A Straight Forward Signal Processing Scheme to

A Straight Forward Signal Processing Scheme to Improve Effect Size of fNIR Signals Md. Asadur Rahman

Mohiuddin Ahmad

Department of Biomedical Engineering Khulna University of Engineering & Technology Khulna-9203, Bangladesh [email protected]

Department of Electrical and Electronic Engineering Khulna University of Engineering & Technology Khulna-9203, Bangladesh [email protected]

Abstract—Functional near-infrared spectroscopy (fNIRS) plays an imperative role for studying hemodynamic measurement of brain. Event related task measurement mostly depends on the effect size (ES) of fNIRS data. The noisy fNIR signal is an obstacle to estimate the precise ES of such measurement. Though Savitzky-Golay and Moving Average filters are often used for denoising the fNIR signal, they have some limitations in measuring ES. In this paper, we have proposed a simple signal processing scheme which contributes to remove noise and evaluates not only proper ES but also overcome the drawback of Savitzky-Golay and Moving Average filter. By this scheme, the filtered signal becomes lower standard deviated than the raw fNIR signal. Else, the scheme maintains the mean of original and filtered signal unchanged. Since, the scheme reduces the standard deviation of the signal notably remaining the mean value unchanged; the ES of interest is improved eloquently. The numerical results and corresponding contrast to noise ratio (CNR) pattern prove the usefulness of the proposed scheme. The numerical results and corresponding contrast to noise ratio (CNR) pattern prove the effectiveness of the proposed scheme. Keywords- Functional near infrared spectroscopy (fNIRS), low pass Butterworth filter (LPBF), Savitzky-Golay filter, moving average filter, contrast to noise ratio (CNR), effect size (ES).

I.

INTRODUCTION

Functional near infrared spectroscopy (fNIRS) is a photonic device to detect changes in the concentration of oxygenated (HbO2) and deoxygenated (Hb) hemoglobin molecules in the blood. The NIRS method works based on neuro-vascular coupling which means a rise of cerebral blood flow is observed in the cerebral region of neuron activation that was first observed by Jöbsis [1]. Over the last two decade, fNIRS has widely extended its applications due to its capacity to quantify oxygenation in blood and organic tissue in a continuous and non-invasive manner [2]. Brain activity localizing in noisy fNIRS data plays an important role for investigating event related hemodynamic of the neuronal sites [3]. There are a number of research article where the researchers use the feature ‘effect size’ (ES) to quantify the difference between two events. Finger tapping task [3], eyes open and eyes closed event [4], resting and active condition [5], activation pattern for regulation of childhood frustration [6], analysis of cortical blood flow with music [7] and many more analysis through fNIRS are quantified and

differentiated by ES. Considering these analytical evidences, we can realize the importance of ES in the context of event classification from fNIRS. This ES solely depends on the mean and standard deviation of the data set of two different states. Therefore, it is very necessary to estimate the proper mean value and standard deviation from the raw fNIR signals. On the other hand, raw fNIR signal is very complex data with a high degree of noisy drift [3], [8]. There are several proposals to remove the noise from original fNIR signal by filtering. In [9], Savitzky-Golay filter, in [10], state-space modelling and in [11], modified statespace modelling by PCA analysis is used to estimate the signal in more simple format by removing noise. In the case of Savitzky-Golay filter, the mean value of the signal remains same but the standard deviation becomes very low which cannot follow the contrast to noise ratio (CNR) of the original signal. Due to this limitation, Savitzky-Golay filter cannot provide the proper information about ES. On the other hand, in [10-11], state-space modelling and PCA based modified state-space modelling are proposed to remove noise for improving the numerical value of CNR. Therefore, in this case, PCA analysis cannot be fruitful. Therefore, the original fNIR signal should be processed in such a way that by removing the noise the numerical value of enhanced ES will be proper and of course, the reconstructed signal after filtration follow the pattern of CNR of original signal. In this paper, a signal processing scheme is proposed which is not only simplify the fNIR signal profile but also reduce the high degree of signal variance without changing the mean value of original fNIR signal. In this scheme, a simple low pass Butterworth filter (LPBF) of appropriate frequency ratio is used to remove the noise. We reconstructed the fNIR signal according to the proposed scheme and a comparison of the features of the original and reconstructed signal is presented. In addition, we have compared our result with Savitzky-Golay and Moving Average filter as well as explained their unfitness for this kind of analysis. It is shown with appropriate numerical results that our proposed scheme can intensify ES in average up to 90%. It is also graphically explained that how the proposed scheme follows the CNR of original signal where Savitzky-Golay often fails to follow that pattern.

