ECG signals reconstruction in subbands for noise suppression

BBE 193 1–13 biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Available online at www.sciencedirect.com

ScienceDirect journal homepage: www.elsevier.com/locate/bbe 1 2 3

Original Research Article

ECG signals reconstruction in subbands for noise suppression

4 5

6

Q1

7

Marian Kotas *, Tomasz Moroń Institute of Electronics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland

article info

abstract

Article history:

In this study, we propose a combination of two methods for ECG noise suppression. The first

Received 5 August 2016

one is the robust principal component analysis, applied to QRS complexes reconstruction.

Received in revised form

The second is the method of weighted averaging of nonlinearly aligned signal cycles. The

2 March 2017

novelty of the approach consists in the way these methods are combined. First, a processed

Accepted 20 March 2017

ECG signal is decomposed into three spectral subbands, of high, medium and low frequency.

Available online xxx

Then both methods are applied in such a way that their operation is prevented from the

Keywords:

frequency subbands added. This makes the method more immune to low frequency artifacts

ECG reconstruction

that can be caused by electrodes motion. Dynamic time-warping is performed on the

most common unfavorable factors. RPCA reconstructs QRS complexes in a medium and high

Noise suppression

medium frequency subband which again prevents the procedure from the unfavorable

QT interval

influence of electrode motion artifacts. After the warping paths have been determined, the weighted addition of nonlinearly aligned signal cycles is executed, separately in the three subbands, with optimal weights estimated in each subband. Finally, by the appropriate addition of the obtained signals, the whole spectrum ECG is reconstructed. In the experimental section, the method was investigated with the use of real and artificially generated signals. In both cases, it allowed for effective suppression of noise, preserving important features of the processed signals. When it was applied to ECG enhancement prior to determination of the QT interval, the measurements appeared to be remarkably immune to different types of noise. © 2017 Nalecz Institute of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences. Published by Elsevier B.V. All rights reserved.

10 8 119 12 13

1.

14 15 16

Introduction

ECG signal is a source of compound diagnostic information. Unfortunately, it is often embedded in high energy noise, making its analysis and interpretation rather cumbersome. Q2

Therefore, during decades, many different methods of ECG noise suppression have been proposed [3,13,23,26,28,30,32]. With the increase of computational power of modern computers and the development of more and more sophisticated methods, new approaches to its diagnostic interpretation are continually being developed. At the end of the 20th

* Corresponding author at: Institute of Electronics, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland. E-mail addresses: [email protected] (M. Kotas), [email protected] (T. Moroń). http://dx.doi.org/10.1016/j.bbe.2017.03.002 0208-5216/© 2017 Nalecz Institute of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences. Published by Elsevier B.V. All rights reserved. Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

17 18 19 20 21 22

BBE 193 1–13

2 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

century a great interest of the medical cardiological community was turned to the significance of the QT interval measurements [24]. The QT interval reflects the time between depolarization and repolarization of the heart ventricles. It is an important electrocardiographic parameter, often used to quantify the duration of ventricular repolarization [24], particularly its spatial dispersion [2] and time (beat-to-beat) variability [6]. Different measures of these types of variability were observed and found to be able to indicate different health disorders [2,6,24]. However, the low amplitude of the T wave, whose maximum or slope is considered as the right limit of the QT interval, makes measurements of this interval rather difficult and highly prone to errors. Thus after the early fascination, more and more doubts were formulated concerning the diagnostic significance of the QT interval variability. In [15] it was even claimed that instead of the true QT variability, the variability of the measurement errors is usually calculated and falsely regarded as an indicator of the patient's health status. Therefore a need appeared to measure the interval with greater accuracy and with sufficient immunity to noise. Thus new methods of ECG noise suppression had to be developed and their application prior to the measurements of this interval was the way to accomplish these requirements. Among others the method of principal component analysis was effectively applied to ECG noise suppression [16]. The method copes successfully with precise reconstruction of the time-aligned QRS complexes but is less effective in reconstruction of the lower amplitude T waves with varying positions within the successive linearly aligned ECG beats. To improve accomplishment of this operation, in [17] it was preceded by nonlinear instead of linear time alignment of the ECG beats. The technique of dynamic time warping was applied for this purpose. After nonlinear alignment, the considered T waves are better synchronized and their reconstruction is much improved. In noisy ECG recordings the classical PCA, which is not robust against outliers, performs poorly. Therefore it was replaced [18] by a robust, projection pursuit based version, and the developed method of ECG reconstruction appeared much more immune to noise. To obtain still better immunity to noise, in [20] averaging of nonlinearly aligned ECG cycles was applied instead of the analysis of their principal components. This method enables significant enhancement of the low amplitude ECG waves but is slightly less accurate in reconstruction of QRS complexes (which are of more variable shape). Therefore the goal of this study is to apply a combination of principal component analysis and of averaging of timewarped ECG cycles for ECG noise suppression. We propose to decompose the signal into subbands and then to apply the mentioned methods in the particularly selected time segments of the respective formed signal subbands. Finally, by a suitable addition of the results produced, the ECG signal is reconstructed. This approach is aimed to make the ECG reconstruction more immune to a very troublesome low frequency noise, so-called electrode motion artifacts which can not simply be filtered out to avoid suppression of diagnostically important features of the ECG (the idea of distinctive processing of ECG in subbands is not knew; e.g. in [4] it was used for prediction based compression of vectocardiograms).

The rest of the paper is organized as follows. Section 2 presents an outline of the method of weighted averaging of nonlinearly aligned cycles (WANAC) whose modifications led to the development of the proposed method of ECG signal reconstruction in subbands. This new method is described in Section 3. Its ability to suppress noise while preserving with high fidelity details of the ventricular complexes morphology is presented visually and quantitatively in Section 4. Finally, conclusions are drawn in Section 5.

