ECG Denoising Based on Total Variation Denoising and Wavelets 3031252667, 9783031252662

Table of contents :
Contents
About the Author
Acronyms
1 Wavelets and Wavelet Transforms
1.1 Introduction
1.2 The Fourier Transform
1.3 Short-Term Fourier Transform (STFT)
1.4 The Wavelets
1.5 Continuous Wavelet Transform (CWT)
1.6 Discrete Wavelet Transform (DWT)
1.7 Wavelet Packet
1.8 Full Wavelet Packet Decomposition
1.8.1 Adaptive Wavelet Packet Systems
1.9 Conclusion
References
2 A Denoising Technique Based on SBWT and WATV: Application for ECG Denoising
2.1 Introduction
2.2 The Stationary Bionic Wavelet Transform (SBWT)
2.3 The WATV-Based Denoising Method
2.3.1 Problem Formulation
2.4 The Proposed ECG Denoising Technique
2.5 Results and Discussion
2.6 Conclusion
References
3 An ECG Denoising Technique Based on LWT and TVM
3.1 Introduction
3.2 The Lifting Wavelet Transform (LWT)
3.3 The Total Variation Minimization (TVM)
3.4 The ECG Denoising Technique Proposed in [19]
3.5 The 1D Double-Density Complex DWT Denoising Method [24]
3.6 The Denoising Approach Based on Non-local Means (NLM) [25, 26]
3.7 Results and Discussion
3.8 Conclusion
References
Index

Citation preview

Synthesis Lectures on Biomedical Engineering

Talbi Mourad

ECG Denoising Based on Total Variation Denoising and Wavelets

Synthesis Lectures on Biomedical Engineering Series Editor John Enderle, Storr, USA

This series consists of concise books on advanced and state-of-the-art topics that span the field of biomedical engineering. Each Lecture covers the fundamental principles in a unified manner, develops underlying concepts needed for sequential material, and progresses to more advanced topics and design. The authors selected to write the Lectures are leading experts on the subject who have extensive background in theory, application, and design. The series is designed to meet the demands of the 21st century technology and the rapid advancements in the all-encompassing field of biomedical engineering.

Talbi Mourad

ECG Denoising Based on Total Variation Denoising and Wavelets

Talbi Mourad Laboratory of Nanomaterials and Systems for Renewable Energies (LaNSER) Center of Researches and Technologies of Energy of Borj Cedria Tunis, Tunisia

ISSN 1930-0328 ISSN 1930-0336 (electronic) Synthesis Lectures on Biomedical Engineering ISBN 978-3-031-25266-2 ISBN 978-3-031-25267-9 (eBook) https://doi.org/10.1007/978-3-031-25267-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

1 Wavelets and Wavelet Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Short-Term Fourier Transform (STFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Continuous Wavelet Transform (CWT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Discrete Wavelet Transform (DWT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Wavelet Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Full Wavelet Packet Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Adaptive Wavelet Packet Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 6 7 10 10 14 17 17

2 A Denoising Technique Based on SBWT and WATV: Application for ECG Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Stationary Bionic Wavelet Transform (SBWT) . . . . . . . . . . . . . . . . . . . . 2.3 The WATV-Based Denoising Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Proposed ECG Denoising Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 22 22 23 25 26 36

3 An ECG Denoising Technique Based on LWT and TVM . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Lifting Wavelet Transform (LWT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Total Variation Minimization (TVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The ECG Denoising Technique Proposed in [19] . . . . . . . . . . . . . . . . . . . . . . 3.5 The 1D Double-Density Complex DWT Denoising Method [24] . . . . . . . .

39 39 40 41 42 43

v

vi

Contents

3.6 The Denoising Approach Based on Non-local Means (NLM) [25, 26] . . . . 3.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 46 52 52

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

About the Author

Talbi Mourad is an Associate Professor of Electrical Engineering (Signal Processing) at the Center of Researches and Technologies of Energy of Borj Cedria, Tunis, Tunisia. He obtained his master’s degree in automatics and signal processing from the National Engineering School of Tunis in 2004. He obtained his Ph.D. in electronics from the Faculty of Sciences of Tunis, Tunis El-Manar University, and his HDR in electronics from the Faculty of Sciences of Tunis.

vii

Acronyms

AGWN ANN ASMF BW BWT CC CCA CWT dB DWT ECG EEMD EMG EMI fBm FT FWT_TI ICA IMFs LDA LMS LWT LWT−1 MAD MAE MN MSE NLM PLI PSD

Additive Gaussian White Noise Artificial Neural Networks Adaptive Switching Mean Filter Baseline Wander Bionic Wavelet Transform Cross-Correlation Canonical Correlation Analysis Continuous Wavelet Transform Decibel Discrete Wavelet Transform Electrocardiogram Ensemble Empirical Mode Decomposition Electro-Myographic Electromagnetic Interferences fractional Brownian motions Fourier Transform Translation-Invariant Forward Wavelet Transform Independent Component Analysis Intrinsic Mode Functions Linear Discriminant Analysis Least Mean Square Lifting Wavelet Transform Inverse of LWT Median Mean Absolute Error Motion and Noise Mean Square Error Non-Local Means Powerline Interference Power Spectral Density ix

x

PSNR PWVD SBWT SBWT−1 SNR STFT SWT SWT-1 TVM WATV WT WVD σ

Acronyms

Peak Signal to Noise Ratio Pseudo Wigner–Ville distribution Stationary Bionic Wavelet Transform Inverse of Stationary Bionic Wavelet Transform Signal to Noise Ratio Short-Term Fourier Transform Stationary Wavelet Transform Inverse of Stationary Wavelet Transform Total Variation Minimization Wavelet/Total Variation Wavelet Transform Wigner–Ville distribution Standard Deviation

1

Wavelets and Wavelet Transforms

1.1

Introduction

One collecting manner of experimental data by engineers and scientists is as sequences of values at regularly spaced intervals in time. These sequences are named time-series. The fundamental problem with the data in the form of time-series is how to process them for extracting meaningful and correct information, i.e., the possible signals embedded in them. If a time-series is stationary, one can think that it can have harmonic components that can be detected by applying the Fourier analysis, i.e., Fourier Transform (FT). Although, it is evident that many time-series are not stationary and their mean properties are variables over time. The waves of infinite support that form the harmonic components are not adequate in the latter case in which one needs waves localized not only in frequency but in time too. They named wavelets and permit a time-scale decomposition of a signal. Considerable progress in understanding the wavelet processing of non-stationary signals was attained. Although, for getting the dynamics that provides a non-stationary signal, it is crucial that in the corresponding time-series a correct separation of the fluctuations from the average behavior, or trend, is performed. Therefore, people should invent new statistical techniques for detrending the data that have to be combined with the wavelet analysis [1]. A bunch of such methods was developed lately for the important class of non-stationary time-series that display multi-scaling behavior of the multi-fractal type. In the rest of this chapter, we will first deal in Sect. 1.2 with Fourier Transform (FT). Then, in Sect. 2, we will deal with Short-Term Fourier Transform (STFT). After that, in Sect. 1.3, we will deal with Wavelets, and in Sect. 1.4, we will deal with Continuous Wavelet Transform (CWT). Then, in Sect. 1.5, we will deal with Discrete Wavelet Transform (DWT). Finally, in Sect. 1.6, we will conclude.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Mourad, ECG Denoising Based on Total Variation Denoising and Wavelets, Synthesis Lectures on Biomedical Engineering, https://doi.org/10.1007/978-3-031-25267-9_1

1

2

1.2

1 Wavelets and Wavelet Transforms

The Fourier Transform

The signal processing task consists in finding the traits of the interest signal. It is well known that the majority of the signals in engineering are provided in the time domain. Although, features of those signals can often be interpreted in the frequency domain, so that the frequency-domain analysis is very important in the signal analysis [1]. Both FT and its inverse permit to connect the time-domain features with the frequency-domain features [1]. These two transforms are expressed as follows: +∞ X( f ) = x(t)e− j2π f t dt

(1.1)

−∞

+∞ x(t) =

X ( f )e j2π f t d f

(1.2)

−∞

In the stationary signal analysis, one can employ the FT and its inverse to establish the mapping relation between both the frequency domain and the time domain. Although, in the practical applications, the FT is not the best tool for non-stationary signal analysis due to the non-stationary and time-varying feature in the most engineering signals, such as noise and engine vibration signals. For those signals, though their frequency elements can be observed from their frequency spectrum, the time of frequency occurrence and frequency change relationship over time cannot be acquired. For further research on those signals, the time–frequency descriptions are introduced. In Fig. 1.1 are illustrated three time–frequency descriptions of the linear frequency modulation signal: (a) the frequencydomain description which loses the time information; (c) is the time-domain description which loses the frequency information; (b) is the time–frequency description which shows the change rule of frequency over time clearly. The main idea of time–frequency analysis consists in developing a joint function for combining the frequency and time factors. The time–frequency analysis, which can describe the signal traits on a time–frequency plane, has become an important research domain. A great number of time–frequency techniques were presented, which can be divided into three sorts which are quadratic, linear, and nonlinear [1]. Both Short-Term Fourier Transform (STFT) and Wavelet Transform (WT) [3–5] belong to the linear type, and the Wigner–Ville distribution (WVD) and pseudo-Wigner–Ville distribution (PWVD) belong to the quadratic type. In the following section, we will deal with the STFT.