This paper is organized as follows: Section II describes the general background of fNIR signal. In Section III, we have presented oximetry calculation procedure and corresponding output. The experimental results are analyzed in section IV with proper discussions. Finally, we conclude our total research work in section V. II.

BACKGROUND STUDY ON FNIR SIGNAL AND FILTERING TECHNIQUES

A. What is fNIRS? Most biological tissues are relatively transparent to light in the near-infrared (NIR) range between 700 to 900 nm because the absorbance of the main ingredients in the human tissue like water, HbO2, and Hb is small in NIR range [1]. When light in NIR range is shone through the human scalp, injected photons follow various paths inside the head. Some of these photons are absorbed by skin, skull and brain. Rest of the photons exit the head after following the so-called "banana" pattern due to scattering effect of the tissue [12]. The main absorbers in the NIR range are blood chromophores of HbO2 and Hb whereas water and lipid are relatively transparent to NIR light. Therefore, changes in the amplitude of backscattered light can be represented as changes in blood chromophore concentrations. This functional measurement is known as fNIR. B. Mathematical Modelling of Oximetry In practice, the detector and the IR emitter diode are placed 3-4 cm apart as Fig. 1. As NIR light enters the cerebrum, it traces a banana-shaped path from emitter to detector like Fig. 1. An array of sources and detectors allow the haemoglobin concentrations measurement at various places in the cerebrum. Using the Beer-Lambert law, the attenuation of light between the source and detector can be formulated as [13], I out = I in 10 −ODλ (1) where, ODλ is the optical density at the wavelength λ. The optical density can be found as, I (2) ODλ = − log out = attenuation = Aλ + Sλ I in Here, Aλ and Sλ are the absorbing and scattering factors, respectively. Therefore, absorption of light can be formulated as, Aλ = ε i , λ Ci Lλ (3)



Figure 1. Position of fNIR source and detector on human scalp [14]

In (5), μa is the absorption coefficient, and μ's, λ is the reduced scattering coefficient at wavelength, λ. To remove the effect of scattering, two successive measurements yield the differential value of optical density and the procedure can be described as,

ODλ = ODλ , final − ODλ ,initial =

In (4), d is the linear difference between the emitter and detector and the differential path factor, DPF is calculated by, 1  3μ ′  (5) DPFλ ≈  s , λ  2  μ a , λ 

(6)

Now, the effect of scattering is cancelled. Since each chromophore has a specific extinction coefficient and differential path length factor, measurement with two wavelengths leads to:

ΔOD = M × ΔC (7) By using blood chromophore concentrations, we can define two parameters, namely, oxygenated blood concentration (OXY) and blood volume (BV) which are determined as, OXY = ΔC HbO2 − ΔC Hb (8) BV = ΔC HbO2 + ΔC Hb

(9)

C. Savitzky-Golay and Moving Average Filter Savitzky and Golay proposed a method for data smoothing and eventually this technique becomes very popular filter for the researchers since 1964 which is known as Savitzky-Golay filter. This filter is designed based on local least-squares polynomial approximation [15]. In short, a sample sequences x[n] of a signal has a moment at n=0 if the sample number is 2M+1. The mean-squared error (MSE) can be minimized for the group of input sequences which is centered at n=0, where the coefficients of the polynomial are given by equation,

i = Hb. HbO2

In (3), εi,λ is the specific extinction coefficient of blood chromophore for wavelength λ, and Ci is the concentration of blood chromophores. Here, Lλ is the path-length of light at λ and expressed in terms of source detector separation as, Aλ = d .DPFλ (4)