83 84 85 86 87 88 89 90 91 92 93

2. An outline of the weighted averaging of nonlinearly aligned cycles The method is aimed to exploit the ECG signals approximate repeatability to suppress noise. It can be regarded as an extension of the widely applied methods of ECG signals weighted averaging [23,26]. However, whereas traditional weighted averaging assumes that the desired component of the ECG cycles under averaging is exactly repeatable, the WANAC method slackens this requirement allowing for some deformation of the time axis of this desired component. Before averaging these deformations should be compensated by the appropriate nonlinear alignment of the signal cycles. To this end, the method uses the technique of dynamic time warping.

94 95 96 97 98 99 100 101 102 103 104

2.1.

105

Dynamic time warping

Let us consider two time signals of possibly different length: x (n), n = 1, 2, . . ., Nx and y(n), n = 1, 2, . . ., Ny. To perform their nonlinear alignment, we calculate the matrix of costs D ¼ i¼N ;j¼Nx whose classical definition is as follows ½di;j i;j¼1y

106 107 108 109 110

di;j ¼ ðyðiÞxð jÞÞ2 :

(1)

Each di,j corresponds to the alignment of y(i) and x(j). The warping path P of the signals y(n) and x(n) consists of the ordered pairs of time indices: P ¼ fðik ; jk Þjk ¼ 1; 2; . . .; Kg, which indicate the elements of the successive aligned pairs: y(ik), x(jk). In the classical approach, applied e.g. in [9,29], the warping path is searched for that minimizes the total cost of the alignment

Q¼

K X dik ;jk :

112 111 113 114 115 116 117 118 119

(2)

k¼1

preserving the boundary conditions: i1 = j1 = 1, iK = Ny, jK = Nx, which force the warping path to start with the pair x(1), y(1) and to end with x(Nx), y(Ny); the continuity conditions: ik+1  ik  1, jk+1  jk  1, which prevent the elements of both signals from being omitted in the warping path; and the monotonicity conditions: ik+1  ik, jk+1  jk, which force the elements of both signals to occur in the warping path in a non-decreasing order. This warping path can be found with the use of dynamic programming [5].

2.2.

The minimized cost function

Let's denote the L signal cycles to be processed by WANAC as {xl(n) | n = 1, 2, . . ., Nl, l = 1, 2, . . ., L} where Nl is the length of the

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

120 121 122 123 124 125 126 127 128 129 130 131 132

BBE 193 1–13 biocybernetics and biomedical engineering xxx (2017) xxx–xxx

133 134 135 136 137 138 139 140 141

lth cycle. Applying a vector notation, let w ¼ ½w1 ; w2 ; . . .; wL  represent a vector of real non-negative weights associated with the respective L signal cycles and let t = [t(1), t(2), . . ., t(Nt)] denote the constructed template (which is a weighted average of the nonlinearly aligned signal cycles). The set of warping paths relating t with the respective cycles is denoted as ðlÞ ðlÞ P ¼ fðik ; jk Þj k ¼ 1; 2; . . .; Kl ; l ¼ 1; 2; . . .; Lg. To derive formulas for calculating the weights and the template, the following cost function was defined [20]

142 J ðw; t; PÞ ¼

L X ðwl Þm Gðt; xl Þ;

where m is a weighting exponent (like in [20], in this study m = 2); G(, ) is a cumulative cost of nonlinear alignment

146 Gðt; xl Þ ¼

Kl Kl X X ðlÞ ðlÞ 2 ðtðik Þxl ðjk ÞÞ ¼ dik ;jk : k¼1

(4)

k¼1

149 147 148 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168

Cost function (3) can be interpreted as a measure of the total dissimilarity between t and the considered set of cycles. Determination of an optimal weighting vector w, an optimal ‘‘warped mean’’ t and an optimal set of warping paths P can be achieved by minimization of this cost function. To this aim, a kind of alternated optimization was applied [20], based on the following reasoning. For a fixed template t and any fixed vector of weights w, the criterion function reaches its minimum when for each cycle the cumulative cost of its alignment with t is minimal. These minimal cumulative costs can be calculated separately for all cycles with the use of dynamic time warping. This way we obtain L warping paths, i.e. set P and L cumulative costs of alignment G(t, xl), l = 1, 2, . . ., L. Having the cumulative costs computed and assuming they are fixed, we can proceed with minimizing (3) to get a formula for optimal weights w. And then, having the warping paths and the weights determined and assuming they are fixed, we can find formula to update the template. Using the formulas derived [20], we perform these calculations repeatedly until the specified criterion has been satisfied.

169

2.3.

170 171 172 173 174 175 176 177 178 179 180 181

In [20] it was shown that nonlinear alignment of the cycles with the alignment costs defined by (1) has a very unfavorable influence on the results of averaging. The ability to repeat each sample of the time-warped signals the required number of times to diminish the squared differences between them results in alignment of not only the desired but also of the noise components of the signals. Thus after this operation the noise components of the cycles under averaging are more correlated than before and as a result they are less suppressed by averaging. To prevent the described unfavorable effects of time warping, the following definition of the alignment costs was proposed [20]

182 183 184 185 186

which means that they are composed of 2v þ 1 successive signal samples (with the nth sample being the central one).

187 188 189

By the proposed redefinition of the alignment costs, it is assured that longer signal intervals are matched to each other instead of single samples of the warped signals. More remarks, concerning application of this definition, can be found in [20].

190 191 192 193

3. ECG signal reconstruction in subbands (ESRS)

194 195

(3)

l¼1

143 144 145

3

Modification of the alignment costs

d0i;j ¼ kxðiÞ yð jÞ k

196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223

3.1.

224

QRS detection

As we can see in the diagram, a very important role is played by a QRS detector. The detection is accomplished with the use of the method based on matched filtering. It is very immune to different types of noise and it produces a set of detection points {rl|l = 1, 2, . . ., L + 1} located at approximately the same position within the respective L + 1 complexes detected. The detection results are used in almost all blocks of the diagram. They allow to select the proper signal segments to be processed.

225 226 227 228 229 230 231 232 233

3.2.

