Dual-tree complex wavelet transform technique-based optimal threshold tuning system to deliver denoised ECG signal

Abstract

The primary output for any health monitoring system that offers telecardiology services is the recovery of the electrocardiographic (ECG) signal from the noised signal. The mechanized investigation of the ECG signal is the most inspiring challenge for accurate detection of cardiac disease. This could be accomplished by eliminating the various noises from the acquired signal. In this paper, a noise reduction approach employing DTCWT is executed on an ECG signal by proposing a noise estimator along with a detailed assessment of the effect of the choice of the threshold value, threshold algorithm and distribution function. The thresholding technique is executed by varying the threshold value (γ) and its function (f_n) applied to the proposed estimator (α*). The proposed estimator is scaled by 2ⁿ factor to study its impact on performance metrics and the nature of the reconstructed signal utilizing different distribution functions. The best combination of threshold function with threshold value selection has been chosen in this work from eight different sets of threshold value selection rules along with six distinct threshold functions. The experimental results show that the proposed noise reduction approach using a universal modified threshold level-dependent threshold with non-negative garrote threshold function for normal distribution with n = 3 delivers 80.72dB SNR with a subsequent reduction in MSE and PRD as compared with other standard techniques. An elaborate empirical analysis for selecting the distribution function for obtaining the best possible threshold function and technique is the prime objective and novelty of this research work.

Keywords

Threshold functions threshold values noise estimator distribution functions signal to noise ratio

Introduction

An electrocardiogram (ECG) signal plays a significant role in the essential finding, anticipation and survival investigation of cardiovascular (CVD) disease (Kirti et al., 2018). CVD has been the largest risk to mankind for many years, and millions of individuals pass away due to its deferred detection and treatment (Kirti et al., 2019). ECG signal represents the graphical depiction of cardiovascular development and its application for the disclosure of distinctive heart illness and irregularities (Dhiman et al., 2016; Karthikeyan et al., 2012). ECG signal is non-stationary and non-linear that resulted in the induction of various artefacts at different frequencies classified as burst noise, baseline wander noise, power-line interference and electromyography noise (Prashar et al., 2019). These artefacts influenced the morphological characteristics (P-QRS-T) of an ECG signal that resulted in the false extraction of its features (Prashar et al., 2018b). Contamination of ECG signals with various noises causes major challenges for cardiac disease identification. Thus, noise removal is a vital step for signal analysis and processing. In all noise suppression strategies, a clean signal is acquired by accepting the estimation of the noise range already present in the raw signal (Sood, 2018). The performance of the enhancement algorithms and quality of the enhanced signal depends upon the noise estimation. If the estimated noise is too low, the result will be residual noise. If it is high, the result will be a loss of intelligibility. In general, noise power range is assessed and modified in the areas based on noise signal measurable qualities; there is a requirement for new noise estimation techniques, in which the noise spectrum is consistently refreshed over a timeframe. Biomedical signal processing assists the clinicians to find new techniques to monitor the major issues faced in the signal processing applications such as noise. Noise can be separated by its time and recurrence space properties. Uniform noise has a constant probability density over a finite interval whereas Gaussian noise is defined over an infinite interval by just two factors: average and spread. As an impact of these noises, the information in a noisy signal will be misinterpreted.

Different pre-processing strategies have been proposed earlier to expel various noises and to extract precise morphological characteristics of an ECG signal (Benmalek et al., 2010; Chang and Liu, 2011; Chavan et al., 2008; Chawla et al., 2008; Martens et al., 2006). In this context, the finite impulse response-based notch filtering technique has been presented to evacuate power line interference (PLI) from an ECG signal using a high filter order that enlarges the computational complexity (Chavan et al., 2005). Many researchers have performed high order filtering based on infinite impulse response to expel different categories of noise such as baseline wander and power line interference (PLI) that induct some shortcomings of large filtering time which restricts its application for non-linear signals (Mbachu et al., 2011). To enhance the fast filtering response an adaptive method has been introduced for the disposal of different noise from an ECG signal with smaller residual errors (Tareen, 2008). Researchers have presented different approaches such as fixed sub-band decomposition using a wavelet-based wiener filter (Chmelka and Kozumplik, 2005) to eliminate EMG noise and a bionic adaptive wavelet transform (Sayadi and Shamsollahi, 2006) to remove baseline wander noise. An adaptive filtering approach using a moving average filter has been introduced to obtain smooth and distortionless ECG signals (Salih et al., 2015). For an accurate estimation of noise a wavelet theory-based signal noise residue algorithm has been developed that restricts the influence of noise with minimum computational assessments (Khan et al., 2011). Some additional methods such as temporal averaging, linear filtering, cubic spline filtering, principal component analysis (PCA) and independent component analysis (ICA) have been introduced in recent years for signal enhancement (El-Dahshan, 2010; Taigang et al., 2006). Implementation of these methods has some shortcomings such as more sensitivity to various levels of noise and extremely vulnerable to small variations in either the signals or the artefacts. To overcome these limitations, discrete wavelet transform (DWT) is a prominent method for ECG signal processing. Kaur and Rajni (2016) presented an ECG denoising approach using Savitzky-Golay filter and discrete wavelet transform to evaluate the performance parameters such as SNR and MSE. Daubechies (DB) technique based on wavelet filtering has been introduced for denoising of an ECG signal using hard and soft thresholding (Yadav and Mehra, 2016). Another strategy of wavelet transform based on modified shrinkage function has been acquainted to get denoised ECG signal with a high signal to noise ratio (Zubair et al., 2018). Further, the implementation of the discrete wavelet transforms for low-frequency ECG signal resulted in some shortcomings such as aliasing, oscillations and lack of directionality (Jain, 2018; Sukanya et al., 2008). To succeed from these drawbacks, dual tree complex wavelet transforms (DTCWT) that incorporate with anti-aliasing effect and shift-invariance properties to obtained enhanced reconstructed signal (Ghombavani and Kiani, 2015; Prashar et al., 2018a; Wang and Ji, 2014). Practically, in all signal handling applications, it is a noteworthy assignment that the noise is eliminated from such signals. In this aspect, probability distribution functions (pdf) played a major role in the noise estimation process. Thomas (2004) presented an optimal bandwidth using Gaussian distribution which minimizes the integral of square error between the estimated density and true density to deliver the smooth estimation. Another method of ECG denoising is based on high order statistics at distinct wavelet bands that give important information about data in time and frequency domain (Sharma et al., 2010). Moreover, setting the threshold limits for non-stationary signals such as ECG without utilizing the model fitting has demonstrated strong variations between data sets from a similar parameter monitored in a different time period. This problem has encouraged authors to examine the application of pdf modelling that are able to minimize the differences between the data from various time periods (Jablonski et al., 2013). From the last two decades, many researchers have been concentrated on expelling the noise from the signal to get the denoised ECG signal. In this context, Donoho and Johnston proposed hard and soft thresholding techniques for ECG signal denoising (Donoho and Johnstone, 1994). These schemes eradicate numerous wavelet coefficients that may contain valuable signal data. Nonetheless, the serious issue with the two strategies is the decision of an appropriate threshold value. The definition of the coefficient independent threshold is given by Donoho and Johnston relies upon the noise power and the size of the signal (Donoho, 1995). As most of the signal shows spatially non-uniform energy distribution, this inspires the decision of a non-uniform threshold. The thresholding technique is a signal estimation approach that has been applied to detail coefficients obtained at each level of wavelet decomposition to achieve signal denoising (Mandala et al., 2017; Priyaa et al., 2016; Sharma and Kashyap, 2017).