1.3

Short-Term Fourier Transform (STFT)

3

Fig. 1.1 Three description techniques of linear frequency modulation signal [1]

1.3

Short-Term Fourier Transform (STFT)

The main idea of STFT, which is presented by Gabor in 1946, consists in cutting out the signal by a window function, in which the signal can be considered as stationary, and analyze the signal to make sure the frequency elements in the window by the FT then move the window function along the time axis in order to obtain the change in the relationship of frequency over time. This is time–frequency analysis process of STFT and the STFT of the signal x(t) is formulated:       STFTx (t, f ) = x t  g ∗ t  − t e− j2π f t dt  (1.3) where g is the short window function. The WVD presented by Wigner in the research of quantum mechanics in 1932 and applied to signal processing by Ville later owns many mathematical properties expected by time–frequency analysis. The WVD of a signal x(t) is expressed as follows [1]:  WVDx (t, f ) = x(t + τ/2) · x(t − τ/2)e− j2π f τ dτ (1.4)

4

1 Wavelets and Wavelet Transforms

Fig. 1.2 The oscillogram of a signal containing four Gauss components [1]

For eliminating the disturbing of the cross term in the WVD, the PWVD that is equivalent to smooth the WVD is proposed. The PWVD of the signal x(t) is formulated as follows [1]:  PWVDx (t, f ) = h(t) · x(t + τ/2) · x ∗ (t + τ/2) · e− j2π f τ dτ (1.5) In Fig. 1.2 is illustrated the oscillogram of a signal containing four Gauss components. The four time–frequency analysis results obtained by applying the four previously mentioned transforms (STFT, WVD, PWVD) are illustrated in Fig. 1.3, where (a), (b), (c), and (d) denote the results of STFT, WVD, PWVD, and WT, respectively. As shown in Fig. 1.3a, the resolution of the STFT is lower and fixed. However, both the WVD and PWVD have higher resolution and time–frequency concentration, and they are disturbed strictly by cross terms as illustrated in Figures. 1.3b and c. Moreover, the resolution of WT is higher than STFT and can change with frequency. There are a good frequency resolution in the low-frequency range and a good time resolution in the highfrequency range. The cross terms in WVD and PWVD disappear. Although the STFT covers the shortage of the FT to some extent in local analysis, its defects can not be overcome. That is, when the window function is determined, the size of windows is fixed and the time resolution and frequency resolution are fixed. As the resolution of the window function is restricted by the Heisenberg uncertainty principle, the frequency resolution is higher and the time resolution is lower when a long window is u, and the situation is reversed when a short window is used. Therefore, the key to the application is how to choose a reasonable window length. In the case of a signal containing a variety of differences in scales, it is not useful to use STFT for analyzing this signal. In the following two sections, we will deal, respectively, with wavelets and wavelet transforms.

1.4 The Wavelets

5

Fig. 1.3 Four time–frequency representations [1]

1.4

The Wavelets

A wave is defined as an oscillating function of space or time, such as a sinusoid. Fourier analysis is wave analysis [6]. It expands signals in terms of sinusoids (or, equivalently, complex exponentials) which has proven to be extremely valuable in mathematics, science, and engineering, precisely for periodic, time-invariant, or stationary phenomena [6]. A wavelet is a «small wave», having its energy concentrated in time to give a tool for the analysis of transient, non-stationary, or time-varying phenomena. It still has the oscillating wave-like characteristic but also has the ability to allow simultaneous time and frequency analysis with a flexible mathematical foundation. This is illustrated in Fig. 1.1 with the wave (sinusoid) oscillating with equal amplitude over −∞ ≤ t ≤ ∞ and, therefore, owning infinite energy and with the wavelet owning its finite energy concentrated around a point [6]. The wavelets are used in a series expansion of signals much the same way as a Fourier series employing the wave or sinusoid for representing a signal [6].

6

1 Wavelets and Wavelet Transforms

Fig. 1.4 a A sine wave and b wavelet: Daubechies wavelet ψ D20 [6]

1.5

Continuous Wavelet Transform (CWT)

The Fourier analysis consists in breaking up a signal into sine waves with diverse frequencies. Similarly, a wavelet analysis is the breaking up of a signal into shifted and scaled versions of the function named the “mother wavelet”. The Continuous Wavelet Transform (CWT) is the sum over time of the signal multiplied by scaled and shifted versions of the mother wavelet. This process produces wavelet coefficients that are a function of scale and position. The integral wavelet transform of a signal x(t) ∈ L 2 with respect to an analyzing wavelet φ is expressed as follows: +∞ Wφ x(b, a) = x(t)φb,a (t)dt −∞

where φb,a is expressed as follows:

(1.6)

1.6

Discrete Wavelet Transform (DWT)

7

  1 t −b , a>0 φb,a (t) = √ φ a a

(1.7)

Both a and b are, respectively, dilation (or contraction) and translation. The normal√ ization factor a is included so we have   φb,a  = φ (1.8) The formula of the inverse wavelet transform is given as follows: 1 x(t) = Cφ

+∞ +∞ −∞ −∞

1 Wφ x(b, a) φb,a (t)dadb 2 a

(1.9)

where Cφ is a constant depending on the selection of the mother wavelet and is expressed as follows:

2





φ(w) ˆ

dw < ∞ (1.10) Cφ = |w| where φˆ is the Fourier Transform of φ.

1.6

Discrete Wavelet Transform (DWT)

The Discrete Wavelet Transform (DWT) consists in sampling the scaling and shifted parameters, though neither the signal nor the transform. This leads to high-frequency resolution at low frequencies and high-time resolution for higher frequencies, with the same time and frequency resolution for all frequencies. A discrete signal x(n) can be decomposed as follows [7]: x(n) =

 k

a j0 ,k φ j0 ,k (n) +

J −1  

d j ,k ϕ j ,k (n)

(1.11)

j= j0 k

where φ(n) designates the scaling function, and we have

j0 φ j0 ,k (n) = 2 / 2 φ 2 j0 n − k

(1.12)

The function φ j0 ,k (n) is the scaling function at a scale of s = 2 j0 shifted by k. The coefficients a j0 ,k are the approximation coefficients at a scale of s = 2 j and d j ,k are the detail coefficients at a scale of s = 2 j . The function ϕ(n) is the mother wavelet and the function ϕ j,n (n) is the mother wavelet at a scale of s = 2 j and shifted by k and expressed as follows:

8

1 Wavelets and Wavelet Transforms

Fig. 1.5 Wavelet tree decomposition (three levels of decomposition) [1]



ϕ j,n (n) = 2 j/2 ϕ 2 j n − k

(1.12)

And N = 2 J , with N as the number of x(n) samples. The scaling function can be considered as an aggregation of wavelets at scales greater than 1. A discrete signal can be obtained by employing a sum of J − j0 details and an approximation to 1 of a signal at a scale of 2 j0 . A quick manner for obtaining the forward DWT coefficients is to employ the filter bank structure shown in Fig. 1.5. The approximation coefficients at a lower level are transferred through a high-pass (h[n]) and a low-pass filter (g[n]), followed by a downsampling by 2 and this is for computing the approximation coefficients (from the low-pass filter) and the details coefficients (from the high-pass filter). The two filters are linked to each other and they are quadrature mirror filters. Low- and High-pass filters are derived from the scaling function and the mother wavelet considered, respectively, in [8, 9]. The different frequency range coverings for the details and the final approximation for a three-level decomposition are illustrated in Fig. 1.6. These are directly related to the bands where the analysis is performed. The shape of the frequency response for those filters are depending on both the sort and the order of the mother wavelet employed in the analysis. For avoiding overlapping between two adjacent frequency bands, a high-order mother wavelet should be employed that results in a high-order frequency filter. For separating the different frequency bands,

Fig. 1.6 Frequency ranges for details and approximation (at Level 3)

1.6

Discrete Wavelet Transform (DWT)

9

Fig. 1.7 Daubechies mother wavelet time evolution for increasing order [1]

there is an obvious trade-off between the order of the mother wavelet and the calculation cost. Therefore, an intensive study is required for adapting the order of the mother wavelet to the requirements. Taking a common wavelet family such as the Daubechies mother wavelet, the mother wavelet time shape shows an evolution if we just change the Daubechies order as illustrated in Fig. 1.7. In Fig. 1.8 is illustrated the frequency response for low-pass and high-pass filters, determining the detail and approximation decomposition for diverse orders. For low orders, the power of one harmonic near the cut frequency could be split into two diverse details. This could give a false impression of the time evolution of the analyzed signal’s frequency component. By increasing the Daubechies order, it is possible to idealize the filters and, therefore, obtain better frequency decomposition. In Fig. 1.9 is illustrated an example of this drawback. A test signal is built with two harmonic components, one at 100Hz and the other one at 45Hz, and the signal is sampled at 1000Hz. The wavelet analysis is made with a Daubechies db3 mother wavelet. Harmonic content due to the 100 Hz superimposed frequency appears on details 2 and 3 when it should only appear on detail 3, corresponding to the analysis band between 62.5 and 125 Hz. A high-order Daubechies mother wavelet is required for preventing this

10

1 Wavelets and Wavelet Transforms

Fig. 1.8 Low- and high-pass filter frequency response corresponding to details [1]

drawback, which is due to the db3-associated filter not being ideal enough for filtering the 100 Hz harmonic content on detail 2.

1.7

Wavelet Packet

The conventional M = 2 wavelet system is resulting in a logarithmic frequency resolution. The low frequencies own narrow bandwidths and the high frequencies own wide bandwidths, as shown in Figure: Frequency Bands for the Analysis Tree (Fig. 1.10). This is named “constant-Q” filtering and is suitable for some applications but not all. The wavelet packet system was introduced by Coifman [10, 11] to permit a finer and adjustable resolution of frequencies at high frequencies. It permits to have a rich structure too. This structure allows adaptation to particular signals or classes of signal. The cost of this richer structure is a calculation complexity of O(N · log(N )), similar to the FFT, in contrast to the classical wavelet transform which is O(N ).

1.8

Full Wavelet Packet Decomposition

For generating a basis system that permits a higher resolution decomposition at high frequencies, we will iterate (split and down-sample) the high-pass wavelet branch of the Mallat algorithm tree as well as the low-pass scaling function branch. Recall that for the DWT, we repeatedly split, filter, and decimate the low-pass bands. The resulting three-scale analysis tree (three-stage filter bank) is illustrated in Fig. 1.11: Three-Stage Two-Band Analysis Tree (Fig. 1.11). This sort of tree results in a logarithmic splitting of the bandwidths and tiling of the time-scale plane, as shown in Fig. 1.12.

1.8

Full Wavelet Packet Decomposition

11

Fig. 1.9 Test decomposition signal with an overlapping effect [1]

Fig. 1.10 Frequency Bands for the Analysis Tree [6]

If we split both the low-pass and high-pass bands at all stages, the resulting filter bank structure is like a full binary tree as in Fig. 1.13. It is this full tree taking O(N · log N ) computation and results in a completely evenly spaced frequency resolution. In fact, its structure is somewhat similar to the FFT algorithm. Notice the meaning of the sub-scripts on the signal spaces. The first integer sub-script is the scale j of that space as shown in Fig. 1.14.

12

1 Wavelets and Wavelet Transforms

Fig. 1.11 Three-stage Two-Band Analysis tree [6]

Fig. 1.12 Two-band wavelet basis [6]

Each following sub-script is a zero or one, depending on the path taken through the filter bank shown in Fig. 1.11: Three-Stage Two-Band Analysis Tree. A “zero” indicates going through a low-pass filter (scaling function decomposition) and a “one” indicates going through a high-pass filter (wavelet decomposition). In Fig. 1.14 is illustrated the signal vector space decomposition for the scaling functions and wavelets. In Fig. 1.15 is illustrated the frequency response of the wavelet packet filter bank. In Fig. 1.16 are illustrated the Haar wavelet packets: An Example of the Haar Wavelet System. This is similar to the Walsh–Haddamar decomposition, and in Fig. 1.17 is illustrated the full wavelet packet system obtained from the Daubechies ϕD8 ‘ scaling function.

