simulated data ( = 0.40)

Time (sample)

(a)

Low−pass

Transients

Clean Noisy

Fig. 8. Example 3: The simulated test data, generated by driving an ARfilter with a sparse sequence and adding noise.

0 100 200 300 400 500 600 700 800 900 1000

−1.5−1

−0.50

0.51

1.5

−0.50

0.5

−1−0.5

00.5

11.5

22.5

Time (sample)

Denoising

IMF 6−8

IMF 1−5

EMD−CIIT (RMSE = 0.107)

0 100 200 300 400 500 600 700 800 900 1000

−1.5−1

−0.50

0.51

1.5

−0.50

0.5

−1−0.5

00.5

11.5

22.5

ETEA (RMSE = 0.086)

Time (sample)

x+f

f

x

0.86 0.88 0.9 0.92 0.94 0.96 0.980.08

0.09

0.1

0.11

0.12

r

RMSE vs. parameter r ETEA EMD−CIIT

Fig. 9. Example 3: Denoising and decomposition results by (a) EMD baseddenoising (EMD-CIIT [32]) and (b) ETEA.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 245

second-order ETEA with r¼0.95, the results for correcteddata and extracted OA are illustrated in Fig. 10(b).

As a comparison, we use multivariate empirical modedecomposition (MEMD) [52] to correct the data. MEMD isa recently developed algorithm extending conventionalEMD. It has been used in different aspects of EEG signalanalysis and applications [9,50,49], including removing theocular artifacts (OA) from multichannel EEG data [53,42].In this example, we use 4 EEG channels (Fp1, Fp2, C3, C4)measured simultaneously as the input, and decompose thehigher-index IMFs (low-frequency subbands) consideredto be artifacts [53, Section V]. More specifically, among all14 IMFs decomposed in this example, we use IMF 1–4 asthe corrected data, and the rest as artifacts. The correcteddata and estimated OA are shown in Fig. 10(c).

From the results in Fig. 10(b) and (c), ETEA estimatesartifacts more clearly than MEMD. In the MEMD result,some small-amplitude higher frequency oscillations leakinto the artifact (e.g., about t¼5.5 and 9.0 s). Moreover,some artifacts are introduced after applying MEMDmethod to correct the data, (e.g., about t¼3.0 and 9.5 s inFig. 10(c)). Some oscillations are generated as transientswhere the abrupt artifacts occur. In contrast, in Fig. 10(b),there are no oscillations introduced either in the estimatedartifacts or the corrected data.

Computational efficiency: Fig. 11 shows the average com-putation time as a function of the signal length. In thisexperiment, we calculate the time of computation of ETEA(22) and second-order ETEA (46) with different filter settings(controlled by parameter d), using input signal lengths from5000 to 105. For each length, we average the computationtime over 10 trials. For each trial, run 40 iterations of eachalgorithm. The experiment is implemented in Matlab 8.4 on aMacBook Pro 2012 with 2.7 GHz CPU.

As shown, the proposed algorithms have a run time oforder O(N). Most of the computation time is consumed by thestep of solving the linear system Q �1b in Table 2, where Q isa banded matrix and we use a fast banded solver.

Additionally, fast iterative shrinkage-thresholding algo-rithm (FISTA) [3,7], which is an acceleration scheme for

iterative shrinkage-thresholding algorithm (ISTA) [6], (aspecial formulation of MM) may be used to further accel-erate the algorithm.

6. Conclusion

This paper proposes a new algorithm for denoising andartifact removal for signals comprising artifacts arising inmeasured data, e.g., neural time-series recordings, usingsparse optimization. The first algorithm, ETEA, assumes

0 1 2 3 4 5 6 7 8 9 10Time (second)

Raw Data

0 1 2 3 4 5 6 7 8 9 10Time (second)

ETEA Artifacts Corrected data

0 1 2 3 4 5 6 7 8 9 10

Time (second)

MEMD Artifacts Corrected data

Fig. 10. Example 4: (a) EEG data with ocular artifacts, and corrected datausing (b) ETEA and (c) MEMD.

signal length (sample) 1043 4 51 2 6 7 8 9 10

time

(sec

ond)

0

1

2

3

4

5

6Computation time of different signal length

ETEA (d=1) ETEA (d=2) ETEA-2nd (d=1) ETEA-2nd (d=2)

Fig. 11. Comparison of computation time with different signal lengths.

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248246

that the signal is composed of a lowpass signal and anexponential transients (Type 1). It is formulated as anoptimization problem regularized by differentiable andsmooth penalty function. The second algorithm is an

extension of ETEA, using a higher-order recursive filter,which is applicable for correction of continuous protu-berance transients (Type 0), and more irregular artifacts.As applications, we have shown that ETEA with differentregularizers (R and R2) are suitable for the suppression ofType 1 artifact in ECoG data and ocular artifacts (OA) (assequential Type 0 artifacts) in conventional EEG data, withdetailed comparisons to some state-of-the-art methods.Both of the above algorithms are computationally efficientbecause they are formulated in terms of banded matrices.A promising future work is to extend the above data cor-recting methods to multichannel data.