Problem formulation

The ECG contamination is a persistent problem that affects the interpretations of the ECG signal at various stages of its processing. The mutilations in signal due to noise are normal during its acquisition, processing, compression, transmission and reconstruction. Commonly, a noise reduction problem can be effectively handled by separating the preferred signal from the contaminated signal by preserving its features. The thresholding techniques are common ways of removal of noises in which the local and global signal characteristics are explored. ECG signal denoising based on the thresholding technique using different distribution functions is exploited in this research paper. To restrict the spurious high-frequency information at the output signal that results in successive elimination of aliasing effect is the open research challenge. The effect of the different probability distribution functions (Gaussian and Normal) has to be studied to reduce the transient noise from the signal.

Contribution

In this paper, a DTCWT technique-based threshold tuning is investigated to deliver denoised ECG signals. Threshold tuning is executed by differing the threshold value (γ) and its function (f_n) at the optimal decomposition level. The contribution of this paper is to propose an estimator that yields a denoised ECG signal with high SNR value when used with the optimal combination of different distribution functions. In this research work, challenges for the noise removal from the contaminated ECG signals using optimal thresholding is investigated to get a denoised signal. The thresholding technique is applied to detailed coefficients. The thresholding technique is implemented and evaluated with the different threshold values and its function for various configuration and signal parameters such as SNR, PSNR (dB) on MIT-BIH arrhythmia ECG databases. The best combination of threshold function with threshold value selection has been used in this work from eight different sets of threshold value selection rules along with six distinct threshold functions. The proposed estimator is scaled by 2ⁿ factor and z parameter based on different distribution functions to study their impact on performance metrics and the nature of the reconstructed signal.

This research paper presents the acquire methodology accompanied by display of computational results utilizing proposed estimator for different samples of ECG and their comparison with the previous state of art techniques that are followed by a conclusion.

Methodology

The raw input ECG signal MIT-BIH arrhythmia V5 of 100 records is derived from the Physio Bank ATM database (PhysioNet, nd). This ECG input signal is composed of 1536 samples having specifications of 360 Hz sampling frequency and 11-bit resolution over 10 mV range. The whole process of the methodology is shown in Figure 1. This work is further extended for all arrhythmia records (i.e. 101–234) to validate its performance credentials for signal denoising.

The prime motivation of this research work is to propose a robust noise estimator that successively eliminates the noise content from an ECG signal by vitally preserving its features. This can be achieved by removing the noise from the amplitude of detail coefficients obtained after performing the DTCWT technique. The proposed noise estimator is scaled by 2ⁿ factor which delivers extraordinary results than other conventional techniques to get a clean signal.

In Figure 1, the input ECG signal is nourished to the filtering stage to dispose of the baseline wander noise. After filtering, the DTCWT technique is associated to perform multilevel decomposition to extricate detail and approximate coefficients. In another step, thresholding procedure is performed on detail coefficients by choosing the distinctive threshold function and threshold value selection utilizing a proposed noise estimator that resulted in modified detail coefficients (D’1, D’2, D’3, D’4) at distinct decomposition level. In the next stage, an inverse DTCWT strategy is performed on the modified detail coefficients and approximation coefficient to get the reconstructed signal. Finally, the parameters of the output signals have been assessed by computing distinctive execution measurements to approve the proposed strategy.

Filtering stage

Filtering is the foremost basic task in signal handling to dispose of the artefacts utilizing different filtering strategies by preserving the shape of diverse complexes. Deformity in beat morphology of the ECG signal has been diminished by removing the baseline wander noise through the high pass FIR filtering (Prashar et al., 2018a)

DTCWT technique

ECG signal predicts non-linear characteristics that restrict its denoising utilizing normal filters. To counter this restriction, the ECG signal has been denoised employing the DTCWT technique that has been empowered by good directional selectivity with the better shift-invariance and reduced spectral aliasing properties that made this technique more suitable to denoise the non-stationary signals than other conventional techniques. This multilevel decomposition employing the DTCWT strategy is broadly depicted in (Prashar et al., 2018b). The block diagram of the DTCWT technique is shown in Figure 2.