1.8

Full Wavelet Packet Decomposition

Fig. 1.13 The full binary tree for the three-scale wavelet packet transform

Fig. 1.14 Vector space decomposition for a M = 2 full wavelet packet system [6]

Fig. 1.15 Frequency Responses for the Two-Band Wavelet Packet Filter Bank [6]

13

14

1 Wavelets and Wavelet Transforms

Fig. 1.16 The Haar wavelet packet [6]

The “prime” is indicating this is the Daubechies system with the spectral factorization selected so that zeros are inside the unit circle and some outside. This permits us to have the maximum symmetry possible with a Daubechies system. The three wavelets increased “frequency”. They are somewhat similar to windowed sinusoids, consequently, the name of wavelet packet.

1.8.1

Adaptive Wavelet Packet Systems

Naturally are considered the outputs of each channel or band as the wavelet transform and from this is obthave a nonredundant basis system. When, although, we consider the signals at the output of each band and at each stage or scale simultaneously, we have more outputs than inputs and clearly have a redundant system. From all of these outputs, we can choose an independent sub-set as a basis. This can be performed in an adaptive manner, depending on the signal characteristics according to some optimization criterion. One possibility is the regular wavelet decomposition illustrated in Figure: Frequency Bands for the Analysis Tree (Fig. 1.18). Another is the full packet decomposition illustrated in Fig. 1.13. Any pruning of this full tree generates a valid packet basis system and permits a very flexible tiling of the time-scale plane. We can select a set of basic vectors and generate an orthonormal basis, such that some cost measure on the transformed coefficients is minimized. Furthermore,

1.8

Full Wavelet Packet Decomposition

15

Fig. 1.17 Wavelet packets generated by ϕD8 [6] Fig. 1.18 Two-stage, two-band analysis tree [6]

if the cost is additive, the best orthonormal wavelet packet transform can be obtained employing a binary searching algorithm [11] in O(N · log N ) time. Some examples of the resulting time–frequency tilings are shown in Fig. 1.19. These plots demonstrate the frequency adaptation power of the wavelet packet transform. Two approaches exist for employing adaptive wavelet packets where the first one consists in choosing a particular decomposition (filter bank pruning) based on the characteristics of the class of signals to be processed. Then employing the transform

16

1 Wavelets and Wavelet Transforms

Fig. 1.19 Examples (a and b) of time–frequency tilings of different three-scale orthonormal wavelet packet transforms [6]

non-adaptively on the individual signals. The other consists in adapting the decomposition for each individual signal. The first is a linear process over the class of signals. The second is not and will not obey superposition. Let P(J ) denote the number of different J -scale orthonormal wavelet packet transforms. T he P(J ) is expressed as follows: P(J ) = P(J − 1)2 + 1, P(1) = 1

(1.13)

Consequently, the number of possible choices is growing dramatically as the scale increases. This is a reason for the wavelet packets to be practically a very powerful tool. As an example, the FBI standard for fingerprint image compression [12, 13] is based on wavelet packet transforms. The wavelet packets are successfully employed for acoustic signal compression [14]. In [15], a rate-distortion measure is employed with the wavelet packet transform for improving image compression performance. M-band DWTs permit to give a flexible tiling of the time–frequency plane. They are associated with a particular tree-structured filter bank, where the low-pass channel at any depth is split into M bands. The combination of the wavelet packet structure and M-band permits to have a rather arbitrary tree-structured filter bank, where all channels are split into sub-channels (employing filter banks with a potentially diverse number of bands), and provide a very flexible signal decomposition. The wavelet analog of this is known as the wavelet packet decomposition [11]. For a given signal or class of signals, one can, for a fixed set of filters, obtain the best (in some sense) filter bank tree topology. For a binary tree, an efficient scheme employing entropy as the criterion has been developed as the best wavelet packet basis algorithm [11, 15].

References

1.9

17

Conclusion

One collecting manner of experimental data by engineers and scientists is as sequences of values at regularly spaced intervals in time. These sequences are named time-series. The fundamental problem with the data in the form of time-series is how to process them for extracting meaningful and correct information, i.e., the possible signals embedded in them. If a time-series is stationary, one can think that it can have harmonic components that can be detected by applying the Fourier analysis, i.e., Fourier Transform (FT). Although, it is evident that many time-series are not stationary and their mean properties are variables over time. The waves of infinite support that form the harmonic components are not adequate in the latter case in which one needs waves localized not only in frequency but in time too. They named wavelets and permit a time-scale decomposition of a signal. Considerable progress in understanding the wavelet processing of non-stationary signals was attained. Although, for getting the dynamics which provides a non-stationary signal, it is crucial that in the corresponding time-series a correct separation of the fluctuations from the average behavior, or trend, is performed. Therefore, people should invent new statistical techniques for detrending the data that have to be combined with the wavelet analysis.

References 1. J. Olkkonen, Discrete wavelet transforms-theory and applications. INTECH open, March, 2011, ISBN 978-953-307-185-5 2. N. Baydar, A. Ball, A comparative study of acoustic and vibration signals in detection of gear failures using wigner-ville distribution. Mech. Syst. Signal Proc. 15(6), 1091–1107 (2001) 3. F.S. Chen, Wavelet transform in signal processing theory and applications (National Defense Publication of China, 1998) 4. I. Daubachies, Ten lectures on wavelets (SIAM, Philadelphia, PA, 1992) 5. Y.S. Wang, C.-M. Lee, L.J. Zhang, Wavelet analysis of vehicle nonstationary vibration under correlated four-wheel random excitation. Int. J. Automot. Technol. 5(4) (2004) 6. C. Sidney Burrus, R. Gopinath, H. Guo, Wavelets and wavelet transforms. http://cnx.org/con tent/col11454/1.5/ 7. S.G. Mallat, A theory of multiresolution image decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 647–693 (1989) 8. S.G. Mallat, A theory for multi-resolution signal decomposition: the wavelet representation, in IEEE Transaction on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, (July 1989), pp. 674–693 9. S. Mallat, Wavelet tour of signal processing (Academic Press, USA, 1998) 10. M.B. Ruskai, Introduction, in Wavelets and their applications (Boston, MA, Jones and Bartlett, 1992) 11. R.R. Coifman, M.V. Wickerhauser, Entropy-based algorithms for best basis selection. IEEE Trans. Inf. Theory 38(2), 7138211, 718 (March 1992)

18

1 Wavelets and Wavelet Transforms

12. J.N. Bradley, C.M. Brislawn, T. Hopper, The fbi wavelet/scalar quantization standard for grayscale ngerprint image compression, in Visual Info. Process. II, volume 1961 (Orlando, FL, SPIE, April 1993) 13. C.M. Brislawn, J.N. Bradley, R.J. Onyshczak, T. Hopper. The fbi compression standard for digitized ngerprint images, in Proceedings of the SPIE Conference 2847, Applications of Digital Image Processing XIX, vol. 2847 (1996) 14. M.V. Wickerhauser. Acoustic signal compression with wavelet packets, in Wavelets: a tutorial in theory and applications (Academic Press, Boca Raton, 1992). Volume 2 in the series: Wavelet Analysis and its Applications. p. 6798211;700 15. K. Ramchandran, M. Veterli, Best wavelet packet bases in a rate-distortion sense. IEEE Trans. Image Process. 2(2), 1608211, 175 (1993)

2

A Denoising Technique Based on SBWT and WATV: Application for ECG Denoising

2.1

Introduction

The Electrocardiogram (ECG) is representing the cardiac activity which is principally an electrical signal by nature. Electrodes (3 or 12 leads) are employed for its record. These electrodes are connected to the surface of the thorax zone, hands, and legs in an external manner [1]. The ECG is a plot of the potentials produced by cardiovascular muscular activities [1]. The ECG signal is employed basically by physicians for the prediction and treatment of various cardiovascular illnesses. The ECG signal is extremely susceptible to disturbances due to Electromagnetic Interferences (EMI) and also to some artifacts such as Powerline Interference (PLI) (with a frequency of 50 or 60 Hz), Electro-Myographic Noise (EMG Noise), Measurement Noise, and Baseline Wander [1]. Those artifacts can cause misinterpretation of the important cardiac parameters or even wrong diagnosis [1]. Therefore, serious implications can be caused. PLI is a high-frequency noise having 50 or 60 Hz as frequency and additive in nature. Furthermore, it owns low amplitude and consequently can completely corrupt the original signal and makes it hard for locating the complexes existing in the ECG signal, which are P, Q, R, and S. Everss Villalba et al. [2] have addressed the severity of artifacts and noise that exist in the ECG signal. According to [2], various sorts of disturbances can affect the ECG signal, and among them, we have the Baseline Wander (BW), Muscle Contraction, and PLI. The latter is the principal cause of the distortion. Therefore, canceling the PLI noise has an important role in ECG monitoring and diagnosis [1]. The PLI has the main role in ECG monitoring and diagnosis [1]. The problem of PLI noise cancelation was divided into the following: the algorithm has to conserve the valuable ECG signal of interest and it should be effective (in terms of computing) for real-time monitoring [1]. A diversity of approaches were applied for eliminating the PLI from the ECG signal. A host of adaptive filters are employed for that purpose [3–6]. Martens et al. [4] have proposed the use of the LMS (Least Mean Square) © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Mourad, ECG Denoising Based on Total Variation Denoising and Wavelets, Synthesis Lectures on Biomedical Engineering, https://doi.org/10.1007/978-3-031-25267-9_2