234

Subband decomposition

(5)

where kk denotes the Euclidean norm, and vectors x(n) and y(n) are defined as xðnÞ ¼ ½xðnvÞ; xðnv þ 1Þ; . . .; xðnÞ; . . .; xðn þ vÞT

Substitution of the classical definition of alignment costs (1) with the vector norm based one (5) caused a very significant improvement of the averaging results [20]. However, experiments on ECG processing in different types of noise showed that it is extremely difficult to suppress the contaminations caused by electrodes motion. The frequency content of this type of noise can reach up to about 5 Hz, and as it overlaps the spectrum of the desired ECG, it can not be suppressed by linear high-pass filtering without disturbing the signal morphology. Unfortunately, as it is illustrated in Fig. 1, this type of noise can completely spoil determination of warping paths. Therefore we propose to decompose the processed signal into 3 subbands (of high, medium and low frequency) and to use the medium frequency one (with electrode motion artifacts significantly suppressed) for determination of warping paths. Later the weighted averaging of nonlinearly aligned cycles can be executed in the respective subbands separately with the use of the warping paths determined. Finally, the results obtained in the respective subbands can be added to form the full spectrum signal. Using this approach, the WANAC method copes well with the enhancement of low amplitude parts of ECG beats; however, it is less effective within QRS complexes which are of more variable shape. Therefore, to enhance this part of the signals, principal component analysis will be applied (its robust, projection pursuit based version [18]). Block diagram of the proposed ESRS method is presented in Fig. 2. Details of its operation are as follows.

(6)

It is generally regarded, following the experiments presented in [1], that using high-pass filters of linear phase response, we can apply the cut-off frequency of about 0.8 Hz for suppression of low frequency baseline wander, without disturbing the morphology of the ECG signals. When an off-line analysis of

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

235 236 237 238 239

BBE 193 1–13

4

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Fig. 1 – Illustration of the influence of low frequency noise on determination of warping paths. For two high quality ECG signals, the smooth almost diagonal warping path is produced. When, however, a low frequency baseline wander is added to one of them, the warping path is much distorted.

240 241 242 243 244

the signals is to be performed, linearity of the phase response can simply be obtained by double filtering: in forward and backward direction. In fact, using this approach we obtain filters with zero phase response. Using a few such filters and adding or subtracting their output signals, we can form filters

of different magnitude response but always with linear phase response. Such filters are favorable because they do not cause phase distortions of the processed signals [14]. This approach (the double filtering) allowed us to apply a low-pass Butterworth filter (with nonlinear phase response) whose flat,

Fig. 2 – Block diagram of the proposed method of ECG signal reconstruction in subbands. RPCA stands for robust principal component analysis, WANAC denotes weighted averaging of nonlinearly aligned cycles. Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

245 246 247 248 249

BBE 193 1–13

5

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

289

alignment costs and minimizing ðrÞ

gl

0

Kl X ¼ G0 ðm tðr1Þ ; m xl Þ ¼ d0ik ;jk ;

l ¼ 1; 2; . . .; L:

(8)

k¼1

Fig. 3 – Bidirectional Butterworth filters based signals decomposition into frequency subbands: Hfi(z) = BFfi(z) BFfi(zS1) (where BFfi(z) denotes a low-pass Butterworth filter with the cut-off frequency equal to fi); NF is a notchfilter for powerline interference suppression, Hf1 with f1 = 0.5 Hz is used to extract and subtract the baseline wander, Hf2 and Hf3 form the subbands of low (lx(n)), medium (mx(n)) and high (hx(n)) frequency.

250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265

monotonically descending magnitude response is very favorable, to ECG signals subband decomposition (see the block diagram in Fig. 3). The low frequency baseline wander is captured in the lowest branch of the diagram. By subtracting this signal from the output of the upper branch, we form a low frequency subband lx(n) (with the baseline wander suppressed but electrode motion artifacts preserved). Similarly we form medium (mx(n)) and high (hx(n)) frequency subbands. By adding the three subbands, we can recover the whole frequency spectrum (with only the lowest frequency baseline wander suppressed). Our primary goal is to form the mx(n) signal that contains the medium frequency components of the ECG preserved but whose electrode motion artifacts are suppressed as much as possible.

266

3.3.

267 268 269 270 271

This block operates on the medium frequency subband signal mx(n). The constructed set of detection points {rl|l = 1, 2, . . ., L + 1}, is used to cut this input signal into L time segments beginning before the detected QRS complexes and ending before the successive ones

3. Use the obtained cumulative alignment costs to calculate elements of the weighting vector w(r) using the formula 0 ! 11 ðrÞ0 1 L X gl ðrÞ A ; s ¼ 1; 2; . . .; L: (9) ws ¼ @ ðrÞ0 l¼1 gs

(10) ðlÞ

ðlÞ

where Gl ðnÞ ¼ fkjik ¼ ng contains indices of all pairs (ik ; jk ) ðlÞ in the lth warping path that satisfy condition ik ¼ n, which means that they relate the nth sample of the template (mt(r1)(n) which is to be updated) with the corresponding ðlÞ ðnÞ samples of the lth cycle (fm xl ðjk Þj k 2 Gl g). 5. If kmt(r)  mt(r1) k/k mt(r1) k > j and r < rmax then r go to (2) else stop.

r + 1 and

Remark. The initial form of a template mt(0) is created in the following way. We group the successive cycles in pairs (if the number of cycles is odd, we neglect the last one). Then we perform nonlinear alignment and averaging of each pair and we choose that pair for which the total cost of the alignment was lowest. The 'warped mean' of the chosen pair of cycles is used as the initial template.

299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318

3.4.

319

WANAC in subbands

Initial WANAC

272

After the operation of initial WANAC has been completed, we can proceed to execute WANAC in the respective signal subbands sx(n) (where left subscript s specifying the subband belongs to the set {h, m, l}). First, like in the block of initial WANAC, the processed signal is cut into segments

¼ m xðrl b þ nÞ;

1nNl ; l ¼ 1; 2; . . .; L;

(7)

273 274 275 276

where Nl = rl+1  rl; b denotes the assumed shift of the segments onsets with respect to the determined detection points (for the sampling frequency of 250 Hz, we used b = 10).