In this research paper, 4-level multilevel decomposition is performed utilizing the DTCWT technique that enables the extraction of a detail coefficient at each level and approximation coefficient at the 4^th level for both real and imaginary tree. Detail coefficients are high-frequency coefficients containing the massive amount of noise content in it whereas, the approximation coefficient is of low frequency which is slightly influenced by noise. In this manner, the thresholding procedure is performed on high-frequency detail coefficients for successive noise elimination.

Thresholding techniques

Thresholding can be performed using two parameters consisting of threshold function and threshold value selection. Hard and soft thresholding are two types of thresholding of ECG signals which are used for the noise removal applications. The hard thresholding method is more sensitive to small variations which leads to less stable than soft thresholding. In particular, the soft thresholding of noisy ECG signals based on the best estimation of the wavelet coefficient that helps in low frequency and high frequency ranges for the effective removal noises without losing the clinically important ECG signals.

Threshold functions

Threshold functions restrict the input signal to the desired range by removing the noise from every detail coefficients depending on a threshold value. Different threshold functions (El. B’charri et al., 2017) along with their mathematical expression are defined in this paper.

1. Hard Thresholding – denoising of an ECG signal using hard thresholding leads to oscillation in the reconstructed signal. The hard thresholding is defined by Eq. (1)

f_{H} = {\begin{matrix} c (n), for | c (n) | ⩾ γ \\ \begin{matrix} 0 & , & otherwise \end{matrix} \end{matrix}}

(1)

2. Soft Thresholding – soft thresholding has been contributed to the denoising process by reducing the amplitude of the ECG signal. The soft thresholding function expressed by Eq. (2)

f_{s =} {\begin{matrix} sign (c (n)) (| c (n) | - γ), for | c (n) | ⩾ γ \\ \begin{matrix} 0 & , & otherwise \end{matrix} \end{matrix}}

(2)

3. Non-Negative Garrote Thresholding – soft thresholding resulted in fewer artefacts than hard thresholding and give smaller estimation bias than the soft thresholding for large coefficients by Eq. (3)

f_{non - ne g} = {\begin{matrix} sign (c (n)) (| c (n) | - \frac{γ^{2}}{| c (n) |}), for | c (n) | ⩾ γ \\ \begin{matrix} 0 & , & otherwise \end{matrix} \end{matrix}}

(3)

4. Hyperbolic Thresholding – hyperbolic thresholding is the soft thresholding of squared detail coefficients. The hyperbolic thresholding is defined by Eq. (4)

f_{hyp} = {\begin{matrix} \begin{matrix} sign (c (n)) \sqrt{\overset{2}{c (n)} - γ^{2}} & , for & | c (n) | ⩾ γ \end{matrix} \\ \begin{matrix} 0 & , & otherwise \end{matrix} \end{matrix}}

(4)

5. Trimmed Thresholding – this type of thresholding depends on α value. If α = 1, then it is equivalent to soft thresholding while $α \to 0$ it is equivalent to hard thresholding. Trimmed thresholding function defined by Eq. (5).

f_{trim} = {\begin{matrix} \begin{matrix} c (n) * \frac{{| c (n) |}^{α} - {| γ |}^{α}}{{| c (n) |}^{α}} \end{matrix} \\ 0, otherwise \end{matrix}}, for c (n) ⩾ γ

(5)

6. Semi-Soft Thresholding – it is non-linear thresholding that has been introduced between hard and soft thresholding. It uses two threshold values, that is, primary threshold and secondary threshold. Semi-soft thresholding defined by Eq. (6).

f_{semi} = {\begin{matrix} c (n), | c (n) | < γ_{2} \\ sign (c (n)) \frac{(c (n) - γ_{1}) * γ_{2}}{γ_{2} - γ_{1}}, \underset{2}{γ} < | c (n) | ⩽ γ_{1} \\ \begin{matrix} 0, & | c (n) | < γ_{1} \end{matrix} \end{matrix}}

(6)

where γ₁ = $\sqrt{γ_{2}}$ for semi-soft thresholding.

c(n) represents the wavelet coefficient and γ is thresholding value for all above-threshold functions.

Threshold value

A threshold value is a minimum value below it, all coefficients tend to zero. The various threshold value selection rules for the ECG signal denoising are well defined in this section. These threshold values are updated for every detail coefficient obtained at the subsequent decomposition level (El. B’charri et al., 2017). Different types of threshold values selection are

Universal Threshold

γ_{u} = α \sqrt{2 \log N}

Universal Threshold Level Dependent

γ_{uld} = α_{j} \sqrt{2 \log n_{j}}

Adaptive Threshold Selection

γ_{adp} = α_{j} \sqrt{w_{b}}

Heursure Threshold Selection (γ_heur ): this threshold selection is the combination of universal threshold and adaptive threshold selection. Dependent on the signal to noise ratio, if it is small, then the universal threshold is used rather than the adaptive threshold selection for better threshold estimation.

Universal Modified Threshold Level Dependent

γ_{umld} = α_{j} \frac{\sqrt{2 \log n_{j}}}{\sqrt{n_{j}}}

Exponential Threshold Level Dependent

γ_{expld} = 2^{(j - 1 / 2)} α_{j} \sqrt{2 \log n_{j}}

Minimax Threshold

$γ_{mini} = 0.3936 + 0 . 1829^{*} (\log n_{j} / \log 2)$

Modified Unified Threshold

γ_{mou} = α_{j} \frac{\sqrt{2 \log (N)}}{\log (j + 1)}

where w_b is the b^th coefficient wavelet square (coefficient at minimal risk) chosen from the vector W= [w₁,w₂,..,w_N]. For all threshold value selection rules, N indicates the original signal length, n_j represents the signal length at j^th scale, while α represents the standard deviation (SD) based on Median Absolute Deviation (MAD). ‘α’ is noise estimator assumed to be proportional to the SD of the coefficients. The conventional estimator α (Naga Prudhvi Raja and Venkateswarlu, 2012) is defined by Eq. (7).

