Indonesian J our nal of Electrical Engineering and Computer Science V ol. 36, No. 2, No v ember 2024, pp. 837 ∼ 845 ISSN: 2502-4752, DOI: 10.11591/ijeecs.v36.i2.pp837-845 ❒ 837 Electr ocardiogram r econstruction based on Hermite inter polating polynomial with Chebyshe v nodes Shashwati Ray 1 , V andana Chouhan 2 1 Department of Electrical Engineering, Bhilai Institute of T echnology , Dur g (C.G.), India 2 Department of Electronics and T elecommunication Engineering, Bhilai Institute of T echnology , Dur g (C.G.), India Article Inf o Article history: Recei v ed No v 6, 2023 Re vised Jul 5, 2024 Accepted Jul 14, 2024 K eyw ords: Chebyshe v nodes Electrocardiogram Equally spaced nodes Hermite Noise Polynomial ABSTRA CT Electrocardiogram (ECG) signals generate massi v e v olume of digital data, so the y need to be suitably compressed for ef cient transmission and storage. Poly- nomial approximations and polynomial interpolation ha v e been used for ECG data compression where the data signal is described by polynomial coef cients only . Here, we propose approximation using hermite polynomial interpolation with chebyshe v nodes for compressing ECG signals that consequently denoises them too. Recommended algorithm is applied on v arious ECG signals tak en from MIT -BIH arrh ythmia database without an y additional noise as the signals are already contaminated with noise. Performance of the proposed algorithm is e v aluated using v arious performance metrics and compared with some recent compression techniques. Experimental results pro v e that the proposed method ef ciently compresses the ECG signals while preserving the minute details of important morphological features of ECG signal required for clinical diagnosis. This is an open access article under the CC BY -SA license . Corresponding A uthor: Shashw ati Ray Department of Electrical Engineering, Bhilai Institute of T echnology Dur g (C.G.), India Email: shashw atiray@yahoo.com 1. INTR ODUCTION Electrocardiogram (ECG or EKG) is a recording of 1-D time series data sequence generated by cardiac muscles. The y help to track and detect abnormalities in the heart rh ythm based on the morphology and frequenc y of heartbeat [1]. 24 hours ECG record with the sampling rate of 360 Hz and 11 bit/sample data resolution requires about 43 MBytes per channel [2]. Therefore, an ef fecti v e data compression scheme is often required for ef cient ECG data storage and transmission o v er telephone line or digital telecommunication netw ork [3]. ECG compression techniques can be broadly classied i nto three major cate gories: direct time domain, parameter e xtraction and transform domain method [4]. In direct ti me domain method compression is achie v ed by nding correlation between the adjace nt samples, i.e., intra-beat redundanc y in a group and encode them into a sma ller sub-group [5]. Some popu- lar algorithms of direct method are: amplitude zone time epoch coding (AZTECH) [6], coordinate reduction time encoding system (COR TES) [7], entrop y coding [8], scan-along polygonal approximation (SAP A) [9] and long–ter m prediction (L TP) [10]. P arameter e xtraction methods e xtract important morphological features from the signal and encode these features to achie v e desired compression [11], e.g., linear prediction [12] and residual encoding methods [13]. In transform domain method signals are transformed to another space using v arious transform schemes [14], in the subsequent steps small transform coef cients are discarded J ournal homepage: http://ijeecs.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