277 278 279 280

Finally, the selected L signal segments undergo time warping and weighted averaging according to the following algorithm.

281 282

Algorithm 1.

283 284 285 286

1. Initialize m t(0). Fix j > 0, rmax  1. Set the iteration index r = 1. 2. Align nonlinearly mt(r1) with respect to the individual signal cycles using the vector norm based definition (5) of the

320 321 322 323 324 325

s xl ðnÞ m xl ðnÞ

295 296 298 297

4. Update elements of template mt(r1) using PL ðrÞ 2 P ðlÞ l¼1 ðwl Þ k 2 Gl ðnÞ m xl ðjk Þ ðrÞ ; n ¼ 1; 2; . . .; Nt ; m t ðnÞ ¼ PL ðrÞ 2 l¼1 ðwl Þ jGl ðnÞj ðlÞ

290 291 293 292 294

¼

s xðrl b

þ nÞ;

1nNl ; l ¼ 1; 2; . . .; L;

(11)

Now, preserving the paths unchanged (see that the operation of DTW is performed in the block of Initial WANAC, only), we can process the signal according to the following algorithm.

327 326 328 329 330 331 332

Algorithm 2.

333 334

1. Apply the paths and the weights determined in the block of initial WANAC to construct the initial form of the template (0) (using (10) with left subscript m replaced by s and with st r = 0). Fix j > 0, rmax  1. Set the iteration index r = 1. 2. Preserving the previously determined warping paths find the costs of aligning template st(r1) with respect to the individual signal cycles, using the classical definition (1) of

335 336 337 338 339 340 341

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

BBE 193 1–13

6 344

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

the alignment costs ðrÞ

gl

Kl X ðlÞ ðlÞ 2 ¼ ðm tðr1Þ ðik Þm xl ðjk ÞÞ k¼1

Kl X ¼ dik ;jk ;

(12) l ¼ 1; 2; . . .; L:

k¼1

345 346 348 347 349 351 350 352 354 353 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374

3. Calculate elements of the weighting vector w(r) using (9) ðrÞ0 ðrÞ with gl replaced by gl . 4. Update elements of template st(r) using (10) with left subscript m replaced by s. 5. If kst(r)  st(r1) k/k st(r1) k > j and r < rmax then r r + 1 and go to (2) else stop.

After the final form of the template has been obtained, we can reconstruct the individual cycles of the subband processed and use them to replace the original noisy ones [20]. The obtained signals are denoted as as x0 ðnÞ where left subscript s stands for the subband processed and superscript a denotes the signal enhanced by averaging (using WANAC). Remark. In Algorithm 2 we do not execute the operation of dynamic time-warping, we only exploit the warping paths determined previously with the use of Algorithm 1 on the basis of the mx(n) subband. Therefore we do not have to apply the modified definition of the alignment costs (5) to prevent the discussed unfavorable effects of nonlinear alignment. Thus, in the second stage of Algorithm 2, we apply the classical definition (1) to calculate the total costs of alignment.

375 376

3.5.

377 378 379 380 381 382 383 384 385 386 387 388 389 390 391

After detection of QRS complexes, the obtained set of detection points is used to select signal segments containing successive complexes and to store them in an auxiliary matrix XgxL = [x1, x2, . . ., xL] where each column vector xi contains an individual complex xTi ¼ ½xðri g=2 þ 1Þ; . . .; xðri þ g=2Þ; g is the assumed width of the segments (it is an even number). As it was proved e.g. in [18], projection of the respective signal segments (column vectors which can be regarded as points in a g-dimensional space) on specifically constructed robust principal subspaces can cause significant suppression of noise with limited reduction of the variability of the desired component shape. Projection onto the q-dimensional robust principal subspace and reconstruction of the observed gdimensional variables can be accomplished according to the following formula

392 393 394 395 396 397

RPCA for QRS complexes reconstruction

^ nÞ þ m ^ n: x0i ¼ EQ n EQ>n ðxi m

(13)

where EQ n ¼ ½eQ n ;1 ; eQ n ;2 ; . . .; eQ n ;q ; eQ n ;k is the kth principal eigenvector determined by maximizing the robust measure of ^ n is the so-called L1-medidispersion (the Qn estimator [31]); m an [11], a robust estimator of location, defined as the point that

minimizes the sum of distances to all observations (column vectors of X). Details of the applied algorithm of robust principal subspace construction can be found in [18] (it is a modification of the projection pursuit based algorithm proposed in [7]). After reconstruction of QRS complexes gathered in matrix X, we use them to replace the original ones in the processed signal. This output signal, containing the reconp structed QRS complexes is denoted as hm x0 ðnÞ where the left superscript p denotes the signal enhanced by robust PCA and the left subscripts hm denote the subbands processed. And indeed, in Fig. 2 we can notice that the described RPCA is applied to the sum of high and medium frequency subbands. Such approach allowed us to limit the very disadvantageous influence of low frequency artifacts on RPCA performance.

398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413

3.6.

414

Weighting function

Using WANAC we obtain the three spectral components enhanced: ah x0 ðnÞ, am x0 ðnÞ and al x0 ðnÞ (where left superscript a stands for averaging). On the other hand, using RPCA we enhance the sum of high and medium frequency subbands within the segments containing QRS complexes. To combine all these signals without disturbing their continuity, we p multiply hm x0 ðnÞ by a weighting function g(n), which is series of Gaussian windows (one of them presented in Fig. 4) and a 0 a 0 h x ðnÞ, m x ðnÞ signals by 1-g(n). Finally, we reconstruct the ECG as the sum

415 416 417 418 419 420 421 422 423 424 425

0

x ðnÞ

p ¼ hm

0

x ðnÞgðnÞ þ fah x0 ðnÞ f1gðnÞg þal x0 ðnÞ;

þam

0

x ðnÞg

(14)

which is illustrated in Fig. 4.