19

20

2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

scheme for showing the SNR improvement of up to 30 dB. Rahman et al. [5] showed the adaptive filters with the use of signs for canceling noise existing in the ECG signal [1]. Various adaptive filter-based approaches are suggested for canceling the PLI noise existing in the ECG signal [6] and they are in number of five. Iterative approximation techniques [1, 7] are used for reconstructing the information which is lost by the filtering. However, these filtering techniques suppose that the ECG signal is both linear and stationary [1]. Artificial Intelligent approaches [8] and Genetic algorithms [9] are also employed but their calculation complexity makes them unsuitable for real-time analysis [1]. The wavelet transforms are also largely employed for canceling noises from the ECG signal [10]. One of the new advances in this domain is in the report with the EMD (Empirical Mode Decomposition) and EEMD (Ensemble Empirical Mode Decomposition). The EMD permits to decompose the input signal into numerous IMFs (Intrinsic Mode Functions) components. The noisy IMFs are eliminated when reconstructing the signal [1]. Lower-order IMFs are discarded during the reconstruction [11] and this is based on the fact that lower-order IMFs agree with high-frequency noise [1]. A major hindrance in EMD analysis is mode mixing [1], and in order to overwhelm that, a noise-assisted based technique, called EEMD (Ensemble EMD), was introduced by Wu and Huang [12]. The implementation of the EEMD-based ECG denoising technique was discussed in [13]. Further, a comparative study was made between a conventional IIR Filter, Wiener filtering, EEMD, and EMD, which has shown that the EEMD outperforms the other techniques. The ICA (Independent Component Analysis), the CCA (Canonical Correlation Analysis), wavelet transforms, and EMD are applied for canceling muscle artifacts from EEG data [14]. Zhidong and Chan [15] have employed the EMD with LMS algorithm-based adaptive filters for denoising IMFs and finally obtaining the denoised ECG signal. The combined band pass filters and EMD have been employed for noise cancelation [16]. Agrawal and Gupta [17] introduced the fBm (fractional Brownian motions) for modeling Baseline Wander (BW) and employing the combination of EMD and wavelets for noise cancelation. The elimination of the Motion and Noise (MN) artifacts when the ECG signal is collected employing Holter monitors was detailed by Lee et al. [18]. The LDA (Linear Discriminant Analysis) [19] is employed for checking the presence of PLI noise in the ECG signal and adaptive filters are employed when the PLI noise exists [1, 19]. Lehmann et al. [20] employed the frequency-domain analysis based on Welch’s PSD (Power Spectral Density) for the ECG signal estimating due to PLI noise distortion. The reduction in calculation complexity of the EMD algorithm is performed by adding a constraint on the ECG segment length [21]. Recently, numerous techniques are proposed for eliminating the noise from the EEG/ECG signal. These techniques employ the deep learning [22–24]. Ge Wang et al. [25] introduced a technique of ECG enhancement employing deep factor analysis. Each and every layer of the used deep neural network is updated based on the factor analysis of the noisy ECG. The degraded ECG signal is also reconstructed by employing deep learning-based auto-encoders [26]. The fully convolutional neural network is employed in order to denoise and compress the noisy ECG signal [1]. In this

2.2 The Stationary Bionic Wavelet Transform (SBWT)

21

chapter, we will detail our ECG denoising technique proposed in [27]. It is based on the application of the Wavelet/Total Variation (WATV) denoising method [33] in the domain of the Stationary Bionic Wavelet Transform (SBWT) [23]. The rest of this chapter is organized as follows: in Sect. 2.2, we will deal with the SBWT. In Sect. 2.3, we will detail the WATV-based denoising technique [10]. In Sect. 2.4, we will detail our ECG denoising technique proposed in [27]. In Sect. 2.5, we will present and discuss the results obtained from the application of the proposed technique [27] and other ones used in our evaluation and proposed literature. We will conclude in Sect. 2.6.

2.2

The Stationary Bionic Wavelet Transform (SBWT)

The Stationary Bionic Wavelet Transform (SBWT) was introduced in [28]. It was applied for speech enhancement and was proposed for solving the problem of perfect reconstruction existing with the Bionic Wavelet Transform (BWT) [28–31]. This transform was also applied for ECG denoising [32]. The applications of the SBWT and its inverse, SBWT−1 , are summarized by the block diagram illustrated in Fig. 2.1. According to Fig. 2.1, the first step of SBWT application to the input signal consists in applying the Stationary Wavelet Transform (SWT) to that signal and we obtain stationary wavelet coefficients. Those coefficients are then multiplied by a K factor for obtaining finally the stationary bionic wavelet coefficients. Also, according to Fig. 2.1, the first step of the application of SBWT−1 consists in multiplying the stationary bionic wavelet coefficients by the factor 1/K for obtaining stationary wavelet coefficients to them is Fig. 2.1 The stationary bionic wavelet transform (SBWT) and its inverse (SBWT−1 )

22

2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

applied the inverse of SWT, SWT−1 , for having finally the reconstructed signal. The K factor is a function of the adaptation factor, T and this is detailed in [28–32].

2.3

The WATV-Based Denoising Method

In [33], Ding et al. proposed a unified Wavelet-TV (WATV) approach which estimates the wavelet coefficients (insignificant and significant) simultaneously through the minimization of a single objective function. In order to induce wavelet-domain sparsity, Ding et al. [33] have used non-convex penalties, and this is due to their strong sparsityinducing properties [33]. In general, when using non-convex penalties, the convexity of the objective function is usually sacrificed. However, in [33], Ding et al. have restricted the non-convex penalty for insuring the strict convexity of the objective function; after that, the minimizer is unique and can be reliably gotten using convex optimization. The WATV denoising approach [33] is relatively resistant to pseudo-Gibbs oscillations and spurious noise spikes. Ding et al. [33] derived a computationally effective, fast-converging optimization algorithm for the proposed objective function.

2.3.1

Problem Formulation

Let x ∈ IR N , a signal corrupted by an additive white Gaussian noise as follows: yn = xn + vn , n = 0, 1, . . . , N − 1

(2.1)

Let W denote the wavelet transform and therefore the wavelet coefficients of x are as follows: w = Wx

(2.2)

Let w j ,k denote the indexes of those wavelet coefficients with k and j , respectively, the time and scale indices. In [33], Yin Ding et al. used the translation-invariant (i.e., undecimated) Wavelet Transform (WT) [34, 35], satisfying the Parseval frame condition expressed as follows: W TW = I

(2.3)

The WATV denoising algorithm proposed in [33] can be employed with any Wavelet Transform, W which satisfies (3). The TV (Total Variation) of a signal x ∈ IR N is as follows: TV(x) := ||Dx||1

(2.4)

2.4 The Proposed ECG Denoising Technique

23

where D is the first-order||difference matrix and ||·||1 is the ℒ1 - norm of x, i.e., ||x1 || = || ∑ ∑ 2 ||x 2 || = |x |. Also, we have n n n |x|n | and for a set of doubly indexed wavelet 2 | ∑ 2 coefficients, we have ||w||22 := j ,k |w j,k | . The matrix D is expressed as follows: ⎡

−1 1 ⎢ −1 1 ⎢ D=⎢ .. .. ⎢ . . ⎣ −1

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(2.5)

1

The WATV denoising technique proposed in [33] finds the coefficients w and this by solving the following optimization problem: ⎫ ⎧ || || ⎬ ⎨ ∑ ) ( 1 || || wˆ = arg min F(w) = ||W y − w||22 + λ j Φ w j,k ; α j + β ||DWT w|| (2.6) ⎩ 1⎭ 2 w j ,k

The estimate of x which is xˆ is consequently obtained from the application of the ˆ so we have wavelet transform inverse to w xˆ = W T wˆ

(2.7)

|| || The penalty term ||DWT w1 || is the TV of xˆ [33]. The regularization parameters are β > 0 and λ j > 0. In [33], Yin Ding et al. allowed the wavelet regularization and the ( ) penalty parameters a j and λ j varying with j.

2.4

The Proposed ECG Denoising Technique

In this work, we detail an ECG denoising technique proposed in [27]. It is based on the application of the Wavelet/Total Variation (WATV) denoising method [33] in the domain of the SBWT [27, 28]. This proposed ECG denoising technique [27] can be summarized by the block diagram illustrated in Fig. 2.2. As shown in Fig. 2.2, the SBWT [27, 28] is first applied to the noisy ECG signal for obtaining two noisy stationary bionic wavelet coefficients, wtb1 and wtb2. Then is estimated the noise level, σ , from the details coefficient, wtb1, and the threshold, thr is calculated as follows: √ (2.8) thr = σ · 2 × log(N ) where N is the samples number in wtb1 and σ is expressed as follows: σ = MAD(|wtb1|)/0.6745

(2.9)

24

2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

Fig. 2.2 The block diagram of the proposed ECG denoising technique

After that, wtb1 is thresholded using a soft thresholding function, Sthr which requires the use of the threshold thr. This function is expressed as follows:  sign(x)(|x| − thr) if |x| > thr xˆ = Sthr (x) = (2.10) 0 if |x| ≤ thr Apart from the soft thresholding function, there are other functions such as hard thresholding which is expressed as follows:  x if |x| > thr (2.11) xˆ = Hthr (x) = 0 if |x| ≤ thr Hard thresholding (Fig. 2.2b) permits to maintain the scale of the signal and introduces ringing and artifacts after reconstruction due to a discontinuity in the wavelet coefficients. However, soft thresholding (Fig. 2.2a) cancels this discontinuity and therefore results in smoother signals but slightly decreases the magnitude of the reconstructed signal [36]. Also, according to Fig. 2.2, the denoising technique based on WATV [33] is applied to wtb2 for obtaining a denoised coefficient, wtd2. This denoising technique [33] requires the employment of the noise level, σ (Eq. 2.9 and Fig. 2.2). The denoised ECG signal is finally obtained from the application of the inverse of SBWT, SBWT−1 to the denoised coefficients, wtd1 and wtd2 (Fig. 2.3).

2.5

Results and Discussion

25

Fig. 2.3. a Soft thresholding function and b Hard thresholding function

2.5

Results and Discussion

In this section, we make a comparative study between the ECG denoising approach proposed in [27] and five other denoising approaches. These approaches are as follow our ECG denoising technique proposed in [37] and based on the Bionic Wavelet Transform (BWT) [29–31] and Translation-Invariant Forward Wavelet Transform (FWT_TI). The second one is the 1D double-density complex DWT denoising method [32, 38]. The third one is the ECG denoising technique based on Non-Local Means [39, 40]. The fourth one is the technique based on wavelets and hidden Markov models [41]. The fifth one is the Wavelet/Total Variation (WATV) denoising technique [33]. All these techniques are applied to a number of noisy ECG signals which are in number of 35. Those 35 noisy ECG signals are obtained in the following manner: each clean ECG signal chosen from seven clean ECG signals belonging to MIT-BIH database is degraded by an additive white Gaussian noise with five different values of SNRi (SNR before denoising) which are −5 dB, 0 dB, 5 dB, 10 dB, and 15 dB. Consequently, for each clean ECG signal, we obtain five different noisy ECG signals. The seven clean ECG signals are 100–106.dat. For this comparative study, we have used as evaluation criteria the Signal to Noise Ratio (SNR), the Peak SNR (PSNR), the Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Cross-Correlation (CC). Those criteria are expressed as follows [42–45]: (

)2 

N −1 )2 1 ∑( x(n) − x(n) ˆ N n=1   ∑N 2 n=1 x (n) SNRdB = 10 · log10 ∑ )2 N ( ˆ n=1 x(n) − x(n)   max(x(n)) PSNR = 20 · log10 √ MSE

MSE = E

x(n) − x(n) ˆ

MAE =

=

N −1 | 1 ∑ || x(n) − x(n) ˆ | N n=1

(2.12)

(2.13)

(2.14)

(2.15)

26

2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

  ˆ − xˆ [x(n) − x] · x(n) CC = / /  2 ∑N 2 ∑N x(n) ˆ − x] x ˆ − [x(n) n=1 n=1 ∑N n

(2.16)

where x(n) and x(n) ˆ are, respectively, the clean and the denoised signals and N is the samples number in x(n). The quantities x and xˆ are, respectively, the mean value of x(n) and the mean value of x(n). ˆ In Tables 2.1, 2.2, 2.3, 2.4, and 2.5 are listed the results obtained from the computations of the SNR after denoising (SNRf), the PSNR, the MAE, the MSE, and the CC and this for all the techniques. According to Tables 2.1, 2.2, 2.3, 2.4, and 2.5, the results in blue color are the best results and they are obtained from the application of the proposed ECG denoising technique. In fact, this proposed technique permits to obtain the highest values of SNR, PSNR, and CC and the lowest values of MAE and MSE. Consequently, it outperforms all the other denoising techniques applied for our evaluation. Figures 2.4, 2.5, 2.6, and 2.7 show examples of ECG denoising by applying the proposed technique [27]. Figures 2.4, 2.5, 2.6, and 2.7 show the performance of the proposed technique. In fact, the noise is considerably reduced and the different waves P-QRS-T of the original signal are practically conserved.