838 ❒ ISSN: 2502-4752 and only critical information is encoded to achie v e the desired compression. T ransform techniques include fourier transforms [15], discrete w a v elet transform (D WT) [16], discrete cosine transform (DCT) [17], discrete Le gendre transform [18] and Karhunen Louv e transform (KL T) [19]. Polynomials nd application in signal processing, viz., for ltering noisy signals, interpolating data se- quence, and for data compression [20]. High compression ratio can be achie v ed by polynomial approximation, since, polynomial coef cients describing the signal are only required for compression [21]. Le gendre poly- nomials is used by [22] for image compression and reconstruction. F or higher order of polynomials the V an- dermonde matrix gets ill conditioned which is a v oided by di viding image matrix into sub-matrices to achie v e desired compression ratio. ECG data compression utilizing Jacobi polynomials is proposed by [23]. ECG signals are rst se gmented into blocks that match with cardiac c ycles before being decomposed in Jacobi poly- nomials bases. Gauss quadratures mechanism for numerical inte gration is used to compute Jacobi transforms coef cients. T o achie v e desired compression, coef cients of small v alues are discarded in the reconstruction stage. As the deri v ed polynomials use recurrence formula, rounding errors pile up during the computation and polynomials gradually denature with increase in order of the polynomial. ECG data compression based on B-spline basis functions is proposed by [24]. The position of the knots are computed using run-length coding. Ho we v er , for f alse R-R comple x ne w sequence of knots has to be sent, thereby increasing the o v erhead data resulting in increasing computational comple xity . Nyg aard and Haugland [25] proposed piece wise polynomial approximation for reconstructing the ECG signal by second order polynomials. Khetk eeree and Chansamorn [26] introduced signal reconstruction based on the second order tetration polynomial. Fundamental signals such as square, sa w-tooth and sine w a v e with v arious sampling resolution were applied to test the interpolation performance. High v alues of peak signal-to-noise ratio (PSNR) were obtai ned for square w a v e and sa w-tooth w a v e. No w a days, long term ECG monitoring is applied for management of cardio v ascular diseases where wireless technology is used to transmit ECG data through communication channels. Channel bandwidth can be optimized by performing ECG compression. ECG signals can be compressed using polynomial interpolation. Here high compression ratio can be achie v ed, since ECG signals can be reconstructed using fe w sampling points. W e propose here an algorithm to obtain an ECG approximating model based on lagrange form of hermite interpolating polynomial with chebyshe v nodes. Hermite interpolating polynomials are more rob ust, since we ha v e higher de gree of freedom with deri v ati v e v alues as additional information which is equi v alent to almost twice the order of the interpolating polynomial. This polynomial smoothly interpolates between the k e y-points and compresses the ECG data, thus f acilitating less data for storage and transmission. In the process it also denoises the ECG signal in an ef cient w ay while preserving the morphologi cal features as required by the cardiologist. The or g anization of the paper is as follo ws: in section 2 we e xplain the research methodology of this w ork, in section 3 we pro vide e xperimental details and in section 4 we present the conclusions dra wn out of our w ork. 2. RESEARCH METHOD In real life applications e xperimental data are in the form of set of discrete data points and functional relation between input and output is nondeterministic. In such situations polynomial interpolation plays an important role in determining a polynomial matching the points. In our research w ork, we approximate an ECG signal f consisting of N ECG samples with lagrange form of hermite interpolating polynomial H p n ( x ) using n chebyshe v nodes. Our research methodology comprises mainly of v e stages: - Obtain ra w discrete ECG data as .mat le from the MIT -BIH arrh ythmia database no w freely a v ailable on Ph ysioNet. - At this sta g e preprocess and normalize the signals. The ECG data is con v erted into ph ysical units (mV), then normalised by reducing the g ain and re-scaled to limit the range within [ − 1 , 1] . - Map the ‘ n ’ chebyshe v nodes, x k , k = 1 , ..., n on the abscissa of time (seconds) with the equi v alent ECG data to obtain them as function v alues f x k , k = 1 , ..., n . - Obtain the deri v ati v es f ′ x k at all k = 1 , ..., n points using numerical dif ferentiation method. - Construct