427 426

This way the lowest frequency subband is enhanced by the weighted averaging of nonlinearly aligned cycles only (using Algorithm 2), whereas in the remaining two subbands a transition from the PCA enhanced signal to the signal enhanced by WANAC (and vice verse) takes place on the borders of QRS complexes.

428 429 430 431 432 433 434

Remark. Eq. (14) is equivalent to the following one

435 436

x0 ðnÞ

¼ ah x0 ðnÞ þam x0 ðnÞ þal x0 ðnÞ þ gðnÞ p fhm x0 ðnÞah x0 ðnÞam x0 ðnÞg;

(15)

which can help us interpret the ESRS operation. By adding the three signal subbands enhanced by WANAC (ah x0 ðnÞ þam x0 ðnÞ þal x0 ðnÞ), we could recover the whole spectrum signal; however, its variability within QRS complexes would almost completely be reduced. Therefore we use RPCA to analyze and enhance the added medium and high frequency subbands (in which we are able to distinguish variability of the desired QRS complexes from noise), and we replace the complexes enhanced by WANAC with the ones enhanced by RPCA simply by adding the difference between them p (fhm x0 ðnÞah x0 ðnÞam x0 ðnÞg) multiplied by the properly located Gaussian windows (g(n)).

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

437 438 439 440 441 442 443 444 445 446 447 448 449 450 451

BBE 193 1–13

7

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Fig. 4 – Illustration of ECG signal reconstruction in subbands. From the left: the signals to be added (results of averaging in subbands and of RPCA), the weighting function by which they are multiplied and the results that undergo the final addition.

452

4.

Numerical experiments and discussion

453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473

The experiments were based on real signals from the Physionet database [8] and on artificially constructed ones generated with the use of the procedure described in [25]. In the first experiment, 6 ECG signals from the QT database [21] of the Physionet resources were used. Since the signals were aimed to simulate the desired ECG, they had to be of very high SNR. Therefore we had carefully performed [16] visual inspection of the beginning 80 s segments of the first channel of many records and this way we selected these signals (of the assumed length of 80 s), assuring their very high quality. These 'desired' signals were additively contaminated by the noise records from the MIT-BIH database [10]. Like in [18,20,27], the muscle artifacts record 'ma.dat' was used for this purpose and additionally the 'em.dat' record, with the noise produced by electrodes motion. Both of these records, originally sampled with the frequency of 360 Hz, were resampled with the frequency of 250 Hz, applied in the QT database. For the purpose of the experiments, these noise components were multiplied by coefficients assuring the required SNR of the simulated noisy ECG signals (the assumed SNR concerned the signals with baseline wander suppressed).

474

4.1.

475 476 477 478 479 480

The spectral subband decomposition (according to the diagram from Fig. 3) was based on the 6th order low-pass Butterworth filters. Whereas the f1 parameter should be chosen so as the requirements specified in [1] were satisfied (we set f1 = 0.5 Hz and it is visible in Fig. 5 that this value corresponds to the cut-off frequency of about 0.7 Hz of the

high-pass filter used to suppress the baseline wander), the f2 and f3 frequencies should assure the proper action of the method proposed. To chose the proper values of these frequencies and of parameters v and Q, we investigated their influence on the noise reduction factor (NRF), defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðxðnÞsðnÞÞ2 ; NRF ¼ P n 2 0 n ðx ðnÞsðnÞÞ

481 482 483 484 485 486

(16)

where x(n), s(n) and x0 (n) denote the processed signal, its desired component and the enhanced signal, respectively.

487 488 489

To assure the proper action of the proposed method in various noise conditions, we selected the parameters for which the highest average NRF was achieved for all test signals corrupted by muscle noise of very different energy (SNR of 0, 10 and 20 dB). For f2 the following values were tested: 1, 1.5, 2, . . ., 6 [Hz]. For f3 we applied f3 = f2 + Df with Df 2 {2, 3, . . ., 20}. For v we applied the values 12, 25, 37 and 50, corresponding to the time intervals ð2v þ 1Þ=f s of about 0.1, 0.2, 0.3 and 0.4 s (for sampling frequency fs = 250 Hz). For dimension Q of the principal subspace, we applied the values 1, 2 and 3. The highest mean NRF of 3.90 was obtained for f2 = 3 Hz, f3 = 11 Hz, v ¼ 50 and Q = 1. The frequency responses of the filters, and the subbands created are presented in Fig. 5.

490 491 492 493 494 495 496 497 498 499 500 501 502

4.2.

503

Selection of the ESRS method parameters Qualitative results

An example of the ECG signal decomposition using the filters selected, is presented in Fig. 6. We can see that high energy electrode motion artifacts are removed from the two higher frequency subbands and are preserved mostly in lx(n). Since it is the mx(n) subband that is used for warping paths

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

504 505 506 507 508

BBE 193 1–13

8

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Fig. 5 – Frequency responses of the bidirectional low-pass Butterworth filters applied to ECG subband decomposition (the upper plot) and of the formed band-pass filters: G1(z) = Hf2(z) S Hf1(z), G2(z) = Hf3(z) S Hf2(z) and G3(z) = 1 S Hf3(z). Because of the double (bidirectional) filtering Hf1( f1) = Hf2( f2) = Hf3( f3) = 1/2. The cut-off frequencies of the created subbands are given in boxes.