2.6

Conclusion

In this chapter, we detailed our ECG denoising technique introduced in literature and based on the application of the WATV-based denoising approach, in the domain of the SBWT. For its performance evaluation, it was compared to five other denoising techniques. These techniques are the 1D double-density complex DWT denoising one, the denoising technique based on Wavelets and Hidden Markov Models, the technique based on Non-Local Means, the proposed technique based on BWT and FWT_TI with hard thresholding, and also WATV-based denoising approach. The results obtained from the calculations of SNR, MSE , MAE, PSNR, and CC show that the proposed technique outperforms the other ones applied for this evaluation. In fact, our proposed approach permits us to obtain the highest values of SNR, PSNR, and CC and the lowest values of MAE and MSE. Moreover, it considerably reduces the noise and the different waves P-QRS-T of the original signal are practically conserved.

SNRf: 11.6926 dB

SNRf: 8.9095 dB SNRf: 3.6629 dB SNRf: 8.5354 dB

SNRf: 7.5563 dB SNRf: 7.2861 dB SNRf: 3.2999 dB

SNRf: 4.3536 dB

WATV denoising technique [19]

The ECG denoising technique proposed in [23]

The 1D double-density complex DWT SNRf: 3.6629 dB denoising method [24] SNRf: 3.9834 dB

The proposed ECG denoising technique

The ECG denoising technique based on non-local means [18, 19]

The tech-nique based on wavelets and hid-den Markov mod-els [27] SNRf: 8.9189 dB

SNRf: 11.2690 dB

SNRi = 0 dB

SNRi = −5 dB

Denoising technique

SNRf: 13.1256 dB

SNRf: 12.3277 dB

SNRf: 12.9861 dB

SNRf: 13.4812 dB

SNRf: 14.9927 dB

SNRf: 15.4771 dB

SNRi = 5 dB

SNRf: 17.6421 dB

SNRf: 16.7405 dB

SNRf: 17.1191 dB

SNRf: 17.6433 dB

SNRf: 18.0846 dB

SNRf: 18.7926 dB

SNRi = 10 dB

SNR: 20.1387 dB

SNRf: 20.7730 dB

SNRf: 20.8943 dB

SNRf: 20.9656 dB

SNRf: 21.0755 dB

SNRf: 21.7291 dB

SNRi = 15 dB

Table 2.1 Comparative study in terms of SNR: results obtained from the computation of the mean of seven values of SNR. This mean is computed for seven clean ECG signals 100 to 106.dat corrupted by Gaussian white noise with different values of SNRi before denoising (varying from −5 dB to 15 dB with step of 5 dB)

2.6 Conclusion 27

PSNR: 19.7287 dB PSNR: 25.3384 dB PSNR: 29.9101 dB PSNR: 29.9101 dB PSNR: 37.3743 dB PSNR: 20.0918 dB PSNR: 24.8436 dB PSNR: 29.4150 dB PSNR: 33.5480 dB PSNR: 37.3232 dB PSNR: 20.8533 dB PSNR: 24.9957 dB PSNR: 28.7880 dB PSNR: 33.2008 dB PSNR: 37.2333 dB

The ECG denoising technique proposed in [23]

The 1D double-density complex DWT denoising method [24]

The ECG denoising technique based on non-local means [18, 19]

The technique based on wavelets and PSNR: 20.4772 dB PSNR: 23.6498 dB PSNR: 27.9770 dB PSNR: 33.8433 dB PSNR: 37.5430 dB hidden Markov models [27]

PSNR: 23.9303 dB PSNR: 27.6979 dB PSNR: 31.4215 dB PSNR: 34.5135 dB PSNR: 37.9067 dB

SNRi = 15 dB

WATV denoising technique [19]

SNRi = 10 dB

PSNR: 23.9852 dB PSNR: 28.1215 dB PSNR: 31.9060 dB PSNR: 35.2214 dB PSNR: 38.1580 dB

SNRi = 5 dB

The proposed ECG denoising technique

SNRi = 0 dB

SNRi = −5 dB

Denoising technique

Table 2.2 Comparative study in terms of PSNR: results obtained from the computation of the mean of seven values of PSNR. This mean is computed for seven clean ECG signals 100 to 106.dat corrupted by Gaussian white noise with different values of SNRi before denoising (varying from −5 dB to 15 dB with step of 5 dB)

28 2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

CC: 0.9648 CC: 0.9377 CC: 0.9292 CC: 0.9323 CC: 0.9168

CC: 0.7830

The ECG denoising technique proposed in [23]

The 1D double-density complex DWT denoising method [24] CC: 0.8005

The ECG denoising technique based on non-local means [18, CC: 0.8165 19]

The technique based on wavelets and hidden Markov models CC: 0.8294 [27]

CC: 0.9685

CC: 0.9730

CC: 0.9754

CC: 0.9788

CC: 0.9852

CC: 0.9914

CC: 0.9905

CC: 0.9908

CC: 0.9920

CC: 0.9931

CC: 0.9963

CC: 0.9961

CC: 0.9962

CC: 0.9962

CC: 0.9967

CC: 0.9969

CC: 0.9136

CC: 0.9934

WATV denoising technique [19]

CC: 0.9864

CC: 0.9174

The proposed ECG denoising technique CC: 0.9670

SNRi = −5 dB SNRi = 0 dB SNRi = 5 dB SNRi = 10 dB SNRi = 15 dB

Denoising technique

Table 2.3 Comparative study in terms of CC: results obtained from the computation of the mean of seven values of CC. This mean is computed for seven clean ECG signals 100 to 106.dat corrupted by Gaussian white noise with different values of SNRi before denoising (varying from −5 dB to 15 dB with step of 5 dB)

2.6 Conclusion 29

MSE: 0.0121 MSE: 0.0103 MSE: 0.0096 MSE: 0.007

The ECG denoising technique proposed in [23]

The 1D double-density complex DWT denoising method [24]

The ECG denoising technique based on non-local means [18, 19]

The technique based on wavelets and hidden Markov models [27] MSE: 0.0035

MSE: 0.0035

MSE: 0.0034

MSE: 0.0031

MSE: 0.0018

MSE: 0.0044

WATV denoising technique [19]

SNRi = 0 dB MSE: 0.0016

SNRi = − 5 dB MSE: 0.0043

Denoising technique

The proposed ECG denoising technique

MSE: 0.0012

MSE: 0.0014

MSE: 0.0012

MSE: 0.0011

MSE: 7.5714e−04

MSE: 6.8571e−04

SNRi = 5 dB

MSE: 4.0000e−04

MSE: 5.0000e−04

MSE: 5.4000e−04

MSE: 4.1429e−04

MSE: 3.7143e−04

MSE: 3.1429e−04

SNRi = 10 dB

MSE: 2.2857e−04

MSE: 2.0000e−04

MSE: 2.0000e−04

MSE: 1.7143e−04

MSE: 1.8571e−04

MSE: 1.5714e−04

SNRi = 15 dB

Table 2.4 Comparative study in terms of MSE: results obtained from the computation of the mean of seven values of MSE. This mean is computed for seven clean ECG signals 100 to 106.dat corrupted by Gaussian white noise with different values of SNRi before denoising (varying from −5 dB to 15 dB with step of 5 dB)

30 2 A Denoising Technique Based on SBWT and WATV: Application for ECG …

MAE: 0.0295 MAE: 0.0193 MAE: 0.0135

MAE: 0.0473

WATV denoising technique [19]

The ECG denoising technique proposed in [23]

MAE: 0.0391 MAE: 0.0233 MAE: 0.0147 MAE: 0.0414 MAE: 0.0247 MAE: 0.0140 MAE: 0.0431 MAE: 0.0282 MAE: 0.0170 MAE: 0.0477 MAE: 0.0258 MAE: 0.0148

MAE: 0.0755

The 1D double-density complex DWT denoising method [24] MAE: 0.0711

The ECG denoising technique based on non-local means [18, MAE: 0.0678 19]

The technique based on wavelets and hidden Markov models MAE: 0.0668 [27]

MAE: 0.0101

MAE: 0.0105

MAE: 0.0103

MAE: 0.0101

MAE: 0.0097

MAE: 0.0092

MAE: 0.0458

The proposed ECG denoising technique

MAE: 0.0283 MAE: 0.0186 MAE: 0.0139

SNRi = − 5 dB SNRi = 0 dB SNRi = 5 dB SNRi = 10 dB SNRi = 15 dB

Denoising technique

Table 2.5 Comparative study in terms of MAE: results obtained from the computation of the mean of seven values of MAE. This mean is computed for seven clean ECG signals 100 to 106.dat corrupted by Gaussian white noise with different values of SNRi before denoising (varying from −5 dB to 15 dB with step of 5 dB)

2.6 Conclusion 31

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2.6

Conclusion

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(c) Fig. 2.6 Third example of ECG denoising by applying the proposed denoising technique: a the clean ECG signal (119.dat), b the noisy ECG signal with SNRi = 5 dB, and c the denoised ECG signal (SNRf = 14.758179 dB, PSNR = 29.715380 dB, CC = 0.983566, MAE = 0.019725, MSE = 0.001068)