the lagrange form of hermite polynomial with f x k and f ′ x k , k = 1 , ..., n . Indonesian J Elec Eng & Comp Sci, V ol. 36, No. 2, No v ember 2024: 837–845 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 839 2.1. Function v alues at inter polating nodes Since the ECG samples are obtained for an arbitrary length, we require to transform them to the designated interv al and compute the function v alues (we consider the ECG signal as a function) at all the interpolating nodes. Let the ECG signal be sampled at a frequenc y F s and the sampled v alues be dened as function v alues f , thus comprising of total N samples. W ith spacing h = 1 /F s , compute the end points of [ a, b ] on the abscissa of time as, a = 1 /F s , b = N /F s compute the n chebyshe v nodes x k , k = 1 , ..., n on [ a, b ] as, x k = a + b 2 − b − a 2 cos 2 k − 1 2 n π , k = 1 , · · · , n nd all the equi v alent function v alues as, f x k = f ( x k [1 : n ]) an y missing function v alue is e v aluated with linear interpolation using the adjacent sampled v alues. No w , we ha v e the data points in the form ( x k , f x k ) , k = 1 , ..., n . 2.2. Lagrange f orm of hermite inter polation Hermite interpolating polynomials require the kno wledge of the deri v ati v es at the interpolating n odes . Since the function v alues are discrete, the deri v ati v es are computed applying numerical dif ferentiation meth- ods. The numerical deri v ati v es are computed using forw ard, central and bac k w a rd dif ferences. Use forw ard dif ference to compute f ′ x k at lo wer points of [ a, b ] , f ′ x k = 1 2 h [ − 3 f x k + 4 f x k +1 − f x k +2 ] use backw ard dif ference to compute f ′ x k at upper points of [ a, b ] , f ′ x k = 1 2 h [ − 3 f x k − 4 f x k − 1 + f x k − 2 ] use central dif ference to compute f ′ x k at intermediate points of [ a, b ] . f ′ x k = 1 2 h [ f x k +1 − f x k − 1 ] F or the data of the form ( x k , f x k ) , ( x k , f ′ x k ) , k = 1 , ..., n , the unique lagrange form of hermite polynomial H p n ( x ) of de gree 2 n + 1 that agrees with f x k and f ′ x k is gi v en by: H p n ( x ) = n X k =1 f x k A n,k ( x ) + n X k =1 f ′ x k B n,k ( x ) where, A n,k ( x ) = [1 − 2( x − x k ) L ′ n,k ( x k )] L 2 n,k ( x ) and B n,k ( x ) = ( x − x k ) L 2 n,k ( x ) where, L n,k ( x ) denotes lagrange basis function of order n dened by , L n,k ( x ) = n Y i =1 ,i ̸ = k ( x − x i ) ( x k − x i ) the error using lagrange form of hermite interpolation with chebyshe v nodes is gi v en by , E ( x ) = | f ( x ) − H p n ( x ) | ≤ 1 2 n ( n + 1)! ( b − a ) 2 ( n +1) max a ≤ ξ ≤ b | f ( n +1) ( ξ ) | if E ≥ ε , where the tolerance ε = 10 − 2 , then n is increased by 10 and the entire procedure is repeated. Electr ocar dio gr am r econstruction based on Hermite interpolating ... (Shashwati Ray) Evaluation Warning : The document was created with Spire.PDF for Python.
840 ❒ ISSN: 2502-4752 3. RESUL TS AND AN AL YSIS The proposed algorithms are implemented in MA TLAB (R2013b) v ersion. Each part of the proposed ECG approximation algorithm is written in the .m le as a subroutine module. All the computations are carried out on ECG signals tak en from MIT -BIH arrh ythmia database a v ailable on Ph ysioNet [27]. W e consider here v arious signals of channel 1, sampled at 360 Hz with a resolution of 11 bits per sample with duration of 5 seconds resulting in 1,800 s amples. These sample points are the ECG signal magnitudes obtained at equal interv als of ‘ 1 / 360 ’ second. W e perform the delity assessment of the propos ed approximation method using the performance or error measures as - root m ean square error (RMS), percentage root mean dif ference (PRD), signal to noise ratio (SNR), and compression ratio (CR). Here, we consider CR as the ratio of the number of bytes in the uncompressed representation to the number of bytes in the compressed representation. T o test the ef cac y of our proposed method we apply the de v eloped algorithms on 12 records sho wn in Figures 1 and 2 with duration of 5 seconds and approximate them in the form of respecti v e polynomials with 300 chebyshe v nodes. All the 12 obtained results are illustrated in Figures 1 and 