α = z^{*} (median | x_{jk} |)

(7)

x _jk are the coefficients of detail at the finest level and z* is the constant value dependent on different distribution functions.

Proposed estimator

A strong correlation coefficient with a high breakdown point reliant on the least median of squares (LMdS) regression approach was proposed in 1990 (Abdullah, 1990). The LMdS correlation coefficient, in general, gives an extremely high estimation of the relationship. To overrule this issue, Abdullah (1990) proposed a correlation coefficient utilizing weighted least squares by consolidating the LMdS estimator with M-estimator. During 2011, another rendition of correlation coefficient dependent on the middle utilizing scale estimator median absolute deviation (MAD) was proposed and known as median-product (MP) correlation coefficient (Shafiullah and Khan, 2011). They supplanted the mean in the traditional correlation coefficient into the median and utilized MAD in this coefficient figuring. Notwithstanding, MAD comprises a couple of downsides, notably this estimator has low efficiency, which is 37% at the Gaussian distribution, and also MAD views a scattering of symmetric conveyance. The upside of the strong correlation coefficient is that it requires less computing time as compared with other schemes. The utilization of MAD in the condition can be improved to another vigorous scale estimator with the objective that this robust correlation coefficient can perform better.

Eq. (7) explains the median of detail coefficient at the finest level taken to reduce the interference of outliers. Outliers are typical values that are notably different from the rest of the data. The detailed coefficients are high-frequency coefficients and are mostly affected by noise. The existence of these artefacts changes the overall amplitude of an ECG signal which implied that the desired information is accommodated by moderately high amplitude noise in the signal. To reduce the effect of such noises, amplitude downscaling is performed which results in the minimization of outliers by taking the median of the resulted coefficients. In this article, a new estimator α* expressed by Eq. (8) has been proposed to denoise the ECG signal that yields better SNR. In the proposed estimator, amplitude downscaling of detail coefficients at the finest level is performed by factor 2ⁿ. The proposed estimator yields better results in terms of performance metrics than the conventional estimator validates in the results and discussions section. The mathematical expression of the proposed estimator is defined in Eq. (8). The median of resulted coefficients is computed to lessen the impact of outliers.

α^{*} = z \times median (\frac{| x_{jk} |}{2^{n}})

(8)

where 2ⁿ is amplitude downscaling factor for detailed coefficients and z is a constant, whose value depends on the different distribution functions. Further, performance parameters of the proposed estimator are evaluated for different values of n =1, 2, 3 and 4 to perceive its significant impact on the nature of the signal. The output of the proposed estimator ( $α^{*})$ is dependent on the two parameters. First, the downscaling of the signal by 2ⁿ factor to restrict the spurious high-frequency information at the output signal that resulted in successive elimination of aliasing effect. Second, the z parameter whose value is based on the different probability distribution functions (Gaussian and Normal) that enable the reduction of the transient noise from the signal.

Effect of different distribution functions on noise estimation

In statistical measurement, probability distribution represents the probability of each estimation of a sporadic variable of discrete nature. For a continuous variable, it is defined as the probability of the esteem falling inside a particular between time. The probability distribution function depicts the extent of possible characteristics that a subjective variable can achieve and the probability that the estimation of the sporadic variable is inside any subset of that run. A probability distribution gives crucial data about the information, how the qualities are changing, paying little heed to whether they are gathered or spread out and whether they are evenly masterminded on the X-hub or not.

Probability distribution functions are defined in numerous shapes with distinctive characteristics as characterized by mean, standard deviation, skewness, and kurtosis. Some of the Probability distribution functions are Gaussian, Normal and Exponential functions. In this research paper, the impact of Gaussian and Normal distribution function on noise estimator is broadly described along with their results pertaining in section 3.

Gaussian distribution function

Gaussian distribution is a continuous function that is mathematically modelled in Eq. (9)

f (x) = \frac{1}{\sqrt{2 π σ^{2}}} e^{\frac{- {(x - a)}^{2}}{2 σ^{2}}}

(9)

Where a is the mean and σ is the standard deviation of Gaussian distribution.

Normal distribution function

The probability density function (pdf) of Gaussian distribution represents the Normal distribution defined by Eq. (10)

\int_{- \infty}^{\infty} g e^{\frac{- {(x - c)}^{2}}{2 s^{2}}} d x = g s \sqrt{2 π}

(10)

For Standardized Normal distribution the value of mean (µ = 0) and variance (σ² = 1) and is expressed by Eq. (11)

p (x) = \frac{1}{\sqrt{2 π}} e^{\frac{- x^{2}}{2}}

(11)

A normal distribution is frequently utilized as a principal approximation to depict real-valued random variables that cluster around a single mean value. The normal distribution is considered the most prominent probability distribution than other distributions for specific reasons. First, the normal distribution emerges from the central limit theorem, which states that under good conditions, the mean of a large number of random variables autonomously drawn from the same distribution is distributed approximately normally, independent of the shape of the original distribution. Second, the normal distribution is exceptionally tractable systematically, that is, a large number of outcomes involving this distribution can be determined in explicit form.

Performance evaluation metrics

Different performance evaluation parameters are evaluated to analyze the efficiency of the proposed methodology. In this context, Performance metrics such as SNR, MSE, PRD and PSNR are computed in this research work using their mathematical expressions.

SNR is defined by Eq. (12)

SNR = 10 \log (\frac{S}{N})

(12)

where S is the signal value and N is Noise value.

The mathematical expression of MSE is defined by Eq. (13).