2.6

Conclusion

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(c) Fig. 2.7 Fourth example of ECG denoising by applying the proposed denoising technique: a the clean ECG signal (101.dat), b the noisy ECG signal with SNRi = 0 dB, and c the denoised ECG signal (SNRf = 12.276751 dB, PSNR = 30.590276 dB, CC = 0.970692, MAE = 0.021001, MSE = 0.000873)

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References 1. A.K. Dwivedi, H. Ranjan, A. Menon, P. Periasamy, Noise reduction in ECG Signal using combined ensemble empirical mode decomposition method with stationary wavelet transform. Circuits, Syst., Signal Process. https://doi.org/10.1007/s00034-020-01498-4. 2. E. Everss-Villalba, F.M. Melgarejo-Meseguer, M. Blanco-Velasco, F.J. Gimeno-Blanes, S. SalaPla, J.L. Rojo-Álvarez, A. García-Alberola, Noise maps for quantitative and clinical severity towards long term ECG monitoring. Sensors (Switzerland) 17(11), 2448 (2017). https://doi.org/ 10.3390/s17112448 3. M. Maniruzzaman, K.M.S Billah, U. Biswas, B. Bain, Least-mean-square algorithm based adaptive filters for removing powerline interference from ECG signal, in Proceedings of IEEE International Conference on Informatics, Electronics and Vision (2012), pp. 737–740 4. S.M.M. Martens, M. Mischi, S.G. Oei, J.W.M. Bergmans, An improved adaptive power line interference canceller for electrocardiography. IEEE Trans. Biomed. Eng. 53(11), 2220–2231 (2006) 5. M.Z.U. Rahman, R.A. Shaik, D.V. Rama Koti Reddy, Efficient sign based normalized adaptive filtering techniques for cancelation of artifacts in ECG signals: application to wireless biotelemetry. Signal Process 91, 225–239 (2011) 6. I. Romero, D. Geng, T. Berset, Adaptive filtering in ECG denoising: a comparative study. Comput. Cardiol. 39, 45–48 (2012) 7. X. Zhou, Y. Zhang, A hybrid approach to the simultaneous eliminating of power-line interference and associated ringing artifacts in electrocardiograms. BioMedical Eng. Online 12, 42 (2013). https://doi.org/10.1186/1475-925X-12-42 8. J. Mateo, C. Sanchez, A. Torres, R. Cervigon, J.J. Rieta, Neural network based canceller for powerline interference in ECG signals. Comput. Cardiol. 35, 1073–1076 (2008) 9. N. Kumaravel, N. Nithiyanandam, Genetic-algorithm cancellation of sinusoidal powerline interference in electrocardiograms. Med. Biol. Eng. Compu. 36, 191–196 (1998) 10. A.R. Al-Qawasmi, K. Daqrouq, ECG signal enhancement using wavelet transform. WSEAS Trans. Biol. Biomed. 2(7), 62–71 (2010) 11. A. J. Nimunkar, W. J. Tompkins EMD-based 60-Hz noise filtering of the ECG, in Proceedings of the 29th Annual International Conference of the IEEE Engineering in Medicine and Biology (2007), pp. 1904–1907. 12. Z. Wu, N.E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adaptive Data Anal. 1, 01–41 (2009) 13. K.M. Chang, Arrhythmia ECG noise reduction by ensemble empirical mode decomposition. Sensors. 10, 6063–6080 (2010) 14. D. Safieddine et al., Removal of muscle artifact from EEG data: comparison between stochastic (ICA and CCA) and deterministic (EMD and wavelet-based) approaches. EURASIP J. Adv. Signal Process. (2012). https://doi.org/10.1186/1687-6180-2012-127 15. Z. Zhidong, M. Chan, A novel cancellation method of powerline interference in ECG signal based on EMD and adaptive filter, in Proceedings of the International Conference on Communication Technology (2008), pp. 517–520 16. M. Suchetha, N. Kumaravel, Empirical mode decomposition-based subtraction techniques for 50 hz interference reduction from electrocardiogram. IETE J. Res. 59(1), 55–62 (2013) 17. S. Agrawal, A. Gupta, Fractal and EMD based removal of baseline wander and powerline interference from ECG signals. Comput. Biol. Med. 43, 1889–1899 (2013)

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18. J. Lee, D.D. McManus, S. Merchant, K.H. Chon, Automatic motion and noise artifact detection in holter ECG data using empirical mode decomposition and statistical approaches. IEEE Trans. Biomed. Eng. 59(6), 1499–1506 (2012) 19. Y.D. Lin, Y.H. Hu, Power-line interference detection and suppression in ECG signal processing. IEEE Trans. Biomed. Eng. 55(1), 354–357 (2008) 20. C. Lehmann, J. Reinstädtler, A. Khawaja, Detection of power-line interference in ECG signals using frequency-domain analysis. Comput. Cardiol. 38, 821–824 (2011) 21. A. Karagiannis, P. Constantinou, Noise-assisted data processing with empirical mode decomposition in biomedical signals. IEEE Trans. Inf Technol. Biomed. 15(1), 11–18 (2011) 22. K. Antczak Deep recurrent neural networks for ECG signal denoising (2018) http://arxiv.org/ abs/1807.11551 23. C.T.C. Arsene, R. Hankins, H. Yin Deep learning models for denoising ECG signals, in Proceedings of the 27th European Signal Processing Conference (2019) https://doi.org/10.23919/ eusipco.2019.8902833 24. S. Kuanar, V. Athitsos, N. Pradhan, A. Mishra, K. R. Rao Cognitive analysis of working memory load from EEG by a deep recurrent neural network, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (2018), pp. 2576–2580 25. G. Wang, L. Yang, M. Liu, X. Yuan, P. Xiong, F. Lin, X. Liu, ECG signal denoising based on deep factor analysis. Biomed. Signal Process. Control (2020). https://doi.org/10.1016/j.bspc. 2019.101824 26. H. Chiang, Y. Hsieh, S. Fu, K. Hung, Y. Tsao, S. Chien, Noise reduction in ECG signals using fully convolutional denoising autoencoders. IEEE Access. 7, 60806–60813 (2019) 27. M. Talbi. A novel technique of noise cancellation based on stationary bionic wavelet transform and WATV: application for ECG denoising. Int. Arab. J. Inf. Technol. 19(3) (2022) 28. T. Mourad, Speech enhancement based on stationary bionic wavelet transform and maximum a posterior estimator of magnitude-squared spectrum. Int. J. Spee. Tech. 20, 75–88 (2017) 29. T.J. Michael, Y. Xiaolong, R. Yao, ‘Speech signal enhancement through adaptive wavelet thresholding. Spee. Comm. 49, 123–133 (2007) 30. J. Yao, An active model for otoacoustic emissions and its application to time-frequency signal processing. Ph.D Thesis (The Chinese University of Hong Kong, 2001) 31. J. Yao, Y.T. Zhang, Bionic wavelet transform: a new time-frequency method based on an auditory model. IEEE Trans. Biomed. Eng. 48(8), 856–863 (2001) 32. T. Mourad, New approach of ECG denoising based on 1-D double-density complex DWT and SBWT. Comput. Methods Biomech. Biomed. Eng.: Im-Aging Vis. (2020). https://doi.org/10. 1080/21681163.2020.1763203 33. Y. Ding, I.W. Selesnick, Artifact-free wavelet denoising: non-convex sparse regularization, convex optimization. IEEE Signal Process. Lett. 22(9), 1364–1368 (2015) 34. R. Coifman, A. Sowa, ‘Combining the calculus of variations and wavelets for image enhancement.’ J. Appl. Comput. Harmon. Anal. 9(1), 1–18 (2000) 35. L. Combettes, J.-C. Pesquet et al., Proximal splitting methods in signal processing, in FixedPoint Algorithms for Inverse Problems in Science and Engineering, ed. by H.H. Bauschke (New York, NY, USA, Springer, 2011), pp. 185–212 36. T. Mourad, S. Lotfi, C. Adnane, Spectral entropy employment in speech enhancement based on wavelet packet. World Acad. Sci., Eng. Technol., Int. J. Electron. Commun. Eng. 1(9) (2007) 37. T. Mourad, Electrocardiogram de-noising based on forward wavelet transform translation invariant application in bionic wavelet domain. Sadhana J. 39(4), 921–937 (2014) 38. S. Ivan, W. Crystal, Double-density wavelet software, Supported by: NSF 39. D. Ambuj, M. Hasnine, Two-stage nonlocal means denoising of ECG signals. Int. J. Advan. Rese. Comput. Sci. 5, 114–118 (2014)

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40. T. Brian, M. Eric, Nonlocal means denoising of ECG signals. IEEE Trans on Biomed. Eng. 59(9), 2383–2386 (2012) 41. M. Crouse, R. Nowak, R. Baraniuk, Wavelet-based statistical signal processing using hidden Markov models. IEEE Trans. Signal Processing 46, 886–902 (1998) 42. S. Aditya, Evaluating performance of denoising algorithms using metrics : MSE,MAE,SNR,PSNR & cross correlation (https://www.mathworks.com/matlabcentral/fil eexchange/52342-evaluating-performance-of-denoising-algorithms-using-metrics-mse-maesnr-psnr-cross-correlation), MATLAB Central File Exchange. Retrieved August 21, 2021 (2021) 43. Z. Dengyong, W. Shanshan, L. Feng, W. Jin, S. Arun, S. Victor, An ECG signal de-noising approach based on wavelet energy and sub-band smoothing filter. Appl. Sci. 9, 4968 (2019). https://doi.org/10.3390/app9224968 44. H. Ibtissem, S. Lamir, S. Fawzi, ECG signal denoising by fractional wavelet transform thresholding. Res. Biomed. Eng. 36, 349–360 (2020). https://doi.org/10.1007/s42600-020-00075-7 45. W. Zhaoyang, Z. Junjiang, Y. Tianhong, Y. Lulu, A new modified wavelet-based ECG denoising. Comput. Assist. Surg. 24(sup1), 174–183 (2019). https://doi.org/10.1080/24699322.2018. 15600