ε i , λ .ΔCi .d .DPFλ i = Hb.HbO2

M

 ( p(n) − x[n])

E=

2

(10)

n=−M

n

Since p (n) =

 n a , we get from (10), i

i

i =0

 n i   n ai − x[n]  E=   n = − M  i =o  M

 

2

(11)

Therefore, the output of point at n=0 is,

Y [0] = p (0) = a0

(12)

Eventually, for N order polynomial the output can be computed by the given convolution form [16],

Y [ n] =

M

 h[k ]x[n − k ]

(13)

k =−M

To achieve the least square polynomial fitting we need to differentiate (11) and from that it is to consider, M N ∂E 2ni ( n k ak − x[n]) = 0 . = ∂ai k = − M k =0





From the above mathematical derivation, it can be understood that the moving average filter is identical to Savitzky-Golay filter with polynomials of order N=0 and M=1, since the Moving Average filter is defined as [15], n+m 1 y[n] = x[m] 2 M + 1 m= n− M



III.

(14)

Where, λ ≥ 0 is the regularization parameter used to control the trade-off between smoothness of xt and the size of the residual, yt− xt. The argument, xt−1 − 2xt + xt+1 is the second difference of the time series at time t. The second term in the objective is zero if and only if xt is in form xt=α + βt for some constants α and β. The weighted sum objective is sternly convex and coercive in x, and it has a unique minimizer, which we denote xhp. We can write the objective as, 2 2 (1 / 2) y − x 2 + λ Dx 2 ← minimize (16) where, u2=

x = ( x1 ,........, xn ) ∈ R n ,

( u ) i

n −1 1 n ( yt − xt ) 2 + λ  ( xt −1 − 2 xt + xt +1 ) 2 ← minimize  2 t =1 t =2

(15)

is the Euclidean, and D ∈ R

. D= . .   .

The proposed processing scheme of the fNIR signal is shown in the flow diagram as Fig. 3. The steps are given below:

One of the famous linear trend filter is H-P filtering [17] where the trend estimate xt is chosen to minimize the weighted sum objective function as,

1/ 2

( n − 2)× n

and is the

second-order difference matrix which can be represented as, . . .  1 −2 1  . 1 −2 1 . .   

PROPOSED SIGNAL PROCESSING SCHEME

A. Removing Linear Trends Removing a linear trend from a vector is usually necessary for FFT processing. This is one kind of filter and it removes the best straight-line fit linear trend from data in vector X and returns the residual in vector Y.

2 i

y = ( y1 ,.........., yn ) ∈ R n ,

. . . .

. . 1 .

. . .   . . .  −2 1 .   1 −2 1

Then H-P trend estimate is written as, x hp = ( I + 2λ DT D) −1 y

(17)

(18)

B. Rectification of the Signal In this scheme, a very simple procedure is applied for the rectification of the signal that is previously removed linear trends. The mathematical function to convert the liner trend removed signal to its rectified form can be equated as, Yx = X (19) where, X is the input vector and Yx is the rectified output of the input signal.

C. Fast Fourier Transformation Fast Fourier transform or FFT algorithm computes the discrete Fourier transform (DFT) of a sequence. Let, x = ( x0 ,.........., xn −1 ) ∈ C N then the DFT of x is defined as, xˆk =

1 N −1 −2π i Njk  xk e , k = 0,........., N − 1 N j =0

(20)

and xˆ = ( xˆ0 ,........., xˆ N −1 ) . This can be written as a matrixvector-multiplication xˆ = WN x , (WN )ij =

1 −2π i Nij e N

(21)

Where, WN is called the Fourier-matrix. It follows that FFT is an efficient method to calculate the DFT algorithm can be used to calculate the inverse DFT, as well [18].