MSE = \frac{1}{N} \sum_{n = 1}^{N} {[x (n) - y (n)]}^{2}

(13)

Similarly, PRD and PSNR are defined by Eq. (14) and Eq. (15).

PRD = \sqrt{\frac{\sum_{n = 1}^{N} {[x (n) - y (n)]}^{2}}{\sum_{n = 1}^{N} x {(n)}^{2}}}

(14)

where x(n) is the original signal and y(n) is the reconstructed signal.

PSNR = 10 * \log (\frac{Max i^{2}}{MSE})

(15)

where Maxi² is the maximum amplitude of the signal and MSE is the mean square error.

Results and discussion

In this paper, different thresholding functions and values were examined on different ECG signal using the proposed estimator. The results showed in this section are well validating our proposed approach. The experiments were conducted on the ECG data from MIT-BIH arrhythmia database. This database is used for the detection and evaluation of arrhythmia for basic research in cardiac dynamics. The database includes 96 excerpts of two-channel ambulatory ECG recordings, obtained from 47 subjects studied by the MIT-BIH Arrhythmia Laboratory between 1975 and 1979. The duration of each record is 30 min and 5.556 seconds. This paper comprises the processing of ECG signals using filtering method, extraction of detail and approximate coefficient using DTCWT, exploring various γ using different f_n for various distribution functions and reconstruction of the signal using IDTCWT. The raw input ECG signal is shown in Figure 3 which is later fed to the preprocessing stage to remove baseline wander noise from the raw signal as shown in Figure 4.

In the second stage, detail and approximate coefficients are extracted up to the fourth decomposition level using the DTCWT technique as shown in Figures 5 and 6 respectively. In Figure 5, we obtain 768 detail coefficients at level 1 followed by 384, 192 and 96 coefficients at consecutive levels (2, 3 and 4). The approximation coefficient is a low-frequency coefficient extracted at 4 levels of multilevel decomposition. It consists of 96 coefficients which are least affected by noise as shown in Figure 6. All the simulations were carried out in MATLAB R2016 a software in Intel (R) Core i5 processor, 64-bit operating system.

In this research paper, we have applied different threshold value selection (universal threshold, exponential threshold level-dependent, universal modified threshold level-dependent, universal threshold level-dependent, minimax threshold, modified unified threshold, adaptive threshold selection, heursure threshold selection) and threshold functions (hard, soft, non-negative garrote, hyperbolic, trimmed, semisoft) to obtain denoised signal. This has been done by using a conventional estimator (α) and the proposed estimator (α*). Extensive empirical analysis has been performed with different threshold values on varying the threshold functions for various ECG signals. However, the results of all MIT-BIH arrhythmia signals are depicted in this research work.

Results using conventional estimator technique

The thresholding technique is applied to all the detail coefficients using a conventional estimator and different threshold functions and values. Performance evaluation parameters are calculated in terms of SNR and are tabulated in Table 1.

Table 1.

SNR evaluation of various threshold value selection and threshold function using conventional estimator.

S.No	Threshold function	Hard	Soft	Non-negative garrote	Hyperbolic	Trimmed	Semi-soft
1	γ _u	28.77	26.76	28.33	14.09	24.31	10.15
2	γ _expld	34.96	28.788	42.59	14.23	26.27	10.07
3	γ _mini	28.21	25.38	27.49	14.15	25.36	10.35
4	γ _mou	30.81	28.22	30.81	14.13	24.98	10.06
5	γ _Heur	26.86	22.55	25.9	13.91	24.31	10.78
6	γ _adp	37.45	31.09	34.93	14.2	27.48	10.05
7	γ _uld	26.86	22.55	25.9	13.91	21.67	10.78
8	γ _umld	43.62	38.31	45.67	14.23	26.9	9.90

The use of bold defines the best value from all other values in the table.

Among all, the best results are obtained using non-negative garrotte threshold function with a universal modified level-dependent threshold value selection having high SNR value. Figure 7 shows the output waveform generated with the combination of different threshold values and threshold functions to produce the denoised ECG signal.

All threshold functions shrink detail coefficients which are greater than the universal threshold tends towards zero except hard threshold. The garrote threshold function shrinks to zero fewer number of detail coefficients; it accommodates more detail coefficients. This accounts for the garrotte function having the highest gain. The shrinkage in soft threshold is related to the noise level µ whereas the shrinkage in other functions is related to noise variance µ². Furthermore, the shrinkage in the soft threshold has a big bias due to the shrinkage of large coefficients which makes it much less sensitive. This accounts for the soft threshold having the least gain. The non-negative garrote estimator has a piecewise linear solution path and is a scaled version of the least square estimate. This estimator is consistent, flexible and eases of computation.

Results with the proposed estimator technique

Real data sets usually contain a fraction of outliers and other contaminations. The conventional estimator often gives misleading results and is much affected by the outliers. Robust methods are designed to consider the majority of the data rather than all the data. Therefore, robust methods give reasonable results even when data contain a fraction of outliers. To achieve robustness and computational efficiency, a new estimator is proposed. The conventional estimator uses a non-robust estimator as the building blocks. In this research paper, the new robust estimator is proposed and analysed when applied to the detail coefficients obtained by DTCWT. The output of the signal is further evaluated for standard distribution functions with distinct values of n as shown in Figure 8. The results are validated by the performance metrics.

Performance evaluation parameters are evaluated for the different threshold functions and threshold values. Table 2 displays the SNR outcome using the proposed estimator technique.

Table 2.

SNR evaluation of various threshold value selection and threshold function using proposed estimator for n = 1 (Gaussian distribution).