3

An ECG Denoising Technique Based on LWT and TVM

3.1

Introduction

In the diagnosis of heart diseases, the employment of the Electrocardiogram (ECG) is wide. The ECG is an electrical signal produced during the depolarization and repolarization of heart muscle cells [1, 2]. The key high-quality ECG signal is crucial for proper diagnosis. Although, many noises, including the Power Line Interference (PLI), the Electromyography (EMG), and the Baseline Wander (BW), often corrupt an ECG signal, which make the diagnosis more difficult [2]. The surface EMG as a high-frequency noise is produced by non-cardiac muscle activities around the electrodes, owning frequency content varying from 0 to 500 Hz [2, 3]. Power Supply Interference (PSI) is another type of high-frequency noise often superimposed to the ECG signal. It is constituting many sinusoids including a 60 Hz (or a 50 Hz in some countries) sinusoid with either interharmonics or harmonics components. Further, another sort of noise, which is the Baseline Wonder (BW), is a low-frequency noise affecting the quality of ECG signals and is produced by diverse sources, including the body movement of the patient, respiration, and also the use of instruments during recording. It continuously appears as a low-frequency artifact adding slow fluctuations to the recorded signal, i.e., DC components [2]. Many algorithms exist in the literature and are proposed for ECG denoising, and among them, we can mention the Artificial Neural Networks (ANN) [4], the Discrete Wavelet Transform (DWT) [5], the Empirical Mode Decomposition (EMD) [6–9], the combined EMD and DWT [10], or the Non-Local Means (NLM) with DWT [11], the approach based on EMD combined with other algorithms including DWT and Adaptive Switching Mean Filter (ASMF) [12]. In the reference [13] was presented a filtering technique based on both ANN and the DWT. The network is trained by comparing its output with the clean ECG signal, and the weights of this network are updated employing the error signal [2]. The

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Mourad, ECG Denoising Based on Total Variation Denoising and Wavelets, Synthesis Lectures on Biomedical Engineering, https://doi.org/10.1007/978-3-031-25267-9_3

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3 An ECG Denoising Technique Based on LWT and TVM

results obtained from computer simulation showed that this technique effectively eliminates varied noises in the ECG signal. As the behaviors of the heart and the related organs change dynamically, ECG signals are non-stationaries [2]. In [2], a pre-recorded noisefree ECG signal is the reference, and the requirement of an appropriate source signal is the major disadvantage of this technique [2]. Pratik Singh et al. [14] have introduced an ECG denoising approach based on an efficient combination of NLM and EMD. Rashmi et al. [15] proposed a denoising approach based on wavelet transform, and the obtained results showed that the Signal to Noise Ratio (SNR) of “sym8” wavelet transform is higher than that of the digital filter with the Blackman window [16]. A hybrid approach including the combination of the Median filter, Savitzky–Golay filter, the extended Kalman filter, and the DWT has concentrated on separating noise from the ECG signal [17]. Zhang et al. [18] introduced an ECG signals denoising approach based on an improved wavelet thresholding algorithm combining the soft and hard thresholding. The obtained results indicate that the proposed algorithm can efficiently filter out the noise in ECG signals. This algorithm has better characteristic information retention of the ECG signal and possesses a higher SNR and attains a better denoising effect [16]. This chapter presents an ECG denoising technique proposed in [19] and based on Lifting Wavelet Transform (LWT) [20–22] and Total Variation Minimization (TVM) [23]. The remaining chapter is organized as follows: part 2 elucidates TVM, part 3 discusses the ECG denoising approach proposed in [19], part 4 discusses the 1D double-density complex DWT denoising technique [24], part 5 details the denoising technique based on NLM [25, 26], in part 6 are presented results and discussion, and, finally, part 7 is devoted to the conclusion.

3.2

The Lifting Wavelet Transform (LWT)

The Lifting Wavelet Transform (LWT) is a powerful tool for signal and image analysis. The LWT owns a faster and effective implementation compared to the DWT. Correspondingly, the LWT leads to better results than the DWT in the image denoising, image compression, and watermarking fields. The LWT permits time and owns a better frequency localization feature, which permits to overcome the DWT shortcomings. The Signal decomposition by LWT requires three stages, which are splitting, prediction, and updating and are as follows: • The Signal Splitting: This step consists of dividing the original signal X (n) into nonoverlapping even and odd samples, which are, respectively, X e (n) and X o (n), and they are formulated as follows:  X o (n) = X (2n + 1) (3.1) X e (n) = X (2n)

3.3 The Total Variation Minimization (TVM)

41

Fig. 3.1 Decomposition and reconstruction of a signal X [n] employing L W T and its inverse, respectively

• The Prediction: This step can be summarized as follows: If the odd and even samples are correlated, then one can be the predictor of the other. In the prediction of even sample (X e (n)), the odd sample (X o (n)) is used as follows: d(n) = X o (n) − P(X e (n))

(3.2)

where d(n) is the difference between the original sample and the predicted value and represents a high-frequency component and P(·) is the predictor operator. • Updating The details signal d(n) and the update operator (U (·)) can update the even samples. The low-frequency components l(n) represent the coarse shape of the original signal, which is as follows: l(n) = X e (n) + U (d(n))

(3.3)

In Fig. 3.1 are illustrated the afore-discussed three steps.

3.3

The Total Variation Minimization (TVM)

The Total Variation Minimization (TVM) consists of solving the Lagrangian TV Minimization [23]: ) ( xtv = arg min x − u 2 + λ1 · T V (u)

(3.4)

u

Here xtv designates the denoised signal and TV is the discrete TV norm 1D, which is as follows: T V (u) =

N −1 ∑

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i

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3.4

3 An ECG Denoising Technique Based on LWT and TVM

The ECG Denoising Technique Proposed in [19]

The ECG denoising technique proposed in [19] is based on LWT [20–22] and TVM [23]. Figure 3.2 illustrates the block diagram of the ECG denoising technique proposed in [19]. According to Fig. 3.2, this technique consists firstly of applying twice the L W T to the noisy ECG signal (where 2 is the decomposition level) for obtaining three wavelet sub-bands, cD1 , cD2 , and c A2 . The two coefficients cD1 and cD2 are details coefficients and denoised by soft or hard thresholding and we obtain two denoised coefficients cDd1 and cDd2 . The coefficient c A2 is an approximation coefficient and denoised by TVM [23] in order to obtain a denoised coefficient, c Ad2 . The denoised ECG is finally obtained by applying twice the inverse of LWT (LWT−1 ) to cDd1 , cDd2 , and c Ad2 . Also, according to Fig. 3.2, the thresholding of cD1 and cD2 needs the estimation of noise level (σ ) and then computing the threshold, thr, as follows: √ (3.6) thr = σ · 2 · log(N ) where N is the number of samples in the details coefficient cD1 at the first decomposition level. The noise level, σ , is estimated as follows: σ = MAD(|cD1 |)/0.6745

(3.7)

Further, as per Fig. 3.2, applying the TVM-based denoising technique [23] to the cA2 needs the employment of the parameter, λ1 (Eq. (3.4)). In [19], this parameter was chosen to be equal to σ (Fig. 3.2 and Eq. (3.7)). In part 3.7 was presented a comparative study between the ECG denoising technique proposed in [19], the denoising method based on TVM [23], the 1D double-density complex DWT denoising technique [24], and the ECG denoising technique based on NLM [6, 27]. In part 3.5 will be detailed the 1D double-density complex DWT denoising method [24]. In part 3.6, the fourth denoising technique based on NLM [25, 26] will be discussed.

Fig. 3.2 The block diagram of the ECG denoising technique proposed in [19]

3.6 The Denoising Approach Based on Non-local Means (NLM) [25, 26]

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Fig. 3.3 Signal denoising using 1D double-density complex DWT denoising technique

3.5

The 1D Double-Density Complex DWT Denoising Method [24]

Figure 3.3 illustrates the block diagram summarizing the 1D double-density complex DWT denoising technique. As shown in Fig. 3.3, the different stages of this denoising method [24] are listed as follows: • The first stage consists in applying the 1D double-density complex DWT to the noisy signal. • The second stage consists in applying the soft thresholding to the sub-bands obtained in the first step. The soft thresholding requires the use of a certain threshold, T. • In order to have the denoised signal, the third stage consists in applying the inverse of 1D double-density complex DWT to the denoised sub-bands obtained in the second step.

3.6

The Denoising Approach Based on Non-local Means (NLM) [25, 26]

Brian H. Tracey and Eric L. Miller applied the Non-local Means for ECG denoising [25, 26]. This technique addresses the problem of recovering the clean signal s from the noisy signal x, which is as follows: x =s+n

(3.8)

where n is an additive noise corrupting the clean signal, s. For a given sample k, the estimate sˆ (k) is a weighted sum of the values at other samples j which belong to some search neighborhood N (k) [25, 26].

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3 An ECG Denoising Technique Based on LWT and TVM

sˆ (k) =

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j∈N (k)

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∑ j

w(k, j ) and the expression for weights is as follows [25, 26]:

 ∑ 2 δ∈∆ (x(k + δ) − x( j + δ)) w(k, j ) = exp − 2L ∆ λ2 

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(3.10)

(3.11)

where λ is a bandwidth parameter and ∆ designates a local patch of samples surrounding k, including L ∆ samples; a patch of similar shape also surrounding j. In [25, 26], d 2 designates the summed, squared point–point difference between samples in the patches cantered on the samples j and k. In [25, 26], each patch is averaged with itself with weight w(k, k) = 1. For obtaining a smoother result, a canter patch correction is commonly applied [25, 26]: w(k, k) =

max

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In the image denoising field and in numerous research works are also used patch-based models and algorithms, and among them, we can mention those proposed in [27–29]. In Fig. 3.4 are shown the results of the denoising algorithms proposed in [27] for the image “Barbara” with σ = 20. Those algorithms are Global Trained Dictionary and Adaptive Dictionary. In Fig. 3.5 are shown the simulated recovery results of the MODIS reflectance products on the Arabian Peninsula, including Harmonic Analysis of Time-Series (HANTS) [27, 28], Multitemporal-Beta Process Factor Analysis (MT-BPFA), and Multitemporal-KSVD (MTKSVD) in consideration of the visual similarity of BPFA and MT-BPFA, KSVD, and MT-KSVD [29]. The distorted data items are from the 186th day of 2012, which is the fifth temporal data item in the group [29]. Subjectively, HANTS introduces false edges to the result, as illustrated in the yellow circled zones in Fig. 3.6c, and MT-BPFA and MT-KSVD both keep pace with the original reflectances and succeed with satisfactory outcomes [29]. HANTS [30] does not get as good an effect as MT-BPFA and MT-KSVD [29]. Subsequently, the same temporal reflectances of Fig. 3.5 are completely degraded by simulated thick clouds (labeled as zeros), as shown in Fig. 3.6b, and are recovered by HANTS and the two approaches proposed in [29] (again, the results of BPFA and KSVD are omitted). As seen from the figures, the results of MT-BPFA display a certain degree of smoothness, and those of MT-KSVD are alleviated a little. In terms of this sort of smoothness, they are very close to the average of all of the multitemporal data because of the average

3.6 The Denoising Approach Based on Non-local Means (NLM) [25, 26]

45

Fig. 3.4 Example of the denoising results for the image “Barbara” with noise level, σ = 20: the original, the noisy, and two restoration results [27]

Fig. 3.5 Simulated thick cloud corruption and the corresponding recovery of the reflectances of MODIS: a original, b simulated thick cloud corruption, c HANTS recovery, d MT-BPFA recovery, and e MT-KSVD recovery [29]

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Fig. 3.6 Simulated thick cloud complete corruption and the corresponding recovery of the reflectances of MODIS: a original, b simulated thick cloud corruption, c HANTS recovery, d MTBPFA recovery, and e MT-KSVD recovery [29]

being set as the reference data of both MT-KSVD and MT-BPFA. Furthermore, as MTKSVD is solved patch by patch, the smoothness is weakened [29]. As far as the visual effect is concerned, HANTS performs the best and benefits from the temporal and quantitative restrictions of the harmonic analysis. Although, for the quantitative applications, the visual effect is of secondary importance, and the primary concern is the objective evaluation, which is stated in [29].