Figure 2. Flow diagram of the signal processing steps

D. Low Pass Filtering The frequency response of the low pass Butterworth Filter (LPBF) approximation function is also often referred to as “maximally flat” (no ripples) response because the pass band is designed to have a frequency response which is as flat as

mathematically possible until the cut-off frequency at -3dB. The generalised equation representing nth order LPBF, the frequency response is given as, ω  1+ ε 2   ω p   

2n

(22)

Where n represents the filter order, ω is equal to 2πƒ and ε is the maximum pass band gain, Amax. If Amax is defined at a frequency equal to the cut-off -3dB corner point fc, ε will then be equal to 1.

E. Statistical Feature Calculation In this paper, our targeting features are mean, standard deviation and effect size or ES. Mean and standard deviation are very common features and therefore, no need to describe it by elaborative mathematical explanation. We will discuss about the ES. The most favourite way to calculate ES is Cohen’s d method. If, x1 and x2 are the mean values of data vector 1 and 2, respectively, then the effect size between x1 and x2 can be evaluated by Cohen’s d method as,

d=

x1 − x 2

(23)

( N 1 − 1) S 12 + ( N 2 − 1) S 22 N1 + N 2

Here, N1 and N2 are the number of data in each vector and S1 and S2 are standard deviation of the data set, respectively. IV.

DATA COLLECTION

The sample data were measured by OXYMON MKIII (24 channels, sampling frequency 9.75 Hz). In this behavior protocol, the following repeating tasks are performed gradually: 1. An initial 12s was for signal equilibrium 2. A 21s period of finger tapping task 3. Then 30s period of rest By repeating the process 2 and 3 it total 696 samples are taken from a subject using 3 types of NIR source wavelength having 860, 857, and 765. Export sample rate of the analyzing data is 2 Hz. The 24 channels of OXYMON MKIII provide the result into three kinds of measurements (HbO2, Hb, and difference between HbO2 and Hb). The channel number and their corresponding hemodynamic action measurement profile is given in Table I. TABLE I. Category 1 2 3

CHANNEL NUMBER AND CORRESPONIDNG ACTION PROFILE OF OXYMON MKIII Data Set No. 1, 4, 7, 10, 13, 16, 19, 22 2, 5, 8, 11, 14, 17, 20, 23 3, 6, 9, 12, 15, 18, 21, 24

Action Oxidized Hemoglobin (HbO2) Deoxidized Hemoglobin (Hb) Difference between HbO2 & Hb

1.2

Signal Amplitude

1

H ( jω ) =

Savitzky-Golay Filter Original Signal Moving Average Filter Proposed Scheme

1.5

0.9

0.6

0.3

0.0

0

20

40

60

80

100

No of Sample

Figure 3. Comparative filtering effects from the original fNIR signal having 100 samples of one channel.

V. RESULTS AND DISCUSSION In this section, at first we would like to present a typical fNIR data of HbO2 of one channel having 100 samples. In addition with that the filtering effect of Savitzky-Golay, Moving Average filter and our proposed scheme are also presented by Fig. 3. To filter the signal by Savitzky-Golay method 5th order polynomial is considered. In addition with that 3 point moving average method is considered which means no polynomial coefficient is used here. From this figure, it is observed that without proper processing, the original fNIR signal is quite complex to understand. Furthermore, it is also found in Fig. 3 that by the proposed scheme, the fNIR signal is more smooth than the others. Since, it is discussed that high degree of variance can deceive us to acquire proper ES estimation, the proposed scheme has a well ability to estimate the ES effectively. Therefore, in this stage, it is necessary to observe the important features of different schemes to estimate the appropriate ES. The numerical value of those features according to the proposed scheme, Savitzky-Golay, Moving Average filter and original signal are compared in Table II & Table III. From Table II, it is observed that the mean values of Moving Average filtering are decreased significantly compared to the original signal where in case of Savitzky-Golay filtering and proposed scheme, the mean values are remained almost same. This result proves that in the purpose of ES improvement Moving Average filter has sufficient limitation. TABLE II. COMPARATIVE MEAN FEATURES OF HBO2 OF ORIGINAL AND RECONSTRUCTED FNIR SIGNALS IN RESTING CONDITION Channel No. 1 2 3 4 5 6 7 8