Acknowledgments

The authors would like to thank Jonathan Viventi of theDepartment of Biomedical Engineering of Duke University,and Justin Blanco of the Electrical and Computer Engi-neering Department of United States Naval Academy, forproviding the data and giving useful comments.

Appendix A. Proof of Proposition 1

Proof. Substitute the variables u and v in (14) by

u¼ffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

p; and v¼

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p: ðA:1Þ

Therefore, for x; zAR, the inequality holds

ϕ0ðffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p ðx2þϵÞþϕðffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p�

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p

2ϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ

Zϕðffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ: ðA:2Þ

The right of the inequality is the majorizer of thesmoothed penalty function ϕð

ffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ.

(1) If za0, we can multiply z to the numerator anddenominator of the first term on the left side of (A.2),

ϕ0ðffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p ðx2þϵÞ ¼ x2

2zϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ zffiffiffiffiffiffiffiffiffiffiffiffi

z2þϵp

� �

þ ϵ

2zϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ zffiffiffiffiffiffiffiffiffiffiffiffi

z2þϵp

� �¼ x2

2zϕ0ϵðzÞþ

ϵ

2zϕ0ϵðzÞ: ðA:3Þ

Multiplying the numerator and denominator of the thirdterm on the left side of (A.2) by

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p,ffiffiffiffiffiffiffiffiffiffiffiffi

z2þϵp

2ϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ ¼ z2

2ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p ϕ0ðffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ

þ ϵ

2ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

p ϕ0ðffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ ¼ z

2ϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ zffiffiffiffiffiffiffiffiffiffiffiffi

z2þϵp

� �

þ ϵ

2zϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffiz2þϵ

pÞ zffiffiffiffiffiffiffiffiffiffiffiffi

z2þϵp

� �¼ z2ϕ0ϵðzÞþ

ϵ

2zϕ0ϵðzÞ: ðA:4Þ

Using the results in (A.3) and (A.4), the inequality (A.2) canbe rewritten as

x2

2zϕ0ϵðzÞþ

ϵ

2zϕ0ϵðzÞþϕϵðzÞ�

z2ϕ0ϵðzÞ�

ϵ

2zϕ0ϵðzÞZϕϵðxÞ; ðA:5Þ

Y. Ding, I.W. Selesnick / Signal Processing 120 (2016) 236–248 247

which can be reorganized into

gϵðx; z;ϕϵÞ ¼ϕ0ϵðzÞ2z

x2þϕϵðzÞ�z2ϕ0ϵðzÞZϕϵðxÞ: ðA:6Þ

(2) If z¼0 and xa0, by Lagrange's Mean Value Theorem[55, Theorem 5.10], since function ϕðxÞ is continuous andϕ″ðxÞr0 on x40, there exists a value ξ in the rangeffiffiffiϵ

p oξoffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pthat ϕ0ð ffiffiffi

ϵp ÞZϕ0ðξÞZϕ0ð

ffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ satisfying

ϕ0ðξÞffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

p� ffiffiffi

ϵp� �

¼ ϕðffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ�ϕð ffiffiffi

ϵp Þ: ðA:7Þ

Moreover, consider the square that is always positive

ðffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

p� ffiffiffi

ϵp Þ2 ¼ x2þϵ�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵðx2þϵÞ

qþϵ40; ðA:8Þ

which implies the inequality

x242ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵðx2þϵÞ

q�2ϵ: ðA:9Þ

Furthermore, we can multiply both sides of (A.9) by apositive term ϕ0ð ffiffiffi

ϵp Þ=2 ffiffiffi

ϵp

, and then adopt the result from(A.7), so that

ϕ0ð ffiffiffiϵ

p Þ2ffiffiffiϵ

p x24ϕ0ð ffiffiffi

ϵp Þ

2ffiffiffiϵ

p 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiϵðx2þϵÞ

q�2ϵ

� �ðA:10aÞ

¼ ϕ0ð ffiffiffiϵ

p Þffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

p� ffiffiffi

ϵp� �

ðA:10bÞ

4ϕ0ðξÞffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

p� ffiffiffi

ϵp� �

ðA:10cÞ

¼ ϕðffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ�ϕð ffiffiffi

ϵp Þ; ðA:10dÞ

which leads to

ϕ0ð ffiffiffiϵ

p Þ2ffiffiffiϵ

p x2þϕð ffiffiffiϵ

p Þ4ϕðffiffiffiffiffiffiffiffiffiffiffiffix2þϵ

pÞ: ðA:11Þ

Because ϕϵ in (13) is differentiable on R, we can find itssecond-order derivative at zero,

limz-0

ϕ″ϵðzÞ ¼

ϕ0ð ffiffiffiϵ

p Þffiffiffiϵ

p ; ðA:12Þ

and by L'Hôpital's rule [55, Theorem 5.13], we have

limz-0

ϕ0ϵðzÞ2z

¼ limz-0

ϕ″ϵðzÞ2

¼ ϕ0ð ffiffiffiϵ

p Þ2ffiffiffiϵ

p ; ðA:13Þ

which implies that (A.11) is in the same form of themajorizer (A.6) at z¼0.(3) If x¼ z¼ 0, then the condition (9) follows imme-

diately.□

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