509 510 511 512 513 514 515 516 517 518 519 520 521 522 523

determination, it is likely that by this decomposition, we can prevent possible severe distortions of the paths created. And, what should be emphasized, using this approach we do not loose the diagnostic information contained in the lowest frequency subband because in this subband we also perform averaging of nonlinearly aligned cycles. An example of ECG enhancement with the use of the proposed method is shown in Fig. 7. For reference the results obtained with the use of the method of Nonlinear State Space Projections (NSSP) [33] are also presented. NSSP was used as the reference because it is regarded as the very efficient tool for ECG noise filtering (it appeared the most effective among modern methods of ECG noise reduction investigated in [12]). Watching Fig. 7, we can see that the ESRS method was much more effective. Whereas NSSP was able to cause only a kind of

smoothing of the severe noise artifacts disturbing the ECG signal, with the ESRS method we achieved a significant suppression of these artifacts. Moreover, comparing the enhanced signal with the desired one, presented below, we can see that not only the noise was suppressed but primarily all important ECG waves were well reconstructed. For practical applications, it is of utmost importance to preserve the characteristic features of the signals processed. To investigate the method with this respect, we checked its influence on measurements of the QT interval, covering the time between depolarization and repolarization of the ventricles [24]. For determination of the interval limits we applied the method developed by Laguna et al. [22]. This method measures the slopes of ECG waves and searches for the time positions where the slopes decrease below certain thresholds. This way it

Fig. 6 – Results of ECG signal subband decomposition with the use of the selected filters. From the top: the noisy ECG signal (after baseline wander suppression) and its three spectral subbands. The arrows indicate significant electrode motion artifacts (in the uppermost subplot) and how they were retained in the lowest frequency subband. Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

524 525 526 527 528 529 530 531 532 533 534 535 536 537 538

BBE 193 1–13 biocybernetics and biomedical engineering xxx (2017) xxx–xxx

9

Fig. 7 – Results of ECG signal enhancement with the use of the proposed (ESRS) and the reference (NSSP) method. The arrows indicate a high energy electrode motion artifact and the results of its suppression. As we can see, NSSP slightly suppressed the wide band EMGnoise, but it preserved the low frequency artifact almost unchanged. The proposed ESRS method, on the contrary, was able to suppress both types of noise significantly.

539 540 541 542 543 544 545 546 547 548 549 550 551

determines the beginning of the QRS complex, which is the left limit of the QT, and the peak or the end of the T wave which are alternatively used as the right limit. Visual results of these operations, obtained for a noisy ECG of very low SNR (of 0 dB) are presented in Fig. 8. We can see that for so high level of EMG noise, the measurements are completely unreliable. Determination of the respective positions in such a noisy environment is rather a matter of luck, particularly within the T wave. However, although in the presented case the noise artifacts were comparable to the QRS complex, and the T wave was completely covered by them, after reconstruction with the use of the ESRS method a high quality signal emerged, allowing for high accuracy of the measurements. This means that the

method was able to suppress noise and reconstruct the desired ECG preserving well its waves and their slopes. The ESRS method ability to reconstruct the desired ECG so accurately in such demanding conditions results from the high immunity to noise of the warping paths determination on the filtered signal (of the medium frequency subband) and from the weighted addition of the aligned signal cycles (with proper weights established in the respective subbands, separately).

552 553 554 555 556 557 558 559

4.3.

560

Quantitative results

To confirm visual results, we applied ESRS, NSSP and a few other reference methods to enhance the selected signals from

Fig. 8 – Results of the QT limits determination in the desired ECG (QRSo, Tp and Te) and in the noisy and the enhanced signal (QRS0o , T0p and T0e ). As we can see in the second subplot, for so high level of EMG noise the positions can be found almost anywhere within the whole noisy signal segment presented. Using the ESRS method for ECG enhancement, we obtained reliable results with small errors only. Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

561 562

BBE 193 1–13

10

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Table 1 – The average NRFs obtained for different levels of EMG noise in the test on the QT database signals. Method

RPCA þ DBAc RPCA þ DBAm RPCA þ WANACm ESRS NSSP0.5

563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598

SNR 0 dB

10 dB

20 dB

1.35 3.73 4.95 6.68 2.25

1.33 3.15 3.60 3.7 2.48

1.07 1.32 1.32 1.32 1.64

the QT database. These additional reference methods can be regarded as the stages in the development of the ESRS method. They all combine RPCA with different methods of averaging of nonlinearly aligned cycles. In the method denoted as RPCAxDBAc, the Dtw Barry center Averaging [29], based on the classical alignment costs is used (the DBA method can shortly be described as the WANAC method with no weighting, i.e. equal weights associated with all the cycles). In RPCAxDBAm the modified alignment costs are applied, and in RPCAxWANACm the WANAC method based on the modified alignment costs is used. All these methods use no subband decomposition, and realize the transition between RPCA and the averaging similarly as in ESRS, with the use of the Gaussian weighting functions. The average NRFs obtained for each of the methods, tested on the signals from the QT database, in different levels of EMG noise, are gathered in Table 1. Since the NSSP method performance depends strongly on the applied radius of so called neighborhoods [33], we checked the following set of its values: e 2 {0.1, 0.2, 0.5, 1, 2, 5, 10}. For e  0.5 almost the same, best results were obtained (they are presented in the table). The noise reduction factors gathered in Table 1 confirm the visual results of ECG enhancement presented in Fig. 7. For low SNR of 0 dB, ESRS outperforms NSSP and the other reference methods. Although the signals selected to simulate the desired ECG are visually of very high quality, they can contain some low level of noise, not discerned during their visual inspection and selection. It is particularly difficult to discriminate the low frequency noise from the low amplitude (and frequency) ECG waves. The presence of such noise in the signals used to simulate the desired ECG limits the accuracy of the NRF estimation for low level of the noise added. It can be one of the reasons of rather low NRFs obtained for SNR = 20 dB. In order to overcome these limits of the experiment, we repeated it with the 'desired' ECGs from the QT database replaced by the artificially generated ones [25]. The results obtained are

Table 2 – The average NRFs obtained for different levels of EMG noise in the test on the artificially generated ECG signals. Method