Original 0.748918 0.517309 0.66847 0.466013 0.905501 1.12764 0.857821 0.821385

S-Golay Filter 0.747172 0.513642 0.671957 0.463438 0.900715 1.126259 0.860461 0.824054

Moving Average Filter 0.71477615 0.48623462 0.65232239 0.45738758 0.87642941 1.08320707 0.83448187 0.801734

Proposed Scheme 0.748099 0.518156 0.659998 0.462523 0.904659 1.131561 0.859384 0.820953

TABLE III. COMPARATIVE STUDY OF STANDARD DEVIATION OF HBO2 OF ORIGINAL AND RECONSTRUCTED FNIR SIGNALS IN RESTING CONDITION

0.571172 0.351812 0.498581 0.40727 0.658747 0.74266 0.727909 0.550317

S-Golay Filter 0.20432 0.205856 0.235744 0.184477 0.277469 0.219382 0.275825 0.20889

Moving Average Filter 0.22392418 0.17093135 0.25780467 0.14607798 0.3418201 0.30401897 0.29389449 0.22082511

Proposed Scheme 0.37187 0.286055 0.33205 0.283349 0.529237 0.443743 0.418837 0.401191

In case of standard deviation of different filtering scheme from Table III, it is observed that the least standard deviation is occurred in case of Savitzky-Golay filtering. Even Moving average filter also shows least standard deviation than our proposed scheme in some case with compared to the original signal. This result is slightly confusing because if the standard deviation of Savitzky-Golay and Moving Average are lower than the proposed scheme then why we are proposing the new scheme. We can discuss this factor mathematically but that will increase complexity rather than to understand it clearly. Therefore, we will present some interesting results graphically to understand the case clearly. In Fig. 4 we have represented a comparative study among Savitzky-Golay filtering, Moving Average filtering and proposed scheme effect of a part of fNIR signal of single channel regarding an event related task (finger tapping). In this figure, from the selected portion of the filtered signal, we can observe that both the Savitzky-Golay and Moving Average filter have a number of unwanted spikes those are often occurred in event related task oriented fNIR signal. These spikes can be a cause of CNR pattern mismatch but if the reconstructed signal having mismatched CNR pattern, then that signal filtering technique cannot be an appropriate ES estimating procedure because it often reduces the classifying efficiency. Furthermore, CNR is actually calculated [10] as, CNR =

mean(task ) − mean(rest ) var(task ) + var(rest )

0

5

1.0

0

1

0

0

1

5

0

2

0

0

2

5

0

3

0

0

3

5

0

4

0

0

Savitzky-Golay Moving Average Proposed Sceme

0.8

Signal Amplitude

Original

0.6

0.4

0.2 0

15

30

45

60

No of Sample

Figure 4. Comparison of spikes after filter in an event related task (Finger Tapping) from fNIR signal Original CNR Savitzky-Golay CNR M oving Average CNR Proposed M odel CNR

0.6

CNR Magnitude

Channel No. 1 2 3 4 5 6 7 8

0.4

0.2

0.0 1

2

3

4

5

6

7

8

N o of Channels

Figure 5. Comparison of CNR pattern of different schemes

(24)

From (24), we find that the variance and mean of signal in case of resting condition and subject engaged in some physical task are the main parameter to calculate the CNR. From the Fig. 4 it is clearly identifiable that the variance is least in case of proposed scheme than the Savitzky-Golay and Moving Average filtering techniques. Eventually, it is necessary to get proposer information about the CNR pattern profile of the filters to take decision about the effectiveness of the filters.