Threshold		ECG samples
Functions	Value selection	100	101	102	103	104	105	106	107
Hard	γ _u	33.21	36.19	33.98	31.50	35.26	28.64	25.44	40.67
	γ _uld	33.76	36.67	34.33	31.92	35.66	29.18	25.84	41.15
	γ _umld	46.27	49.90	47.12	48.13	47.41	47.39	32.81	42.68
	γ _expl	41.00	45.97	43.64	41.16	43.84	39.43	25.66	42.57
	γ _min	30.78	33.92	32.56	30.60	33.16	27.34	18.30	37.32
	γ _mon	38.62	41.20	38.52	36.63	40.21	33.86	29.52	45.75
	γ _adp	44.76	44.80	48.89	38.54	49.33	51.11	44.27	44.45
	γ _Heur	28.34	31.38	30.18	27.96	30.95	22.36	14.01	33.75
Soft	γ _u	26.98	31.94	30.14	28.56	31.34	25.15	18.57	36.51
	γ _uld	29.31	32.17	30.36	28.74	31.57	25.35	18.88	36.75
	γ _umld	44.17	52.07	50.56	48.71	51.26	45.55	35.59	51.62
	γ _expl	33.50	36.95	35.13	32.84	35.13	30.36	18.28	35.18
	γ _min	27.91	30.83	29.30	27.66	30.06	24.37	14.03	32.51
	γ _mon	32.02	34.52	32.56	30.79	33.83	27.55	21.65	39.13
	γ _adp	35.97	36.84	38.09	31.15	40.33	39.34	28.55	36.12
	γ _Heur	25.74	28.71	26.80	25.42	27.49	24.58	11.84	28.93
Non-negative garrote	γ _u	30.50	33.49	31.81	29.72	32.90	26.38	21.50	38.42
	γ _uld	27.74	30.63	29.35	27.29	30.25	23.68	12.76	31.93
	γ _umld	54.98	58.22	55.43	53.03	54.61	50.99	40.83	55.20
	γ _expl	38.04	42.26	40.72	37.86	40.87	35.35	21.67	39.88
	γ _min	29.35	32.22	31.10	29.11	31.85	25.65	15.51	35.47
	γ _mon	34.40	37.20	35.01	32.87	36.37	29.86	25.69	41.93
	γ _adp	41.01	41.64	44.37	35.09	46.86	46.45	37.95	41.15
	γ _eur	27.74	30.63	29.35	27.29	30.25	23.68	12.76	31.93
Hyperbolic	γ _u	14.16	12.83	14.05	7.46	17.47	16.12	11.90	19.60
	γ _uld	14.08	12.81	13.98	7.43	17.33	15.76	10.32	19.40
	γ _umld	14.23	12.87	14.11	7.48	17.57	16.42	12.36	19.65
	γ _expl	14.21	12.86	14.10	7.47	17.53	16.35	11.94	19.56
	γ _min	14.14	12.83	14.04	7.45	17.44	16.08	11.09	19.50
	γ _mon	14.20	12.85	14.08	7.47	17.52	16.28	12.14	19.63
	γ _adp	14.22	12.86	14.11	7.47	17.56	16.42	12.34	19.59
	γ _Heur	14.08	12.81	13.98	7.43	17.33	15.76	10.32	19.40
Trimmed	γ _u	25.32	29.01	24.73	25.57	26.08	22.14	15.19	31.73
	γ _uld	23.68	27.71	23.33	25.12	24.39	20.63	11.44	28.00
	γ _umld	27.27	31.07	26.19	27.55	26.31	24.36	17.25	33.24
	γ _expl	26.25	30.31	25.42	26.79	26.61	23.24	14.84	31.23
	γ _min	24.82	28.64	24.41	25.52	25.59	21.74	12.92	30.11
	γ _mon	26.06	29.74	25.30	26.02	26.67	22.92	16.15	32.41
	γ _adp	26.65	30.36	25.79	26.79	27.25	24.31	17.20	31.51
	γ _eur	23.68	27.71	23.33	25.12	24.39	20.63	11.44	28.00
Semi-soft	γ _u	10.02	10.00	11.20	4.46	14.81	15.49	12.45	16.83
	γ _uld	10.26	10.12	11.52	4.58	15.06	16.76	18.80	16.86
	γ _umld	9.87	9.92	11.00	4.41	14.34	14.68	11.19	16.60
	γ _expl	9.97	9.94	11.08	4.45	14.57	15.19	13.55	16.60
	γ _min	10.14	10.03	11.24	4.50	14.77	15.65	15.20	16.74
	γ _mon	9.98	9.96	11.16	4.44	14.63	15.06	11.57	16.71
	γ _adp	9.89	9.94	11.04	4.46	14.44	14.65	11.09	16.62
	γ _Heur	10.26	10.12	11.52	4.58	15.06	16.76	18.80	16.86

ECG: electrocardiographic

Table 2 shows the SNR values of a combination of different threshold value selection and threshold functions are evaluated to achieve the optimum threshold tuning. Analysis of eight different set of threshold value selection (universal threshold, exponential threshold level-dependent, universal modified threshold level-dependent, universal threshold level-dependent, minimax threshold, modified unified threshold, adaptive threshold selection, heursure threshold selection) with six distinct threshold functions (hard, soft, non-negative garrote, hyperbolic, trimmed, semi-soft) observed that universal modified threshold level-dependent threshold with non-negative garrote threshold function well suited for denoising.

This work is further extended to evaluate the performance metrics of the conventional and proposed estimator with Gaussian and Normal distribution function utilizing the universal modified threshold level-dependent threshold with non-negative garrotte threshold function for 100–107 records tabulated in Table 3. After performing amplitude downscaling, the median of resulted coefficients was considered to minimize the impact of outliers.

Table 3.

SNR Evaluation for 100–107 records with a proposed estimator for n =1 for Gaussian and Normal distribution.