2, Figure 1(a) #100, Figure 1(b) #104, Figure 1(c) #108, Figure 1(d) #112, Figure 1( e) #115, Figure 1(f) #117, and Figure 2(a) #122, Figure 2(b) #201, Figure 2(c) #205, Figure 2(d) #207, Figure 2(e) #214, Figure 2(f) #220 depicting the original signals and the reconstructed polynomials in red and blue colours respecti v ely . (a) (b) (c) (d) (e) (f) Figure 1. Original noisy ECG signals (pink) and reconstructed samples by proposed method (blue): (a) ECG record # 100 , (b) ECG record # 104 , (c) ECG record # 108 , (d) ECG record # 112 , (e) ECG record # 115 , and (f) ECG record # 117 Indonesian J Elec Eng & Comp Sci, V ol. 36, No. 2, No v ember 2024: 837–845 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 841 (a) (b) (c) (d) (e) (f) Figure 2. Original noisy ECG signals (pink) and reconstructed samples by proposed method (blue): (a) ECG record # 122 , (b) ECG record # 201 , (c) ECG record # 205 , (d) ECG record # 207 , (e) ECG record # 214 , and (f) ECG record # 220 Since the proposed method is an e xtension of the method propose d by Y ada v and Ray [21], we deem it t to compare the performance statistics of the proposed method with the latter . Y ada v and Ray [21], ha v e approximated all these records of same duration with assorted noise le v els using lagrange interpolating poly- nomial with 400 chebyshe v nodes respecti v ely . The performance statistics of the method by Y ada v and Ray , and the proposed method are reported in T able 1. It is w orth mentioning that in the process of approximating, both the methods e v entually denoise the ECG signals. Comparing the respecti v e entries of T able 1 for each ECG record, we observ e that lo wer order hermit e form of interpolati n g polynomial outperforms the lagrange form in all performance metrics. Lo wer v alues of RMS are indicati v e of least distortion and better approximation. The v alues of CR in both the methods remain constant for all the signals, because the number of samples and the respecti v e number of interpolating nodes are persistent in all the signals. Electr ocar dio gr am r econstruction based on Hermite interpolating ... (Shashwati Ray) Evaluation Warning : The document was created with Spire.PDF for Python.
842 ❒ ISSN: 2502-4752 The tw o important features of a compression algorithm are the compression measure and the recon- struction error . Not man y approximating methods are a v ailable in the e xisting literature for ECG signals. T o ha v e a comprehensi v e re vie w of the proposed method, we abstractly compare the performance of the proposed method with tw o e xisting recent w orks on ECG compression. The identied methods are: Y ang et al. [14] us- ing empirical mode decomposition (EMD) and Hamza et al. [28] based on discrete w a v elet transform (D WT) and dual encoding technique. T able 1. Comparison of the proposed method with Y ada v and Ray [21] for signal length of 5 sec Record Y ada v and Ray [21] with n = 400 Proposed method with n = 300 RMS PRD SNR CR RMS PRD SNR CR 100 0.11 15.75 8.45 4.49 0.04 12.27 11.44 6.00 104 0.06 18.92 12.34 4.49 0.07 19.85 11.86 6.00 108 0.02 6.39 16.76 4.49 0.03 7.81 15.33 6.00 112 0.02 2.55 17.09 4.49 0.02 1.70 21.35 6.00 115 0.09 14.72 10.57 4.49 0.04 6.21 18.07 6.00 117 0.03 3.71 16.81 4.49 0.02 2.73 19.71 6.00 122 0.04 4.99 17.85 4.49 0.03 3.53 21.21 6.00 201 0.02 9.22 17.29 4.49 0.02 8.14 19.68 6.00 205 0.05 11.43 10.51 4.49 0.02 4.88 18.33 6.00 207 0.02 5.10 23.56 4.49 0.02 6.97 21.93 6.00 214 0.04 8.79 19.57 4.49 0.03 7.67 21.96 6.00 220 0.11 15.75 8.70 4.49 0.06 8.60 14.07 6.00 F or comparison with Y ang et al. [14] method, we choose 8 MIT -BIH arrh ythmia data sets [27] as test signals with time period as 4.2 seconds and sampling rate as 360 Hz for all the signals. The 8 records are referred as # 100 , #103, # 107 , # 109 , # 116 , # 117 , # 119 , and # 200 . T o e v aluate the quality of the pro- posed algorithm we use RMS and CR as the perf ormance measures and illustrate the results as bar graphs in Figure 3 and Figure 4. From the comparisons we can easily infer that the proposed method f airs v ery well in RMS metric and compares well in CR metric. F or comparison with Hamza et al. [28] method, we choos e 5 MIT -BIH arrh ythmia data sets as test signals with time period as 10 seconds and sampling rate as 360 Hz for all the signals resulting in 3,600 samples. The 5 records are referred as # 100 , # 109 , # 115 , # 119 , and # 200 . T o e v aluate the quality of the proposed algorithm we use RMS as the performance measure and depict the results in Figure 5 fr om where we observ e that the proposed method performs v ery well. Here, we ha v en’ t considered the comparison of CR since we dif fer in our denitions. Figure 3. Bar graph of RMS v alues of v arious records with signal length of 4 . 