RPCA þ DBAc RPCA þ DBAm RPCA þ WANACm ESRS NSSP0.5

SNR 0 dB

10 dB

20 dB

1.21 3.75 4.37 6.64 2.07

1.21 4.25 4.87 6.16 2.43

1.15 3.02 3.06 3.27 3.25

gathered in Table 2. And indeed, we can notice that for SNR = 0 dB the results from both tables are rather similar, but for SNR = 20 dB those in Table 2 are much better. We can also see that in this experiment, the ESRS method outstripped the other ones for all levels of noise added. Its superiority was particularly significant for the heavily corrupted signals (SNR = 0 dB). To investigate the methods ability to restore the important features of the processed noisy signals, we studied quantitatively their influence on the measurements of the QT interval limits. Accuracy of the QT interval measurements is a crucial factor that limits our ability to assess the interval variability in a series of consecutive heart beats. Since most of the measurement methods determine this interval as multitudes of sampling periods, there is a necessity to apply a sufficiently high sampling frequency of the analyzed signals to achieve a satisfactory precision of the measurements. In [34] it was shown that satisfactory results can also be achieved if this frequency is increased just before the measurements with the use of some resampling methods. We have applied a similar approach in this study. For the ESRS method it is as follows. First, the signals are decomposed into subbands in the original sampling frequency of 250 Hz (with the use of the filters whose frequency responses are presented in Fig. 5). Next, the three spectral components are resampled with the frequency of 1000 Hz, and then they are processed and reconstructed with the use of the proposed method (with Q = 1, and v ¼ 200 corresponding to a time interval of about 400 ms). For the methods with no subband decomposition, the signals are simply resampled before being processed. During the experiment, we performed measurements of the QRS complex onset: QRSo, the T wave peak: Tp; and the T wave end: Te on the desired ECG signals (from the QT database), on the noisy ones, and on the signals enhanced with the use of the proposed or the reference methods. We investigated the influence of EMG noise (record 'ma.dat') and of the electrode motion artifacts (record 'em.dat') on the precision of these measurements. For both types of noise, we separately calculated the root mean squared errors (RMSE) of the respective characteristic points measurements. Since even sporadic large errors can substantially increase such a quantity (RMSE), and because such large errors can rather easily be discriminated and rejected during real investigations, before calculating these RMS errors, we rejected the greatest error for each signal, each SNR and each method tested. The results are presented in Fig. 9. As we can see, a very high signal to noise ratio is required to determine the QT limits precisely: the measurements on the unfiltered signal are of too low accuracy even for relatively high SNR of 20 dB. Applying a combination of RPCA and of DBAc (based on the classical alignment costs) to ECG enhancement, we failed to improve the measurements: only a very slight decrease of RMS errors was obtained. When however the DBA method of averaging was based on the modified alignment costs, a significant improvement of the measurement accuracy was achieved. For moderate and high level of noise, for this method much lower RMS errors were obtained than for NSSP. As we have already mentioned, the DBA method is similar to the original WANAC method, with one exception only: it does not apply any weighting (its action could be described as that of

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658

BBE 193 1–13 biocybernetics and biomedical engineering xxx (2017) xxx–xxx

11

Fig. 9 – Illustration of the investigated methods influence on the precision of QRSo, Tp and Te determination in muscle noise environment (record ma.dat). The RMS errors are presented as functions of SNR.

659 660 661 662 663 664 665 666

WANAC with one simplification: using the same weights for all cycles under averaging). Thus, by applying the combination of RPCA and the WANAC method, we actually improve RPCAxDBAm by introducing one modification only: estimation and the use of the weights. And this modification results in significant improvement of the measurements accuracy for highly corrupted signals (with SNR equal to 0 or to 5 dB). However, applying all the modifications described (in the proposed ESRS

method), we still managed to improve these measurements visibly. Moreover, as we can see in Fig. 10, for the electrode motion artifacts the improvement was even more significant. Since this type of noise is of extremely high level in fetal ECG recordings extracted from the maternal abdominal signals [19], it seems that the method could effectively be applied even to the enhancement of such demanding signals. This, however, requires further investigations.

Fig. 10 – Illustration of the investigated methods influence on the precision of QRSo, Tp and Te determination in electrode motion artifacts environment (record em.dat). Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

667 668 669 670 671 672 673 674

BBE 193 1–13

12

675

676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712

5.

biocybernetics and biomedical engineering xxx (2017) xxx–xxx

Conclusions

We have proposed a new approach to ECG signals enhancement. It is based on signals decomposition into three subbands (of high, medium and low frequency) and specific processing of the respective of them. For noise suppression within the segments containing QRS complexes, we apply the method of projection pursuit based robust principal component analysis. To prevent RPCA from the unfavorable influence of the low frequency electrode motion artifacts, which can be of high energy and can dominate determination of the principal directions, we use the method to process only the two higher frequency subbands added. The second method used is the weighted averaging of nonlinearly aligned signal cycles. It is applied in the respective subbands separately; however, for determination of warping paths, the medium frequency subband is used, which again prevents the method from the unfavorable influence of low frequency artifacts. The results obtained with the use of the both methods are combined to reconstruct the original signals processed. The proposed method was adjusted by maximizing the degree of the noise suppression, as measured by the defined noise reduction factor. Compared to some reference methods, it prevailed over them obtaining much higher values of the factor for signals heavily distorted by noise. Then it was applied to ECG signals enhancement prior to the measurement of the QT interval. It has been presented visually that even for very low signal-to-noise ratio the method is able to reconstruct a high quality signal. It allowed for an accurate localization of the QT interval limits in the case when the muscle noise amplitude was comparable to the height of the QRS complex. These visual results were confirmed by the quantitative ones. The method was able to determine the left and the right limits of the QT interval with rather good accuracy even for SNR = 0 dB. This opens new possibilities for a reliable analysis of the QT interval variability in signals of very poor quality (e.g. Holter records).

713 714

Acknowledgments

715 716 717 718 719

This research was supported by statutory funds (BK-2016 and BKM-2016) of the Institute of Electronics, Silesian University of Technology. The work was performed using the infrastructure supported by POIG.02.03.01-24-099/13 grant: GeCONiI–Upper Silesian Center for Computational Science and Engineering.