According to aforesaid discussion the limitations of Savitzky-Golay and Moving Average filters are revealed. Due to those limitations to estimate the proper ES between resting condition and finger tapping task, the proposed scheme is used to overcome their limitations. By the proposed scheme, we have calculated the required features to find the ES from 3 subjects according to the Cohen’s d method. The results are given by Table IV. From the result of % improvement in Table IV, we get that our scheme can improve the ES 50% in average and up to 90%.

Therefore, a comparative pattern of CNR profile of finger tapping task and resting condition among Savitzky-Golay, Moving Average, and proposed scheme are presented with compared to the original fNIR signal in Fig. 5. From this figure, one can easily understand the matching pattern and non-matching pattern compared to the original signal. In Fig. 5, it is very clear that only our proposed scheme follows the original signal's CNR pattern but Savitzky-Golay does not follow the CNR pattern in case of all channels (see selected area).

According to the Cohen’s d method we calculated the ES between the HbO2 of resting condition and finger tapping task for every channel, individually. From the result, we can observe a wide variation between the ES of original and reconstructed signal. This result helps us to realise the underestimation of ES measurement from noisy fNIR signal. Therefore, this underestimation of ES may be the acute cause for the failure of event detection. The proposed scheme solved this underestimation problem and enhance the value of ES for correctly detecting the events of interest.

TABLE IV. NUMERICAL RESULTS OF ABSOLUTE % ES IMPROVEMENT REGARDING THE PROPOSED SIGNAL PROCESSING SCHEME Ch. No 1

2

3

4

5

6

7

8

Subjects 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

ES (Original Signal)

ES (Proposed Scheme)

-0.1078048 -0.0952144 -0.1012806 0.09137117 0.07165248 0.11265401 0.15195534 0.18265418 0.20010456 0.40859871 0.38514672 0.43256102 -0.265527 -0.241598 -0.2847154 -0.2430526 -0.2147590 -0.1985421 0.33626536 0.306253602 0.32325610 0.25800122 0.23154602 0.26512495

-0.15489018 -0.13287405 -0.15110210 0.110897521 0.100589102 0.148471024 0.239015451 0.314625101 0.48700142 0.52756732 0.489512012 0.684792078 -0.36894844 -0.32471698 -0.41208588 -0.35900502 -0.33628944 -0.28978945 0.588457375 0.5109876422 0.6014018099 0.311904397 0.314568475 0.4179856021

Average absolute % ES improvement 44.1401805

31.1829534

90.9728301

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38.1347963

VI. CONCLUSIONS In this paper, with proper explanation of the limitation of Savitzky-Golay and Moving Average filter, we have proposed a simple and efficient fNIR signal processing scheme that can improve the numerical value of the ES correctly. Significant difference took place in reconstructed signal with compared to the original signal for applying the proposed scheme. Reconstructed signal is of reduced noise and consequently standard deviation of the original signal is abridged significantly remaining the mean value unchanged. Due to this impact, the reconstructed effect size of interest is transpired compared to the original noisy fNIR signal. Numerically, this improvement arises up to 90%. Therefore, this scheme can be used for processing the noisy fNIR signal for investigating the effect size of event related hemodynamic measurement. In addition, our proposed scheme also maintains the equivalent CNR pattern with original fNIR signal. The proposed scheme is only beneficiary for ES calculation. We cannot assure about the proposed scheme for other features, those are not directly related to ES. This issue can be considered as the limitation of our proposed scheme. In our feature work, we will find the effectiveness of the proposed scheme for the other features.

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ACKNOWLEDGMENT This work was supported by Higher Education Quality Enhancement Project (HEQEP), UGC, Bangladesh under Subproject “Postgraduate Research in BME”, CP#3472, KUET, Bangladesh.

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