S.No.	MIT-BIH arrhythmia record	SNR (dB)(Conventional)	SNR (dB)(Gaussian)	SNR (dB)(Normal)
1	100	45.67	54.98	58.23
2	101	48.62	58.22	61.33
3	102	47.15	55.43	59.34
4	103	44.65	58.73	65.69
5	104	46.84	62.09	65.92
6	105	41.95	69.77	71.23
7	106	30.91	52.23	54.27
8	107	48.04	55.92	57.11

Comparative table of performance metrics with conventional and proposed estimator techniques using different distribution functions (Gaussian distribution and Normal distribution) for the efficient results is tabulated in Table 3 which shows a remarkable performance. The result shows that non-negative garrote function with universal threshold level-dependent value selection yields more desirable results in terms of SNR using the proposed estimation technique (α^*) than conventional estimator (α). On comparing the output values of SNR for different distributions with the conventional and proposed estimator, normal distribution function using the proposed estimator gives the 58.23 dB as compared to 54.98dB with the Gaussian distribution function.

The effect of different distribution functions (Gaussian, and Normal) on the conventional and proposed estimator technique is tabulated in Table 4. The results in Table 4 are obtained by putting the distinct value of constant (z) for each distribution function in Eq. (7) and Eq. (8). The proposed estimator distribution gives significant results in terms of MSE and PRD. These parameters are calculated and compared to validate the previously obtained SNR results.

Table 4.

Comparison table of performance metrics using different distribution functions.

Non negative Garrote with universal modified threshold level dependent	MSE		PRD
	Conventional	Proposed	Conventional	Proposed
Gaussian distribution	1.73e-6	2.03e-07	0.005	0.003
Normal distribution	8.75e-07	9.63e-08	0.003	0.001

The non-negative garrotte thresholding is propelled with characteristics that with probability tending to one, the solution path having an estimate that correctly identifies the set of prime variables and is reliable for the coefficients of the prime variables. Such property is substantial for another desired variable selection method. Simulation results demonstrate that our proposed strategy is well focused regarding the signal denoising as shown in Figure 9.

The outcomes obtained by our proposed estimator are compared with the existing research techniques for ECG signal denoising and are tabulated in Table 5. The proposed technique is executed on different records of MIT-BIH arrhythmia. The outcome demonstrates the remarkable increment in SNR utilizing the proposed method with Gaussian and Normal distribution functions.

Table 5.

Comparison of the SNR (dB) using 100 to 107 records for MIT-BIH arrhythmia.

Record	Charriet al. (2017)DWT	Charriet al. (2017)DT-WT	Proposed techniqueDTCWT(Gaussian)	Proposed techniqueDTCWT(Normal)
100	42.65	45.83	54.98	58.23
101	48.41	49.39	58.22	61.33
102	45.49	49.87	55.43	59.34
103	52.20	55.41	58.73	65.69
104	54.13	57.51	62.09	65.92
105	62.82	66.18	69.77	71.23
106	47.45	51.89	52.23	54.27
107	54.57	55.52	55.92	57.11

DTCWT: dual tree complex wavelet transforms; DWT: discrete wavelet transform; DT-WT: Dual tree wavelet transform.

Further, to validate and improvise the proposed estimator using (n=1) for Normal and Gaussian distributions, the performance metric SNR was calculated for all the records (96 records) of MIT-BIH arrhythmia database constituted of different leads as shown in Figure 10 and Figure 11.

The proposed estimator using (Normal and Gaussian) probability distribution functions give better SNR as it successively reduces the ringing effect in the form of transient noise as the reaction to a single impulse visible as R waves on the ECG signals that oscillate in the filtered signal.

From Figures 10 and 11, the best value of SNR (70.14) is obtained for MLII 124 record of MIT-BITH arrhythmia database using normal distribution corresponding to SNR (66.60) for the same record using Gaussian distribution similarly, the least value of SNR (39.79) is obtained for V1 209 record using normal distribution corresponding to SNR (36.77) for the same record. The obtained results signify that the best denoising of the signal is achieved by the proposed estimator using normal distribution in comparison with Gaussian distribution.

Proposed noise estimator for distinct values of 2ⁿ using normal distribution

From the above empirical results, it has been observed that the universal modified threshold level-dependent threshold with non-negative garrote threshold function yields the better result in terms of SNR and MSE with n =1 for Normal distribution. Regarding the proposed estimator as in Eq. (8) the effect of downscaling factor 2ⁿ is applied to the chosen threshold function. Table 6 tabulates the impact of the proposed noise estimator downscaled by the 2ⁿ factor for distinct value n on performance parameters for the normal distributions.

Table 6.

Proposed noise estimator downscaled by 2ⁿ using the normal distribution.

S.No.	2ⁿ Factor	SNR	MSE	PRD	PSNR
1.	2¹	58.23	9.64e-08	0.001	70.160
2.	2²	68.75	8.566e-09	3.65e-04	80.68
3.	2 ³	80.72	5.45e-10	9.21e-05	92.63
4.	2⁴	92.75	3.41e-11	2.30 e-05	104.68

From Table 6 it has been observed that implementing amplitude downscaling factor 2ⁿ for n varies from 1 to 3 there has been a considerable improvement in the quality of the signal by incrementing the value of n but for value n as 4 the range of PSNR exceeds the normal range (1 to 100 dB) which indicates that the originality of signal vanishes.

On assessing the outcomes of the proposed estimator with 2³ factor using Normal distribution function that demonstrated high SNR (80.720dB), small MSE (5.45e-10), low PRD (9.21e-05) and sharp PSNR (92.63dB) as compared to the performance of proposed estimator with 2¹ factor having SNR (58.23dB), small MSE (9.64e-08), PRD (0.001) and PSNR (70.160dB). These performance parameters signify the considerable reduction of noise by subsequently improving the quality of the signal with 38.63% SNR improvement as shown in Table 6. For further performance validation of our proposed estimator with 2³ factors, authors have evaluated the PSNR values for 100–107 records for different distribution functions in Table 7. The results show that the best PSNR value (93.24) is obtained for 105^th record using normal distribution function.