2 sec obtained by Y ang et al. [14] and the proposed method Indonesian J Elec Eng & Comp Sci, V ol. 36, No. 2, No v ember 2024: 837–845 Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 843 Figure 4. Bar graph of CR v alues of v arious records with signal length of 4 . 2 sec obtained by Y ang et al. [14] and the proposed method Figure 5. Bar graph of RMS v alues of v arious records with signal length of 10 sec obtained by Hamza et al. [28] and the proposed method 4. CONCLUSION In this w ork, the superiority of the proposed approximation model of lagrange form of hermite poly- nomial interpolation with chebyshe v nodes is established by applying on v arious ECG signals tak en from MIT/BIH arrh ythmia database and comparing with fe w e xisti ng methods tak en from recent literature. From all the analysis we infer that the proposed method of approximation outperforms all the methods in most of the metrics. The proposed method compresses ECG signal thus reducing the memory requir ement. Apart from this, the proposed scheme not only eliminates noise, b ut also preserv es important morphological features re- quired for analysis of v arious conditions lik e arrh ythmias, inadequate coronary artery blood o w , electrolyte disturbances, and cardiomyopath y . Most signicant is that the proposed method is able to con v ert the ECG signal into a polynomial; and all polynomial operations emphasize can be applied to e xtract v arious morpho- logical features for the diagnosis of v arious diseases that are re ected in the ECG. The proposed model can also be e xtended in approximating other time series data such as economic and sales forecasting, b udgetary and stock mark et analysis, yield projections, process and quality control, to predict the future price of the stock mark et, and e xchange rate forecast. Furthermore, the proposed method is riddled with certain challenges. In case of critical base line w ander additional preprocessing step has to be applied. Moreo v er , detrending of time series data is necessary whene v er there is a base line drift in the signal. Electr ocar dio gr am r econstruction based on Hermite interpolating ... (Shashwati Ray) Evaluation Warning : The document was created with Spire.PDF for Python.
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Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 ❒ 845 BIOGRAPHIES OF A UTHORS Dr . Shashwati Ray recei v ed her B.Sc. (Engg.) de gree in Electrical Engineering and M.T ech. de gree in Control Systems from National Institute of T echnol ogy , K urukshetra, India, and her Ph.D. de gree from Indian Institute of T echnology , Bombay , India in 2007. She has been a professor in the Department of Electrical Engineering, Bhilai Institute of T echnology , Dur g, India. Her research interests include interv al analysis techniques, numerical analysis, optimization, po wer system control, rene w able ener gy sources, rob ust control, and signal processing. She can be contacted at email: shashw atiray@yahoo.com. V andana Chouhan recei v ed her B.E. de gree in electronics engineering from CEC, Chandrapur , India. M.T ech. de gree in Instrumentation and M.E in En vironmental from CSVTU, Bhilai. She is pursuing her Ph.D. de gree from CSVTU, Bhilai. Her research interests include biomed- ical and digital signal processing. She ca n be contacted at email: v andanachouhan2212@yahoo.com. Electr ocar dio gr am r econstruction based on Hermite interpolating ... (Shashwati Ray) Evaluation Warning : The document was created with Spire.PDF for Python.