720

references

721 722 723 724 725 726 727

[1] van Alsté JA, van Eck W, Herrmann OE. ECG baseline wander reduction using linear phase filters. Comput Biomed Res 1986;19:417–27. [2] Antzelevitch C. Role of spatial dispersion of repolarization in inherited and acquired sudden cardiac death syndromes. Am J Physiol Heart Circ Physiol 2007;293(4):2024–38.

[3] Augustyniak P. Time-frequency modelling and discrimination of noise in the electrocardiogram. Physiol Meas 2003;24:753–67. [4] Augustyniak P. Optimal coding of vectorcardiographic sequences using spatial prediction. IEEE Trans Inf Technol Biomed 2007;11(3):305–11. [5] Bellman R, Dreyfus S. Applied dynamic programming. New Jersey: Princeton Univ. Press; 1962. [6] Berger RD, Kasper EK, Baughman KL, Marban E, Calkins H, Tomaselli GF. Beat-to-beat QT interval variability novel evidence for repolarization liability in ischemic and nonischemic Kohn87 dilated cardiomyopathy. Circulation 1997;96:1557–65. [7] Croux C, Ruiz-Gazen A. A fast algorithm for robust principal components based on projection pursuit. Compstat: Proceedings in Computational Statistics; 1996. p. 211–7. [8] Goldberger AL, Amaral LA, Glass L, Hausdorff JM, Ivanov PC, Mark RG, et al. PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 2000;101(23):E215–20. [9] Gupta L, Molfese DL, Tammana R, Simos PG. Nonlinear alignment and averaging for estimating the evoked potential. IEEE Trans Biomed Eng 1996;43(4):348–56. [10] Hamilton PS, Tompkins WJ. Quantitative investigation of QRS detection rules using the MIT/BIH arrhythmia database. IEEE Trans Biomed Eng 1986;33:1157–65. [11] Hossjer O, Croux C. Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter. Nonparamet Stat 1995;4:293–308. [12] Hu X, Nenov V. A single-lead ECG enhancement algorithm using a regularized data-driven filter. IEEE Trans Biomed Eng 2006;53:347–51. [13] Jenkala W, Latifa R, Toumanaria A, Dlioua A, El B'charria O, Maoulainine FMR. An efficient algorithm of ECG signal denoising using the adaptive dual threshold filter and the discrete wavelet transform. Biocybern Biomed Eng 2016;36:499–508. [14] Kohn AF. Phase distortion in biological signal analysis caused by linear phase FIR filters. Med Biol Eng Comput 1987;25:231–8. [15] Kors JA, van Herpen G. Measurement error as a source of QT dispersion: a computerised analysis. Heart 1998;80:453–8. [16] Kotas M. Projective filtering of time-aligned ECG beats for repolarization duration measurement. Comput Methods Progr Biomed 2007;85(2):115–23. [17] Kotas M. Projective filtering of time-warped ECG beats. Comput Biol Med 2008;38:127–37. [18] Kotas M. Robust projective filtering of time-warped ECG beats. Comput Methods Progr Biomed 2008;92(2):161–72. [19] Kotas M. Combined application of independent component analysis and projective filtering to fetal ECG extraction. Biocybern Biomed Eng 2008;28(1):75–93. [20] Kotas M, Pander T, Leski J. Averaging of nonlinearly aligned signal cycles for noise suppression. Biomed Signal Process Control 2015;21:157–68. [21] Laguna P, Mark RG, Goldberg A, Moody GB. A database for evaluation of algorithms for measurement of QT and other waveform intervals in the ECG. Comput Cardiol 1997;24:673–6. [22] Laguna P, Thakor NV, Caminal P, Jane R, Yoon H. New algorithm for QT interval analysis in 24 h Holter ECG: performance and applications. Med Biol Eng Comput 1990;28:67–73. [23] Leski JM. Robust weighted averaging. IEEE Trans Biomed Eng 2002;49(8):796–804. [24] Malik M, Batchvarov V. Measurement, interpretation and clinical potential of QT dispersion. J Am Coll Cardiol 2000;36:1749–66.

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795

BBE 193 1–13 biocybernetics and biomedical engineering xxx (2017) xxx–xxx

795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810

[25] McSharry PE, Clifford GD, Tarassenko L, Smith LA. A dynamical model for generating synthetic electrocardiogram signals. IEEE Trans Biomed Eng 2003;50.3:289–94. [26] Momot A. Methods of weighted averaging of ECG signals using Bayesian inference and criterion function minimization. Biomed Signal Process Control 2009;4(2):162–9. [27] Moroñ T. Averaging of time-warped ECG signals for QT interval measurement, information technologies in medicine. Springer International Publishing; 2016. p. 291– 302. [28] Pander T. A new approach to robust, weighted signal averaging. Biocybern Biomed Eng 2015;35:317–27. [29] Petitjean F, Ketterlin A, Gancarski P. A global averaging method for dynamic time warping, with applications to clustering. Pattern Recognit 2011;44:678–93.

13

[30] Poornachandra S. Wavelet-based denoising using subband dependent threshold for ECG signals. Digit Signal Process 2008;18:49–55. [31] Rousseeuw PJ, Croux C. Alternatives to the median absolute deviation. J Am Stat Assess 1993;88:1273–83. [32] Sharma LN, Dandapat S, Mahanta A. ECG signal denoising using higher order statistics in wavelet subbands. Biomed Signal Process Control 2010;5(3):214–22. [33] Schreiber T, Kaplan D. Nonlinear noise reduction for electrocardiograms. Chaos 1996;6:87–92. [34] Tikkanen PE, Sellin LC, Kinnunen HO, Huikuri HV. Using simulated noise to define optimal QT intervals for computer analysis of ambulatory ECG. Med Eng Phys 1999;21:15–25.

811

Please cite this article in press as: Kotas M, Moroń T. ECG signals reconstruction in subbands for noise suppression. Biocybern Biomed Eng (2017), http://dx.doi.org/10.1016/j.bbe.2017.03.002

812 813 814 815 816 817 818 819 820 821 822 823 824 825 826