Table 7.

PSNR evaluation of proposed estimator using 2³ factor for distinct MIT-BIH arrhythmias databases.

Database	100	101	102	103	104	105	106	107
PSNR (dB) (Gaussian distribution)	88.81	86.54	87.42	86.57	85.37	89.47	82.78	74.61
PSNR (dB)(Normal distribution)	92.63	89.04	91.23	90.35	89.11	93.24	85.34	77.70

Table 7 shows the PSNR values of 100 to 107 records taking the proposed estimator with different distribution functions. For all cases, the best results are obtained by the proposed estimator using a 2³ factor with normal distribution function.

Table 7 shows the evaluation of PSNR for (100–107) records of MIT-BIH arrhythmia database using both Gaussian and normal distribution for a proposed estimator. The result shows the significant improvement in the quality of signals using Normal distribution with the best PSNR value (93.24) as compared to Gaussian distribution having PSNR values as 89.47.

Comparison with existing research paper

The proposed technique is further compared with the previous research papers as shown in Table 8.

Table 8.

Comparison of our proposed estimator with existing techniques.

S.No.	Authors	Techniques	SNR (dB)	MSE	PRD
1.	Zubair AR et al.,2018)	DWT + Garrote	42.50	-	-
2.	(Prashar N. et al.,2019)	Cascade Digital Filter	2.2226	-	-
3.	(Kaur H and Rajini.,2016)	DWT	7.452	0.002	-
4.	(Yadav T. and Mehra R.2016)	Daubechies (DB) technique based on wavelet filtering.	52.07	-	-
5.	(Salih K.et al.,2015)	Adaptive filtering using Moving average filter.	-	4.9e–04	-
6.	(Priyaa KD et al., 2016)	Wavelet wiener filter	-	0.1440	-
7.	(Prashar N. et al., (2018(a))	DTCWT with conventional thresholding	27.750	1.077e-04	0.0410
8.	Proposed Technique	2 ¹ factor with Normal Distribution	58.23	9.63e-08	0.001
9.	Proposed Technique	2 ³ factor with Gaussian Distribution	76.890	1.313e-09	1.43e-04
10.	Proposed Technique	2 ³ factor with Normal Distribution	80.720	5.44e-10	9.21e-05

Bold values represent the output results obtained by our proposed technique.

The performance of the proposed estimator using both Gaussian and normal distribution functions is evaluated in terms of SNR, MSE, and PRD. The computational assessment of these performance metrics for the proposed estimator using both distribution functions is displayed in Table 8. On analysing the results, it has been found that our proposed technique has high SNR, smallest PRD and low MSE than other existing techniques by eliminating the aliasing and ringing effect from the ECG signal, subsequently improving the quality of signal with high PSNR value by focusing on two parameters, that is, 2ⁿ downscaling factor and z parameter in a proposed estimator. The best results are obtained by enhancing the proposed estimator with 2³ downscaling factor and z value obtained by the Normal distribution function. The result shows that the proposed estimator using normal distribution function delivers a significant improvement in signal denoising as compared to other existing methods.

Conclusion

The motivation of this research is to propose an efficient noise estimator that successively removes the noise from the ECG signal and to yield a good quality of the signal. The main objective of this research is to incorporate the best combination of the threshold value and its function using the proposed estimator for different distribution functions to reduce the noise present in ECG signals. Unlike, conventional noise estimation method, a proposed noise estimator technique is applied for the rejection of high-frequency noises equally preserving their complexes. This methodology is effectively executed utilizing universal modified threshold level-dependent threshold value selection with a non-negative garrote threshold function. Distinct performance parameters SNR, PSNR, MSE, and PRD are considered for accessing the proficiency of the proposed methodology. This objective has been achieved by our proposed estimator that delivers high performance by focusing on two parameters, that is, 2ⁿ downscaling factor and z value that depends on different distribution functions. The distinct performance metrics are computed such as SNR, MSE and PRD that signify the denoising procedure of the signal by implementing the finer downscaling factor and z value of best distribution function and collaterally examining the quality of the signal by evaluating the PSNR value. The best results are obtained by implementing a proposed estimator with 2³ downscaling factor using Normal distribution that yields a high SNR, 80.72 dB; Sharp PSNR, 92.63dB, low MSE 5.45e-10 and low PRD 9.21e-05 suggest that the proposed noise estimator approach outflanks the existing techniques. An in-depth inspection of qualitative and quantitative analysis recommends that the proposed noise estimator approach will be beneficial for a computer-based automated diagnostic system. On comparing the performance metrics of the proposed estimator with other existing methods, the proposed estimator outperforms in all aspects than other methods.

Footnotes

Authors’ Note

Meenakshi Sood is working as Associate Professor with National Institute of Technical Teachers Training & Research, Chandigarh, MHRD, India.

Declaration of conflicting interest

The authors declare that there is no conflict of interests with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Shruti Jain

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Dual-tree complex wavelet transform technique-based optimal threshold tuning system to deliver denoised ECG signal

Abstract

Keywords

Introduction

Problem formulation

Contribution

Methodology

Filtering stage

DTCWT technique

Thresholding techniques

Threshold functions

Threshold value

Proposed estimator

Effect of different distribution functions on noise estimation

Gaussian distribution function

Normal distribution function

Performance evaluation metrics

Results and discussion

Results using conventional estimator technique

Results with the proposed estimator technique

Proposed noise estimator for distinct values of 2n using normal distribution

Comparison with existing research paper

Conclusion

Footnotes

Authors’ Note

Declaration of conflicting interest

Funding

ORCID iD

References

Proposed noise estimator for distinct values of 2ⁿ using normal distribution