3.7

Results and Discussion

This section performs a comparative study between our ECG denoising technique proposed in [19] and using soft thresholding and other denoising ones, including a method based on TVM [23], the 1D double-density complex DWT denoising technique [24], and the ECG denoising technique based on NLM [25, 26]. This study is made through the computation of Signal to Noise Ratio (SNR) and Mean Square Error (MSE). These different denoising techniques are applied to 35 noisy ECG signals. Those signals are obtained by artificially corrupting each of seven clean ECG signals by an Additive White Gaussian Noise (AWGN) with five different values of SNR before denoising (SNRi). These values are − 5 to 15 dB with a step size of 5 dB and the seven clean ECG signals are taken from the MIT-BIH database. Those signals are (105, 107, 109, 123, 124, 200, and 201).dat. In Tables 3.1 and 3.2 are listed the results obtained from the computation of the SNR after denoising, SNRf, and the MSE between the clean and the denoised ECG signals. Those results are the mean values where each of them is computed from seven values of SNRf /MSE and this for each value of SNRi. According to Table 3.1, the highest S N R f values are obtained by applying the proposed ECG denoising approach with soft thresholding [19]. Therefore, this approach [19] outperforms the other three denoising techniques [23–26], used for this evaluation and studied above. According to Table 3.2, the lowest MSE values are obtained by applying

3.7

Results and Discussion

47

Table 3.1 A comparative study in terms of Signal to Noise Ratio (SNR): results obtained from the mean computations of the seven SNRf values (SNR after denoising) [19] SNRi(dB) The denoising technique

−5

0

5

10

15

SNRf obtained by the denoising technique proposed in [19]

SNRf = 6.9876

SNRf = 11.0423

SNRf = 14.9601

SNRf = 18.4743

SNRf = 21.3775

SNRf obtained by the TVM-based denoising Technique [23]

SNRf = 2.4137

SNRf = 6.4391

SNRf = 10.9663

SNRf = 15.6451

SNRf = 20.3097

SNRf obtained by 1D double-density complex DWT denoising method [24]

SNRf = 4.4703

SNRf = 8.7680

SNRf = 13.1836

SNRf = 17.2554

SNRf = 21.0807

SNRf obtained by the ECG SNRf = denoising technique based 4.8513 on NLM [25, 26]

SNRf = 9.1454

SNRf = 13.0091

SNRf = 16.9997

SNRf = 21.1550

Table 3.2 A comparative study in terms of Mean Square Error (MSE between original and denoised signals): results obtained from the mean computations of the seven MSE values [19] SNRi(dB) The denoising technique

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10

15

MSE obtained by the denoising technique proposed in [19]

MSE = 0.0087

MSE = 0.0035

MSE = 0.0015

MSE = 6.7143e − 04

MSE = 3.4286e − 04

MSE obtained by the MSE = TVM-based denoising 0.0238 Technique [23]

MSE = 0.0098

MSE = 0.0035

MSE = 0.0012

MSE = 4.0000e − 04

MSE obtained by 1D MSE = double-density 0.0150 complex DWT denoising method [24]

MSE = 0.0058

MSE = 0.0022

MSE = 8.7143e − 04

MSE = 3.7143e − 04

MSE = 0.0151

MSE = 0.0055

MSE = 0.0021

MSE = 8.6667e − 04

MSE = 3.1429e − 04

MSE obtained by the ECG denoising technique based on NLM [25, 26]

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3 An ECG Denoising Technique Based on LWT and TVM

Fig. 3.7 First example of ECG denoising by applying the proposed ECG denoising technique: (100.dat corrupted by Gaussian white noise with SNR = 10 dB)

the proposed ECG denoising approach with soft thresholding [19] and the SNRi varied from −5 to 10 dB. Although, for SNRi equals to 15 dB, the ECG denoising technique based on NLM [25, 26] permits to give the lowest MSE value. Figures 3.7 and 3.8 illustrate two examples of ECG denoising by applying the proposed ECG denoising technique [19]. In Fig. 3.9 are illustrated the clean ECG signal (in Blue color and in strong line), the noisy ECG signal (in red color), and the denoised ECG signal (in green color and thin line). This denoised ECG signal is obtained by applying the proposed ECG denoising technique. According to Figs. 3.7, 3.8, and 3.9, the proposed ECG denoising technique [19] permits and performs to cancel noise and conserves the different waveforms of the original ECG signal. In Fig. 3.10 are illustrated the different denoised ECG signals obtained by applying the proposed technique [19] (based on LWT and TVM), the denoising technique based on TVM [23], the 1D double-density complex DWT denoising method [24], and the ECG denoising technique based on NLM [25, 26], to the same noisy ECG signal (105.dat corrupted by an Additive White Gaussian Noise (AWGN) with SNR before denoising, SNRi = 15 dB). According to plots (b) and (c), the clean ECG signal and the denoised one obtained by applying the proposed denoising technique have a significant similarity. Although, in the plot (d), the denoised ECG obtained by applying the TVM-based denoising technique

3.7

Results and Discussion

49

Fig. 3.8 Second example of ECG denoising by using the proposed ECG denoising technique: (123.dat corrupted by Gaussian white noise with SNR = 5 dB)

Fig. 3.9 Blue signal is the clean ECG signal (105.dat), red signal is the noisy ECG signal (clean ECG signal, 105.dat, corrupted by an additive white Gaussian noise (AWGN) with SNRi = 15 dB), and green signal is the denoised ECG signal obtained by applying the proposed ECG denoising technique to the noisy ECG signal (red signal)

50

3 An ECG Denoising Technique Based on LWT and TVM

(a) Noisy ECG signal: Clean Signal, 105.dat corrupted an Addi ve Gaussian with SNRi=15dB

(b) Clean ECG signal (105.dat taken from MIT-BIH database)

(c) Denoised ECG signal obtained by applying the proposed ECG denoising technique, ( , MSE=2.3878e-04)

,

(d) The denoised ECG signal obtained by applying the denoising technique based on TVM [23] with the parameter ) (Eq. 3.7), ( , ,

Fig. 3.10 An example of ECG denoising: a noisy 105.dat, b clean 105.dat, c noisy 105.dat denoised by applying the proposed ECG denoising app, d noisy 105.dat denoised by applying the denoising technique based on TVM [23] with the parameter λ1 = σ (Eq. 3.7), e noisy 105.dat denoised by applying the denoising technique based on TVM [23] with the parameter λ1 = thr (Eq. 3.6), f noisy 105.dat denoised by applying the 1D double-density complex DWT denoising method [24], and g noisy 105.dat denoised by applying the ECG denoising technique based on NLM [25, 26]

3.7

Results and Discussion

51

(e) The denoised ECG signal obtained by applying the denoising technique based on TVM [23] with the ) , , parameter

(f) The denoised ECG signal obtained by applying the 1-D double-density complex DWT denoising method [24], ( , , MSE=3.2204-04)

(g) The denoised ECG signal obtained by applying the ECG denoising approach based on NLM [25, 26], ( , , )

Fig. 3.10 (continued)

[23] with λ1 = σ (Eq. 3.7) contains some residual noise. In plot (e), the denoised ECG obtained by applying the TVM-based denoising approach [23] with λ1 = thr (Eq. 3.6) is much smoothed and including many degraded peaks (surrounded) compared to the clean ECG signal. In plot (f), the denoised ECG signal obtained by applying the 1D double-density complex DWT denoising method [24] also contains some residual noise and many degraded peaks compared to the clean signal. In plot (g), the denoised ECG signal obtained by applying the ECG denoising technique based on NLM [25, 26] is less smoothed compared to the denoised ECG signal obtained by applying the proposed ECG denoising technique.

52

3.8

3 An ECG Denoising Technique Based on LWT and TVM

Conclusion

In this chapter, we detail our technique of ECG denoising proposed in the literature and based on LWT and TVM. This technique consists at the first stage in applying twice the LWT to the noisy ECG signal in order to obtain three wavelet coefficients, cD1 (at level 1), cD2 , and c A2 (at level 2). The two coefficients cD1 and cD2 are details coefficients denoised by soft thresholding in order to obtain two denoised coefficients, cDd1 and cDd2 . The coefficient c A2 is an approximation coefficient denoised by TVM-based denoising technique in order to obtain a denoised coefficient, c Ad2 . The last stage consists in applying twice the inverse of LWT, LWT−1 , to cDd1 , cDd2 , and c Ad2 , for obtaining the denoised ECG signal. This proposed technique is evaluated by comparing it to three other denoising ones existing in the literature. The first one of these techniques is based on TVM, the second one is the 1D double-density complex DWT denoising method, and the third one is the NLM-based ECG denoising technique. All these denoising techniques (including our proposed technique) are applied to many ECG signals taken from MIT-BIH database and artificially corrupted by an Additive White Gaussian Noise at different SNR values, SNRi. The results obtained from the computation of the SNRf (after denoising) and the Mean Square Error (MSE)infer that this proposed technique outperforms the other ones applied in the evaluation.

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Index

C Continuous wavelet transform, 1, 6 Convex optimization, 22

D Denoising, 20–35, 39, 40, 42–44, 46–48, 50–52 Discrete wavelet transform, 1, 7, 8, 10, 16, 25, 31, 39, 40, 42, 43, 46–48, 50–52

E Electrocardiogram (ECG), 19–21, 23, 25, 26, 29–35, 39, 40, 42, 43, 46–52

F Fourier Transform, 1, 2, 7, 17

L Lifting wavelet transform, 40, 48, 52

S Stationary bionic wavelet transform, 21

T Thresholding, 24, 40, 42 Total variation minimization, 40–42, 46–48, 50, 52

W Wavelet packet, 10, 12–16 Wavelets, 1, 4–10, 12, 14, 17, 20–26, 40, 42 Wavelet/Total-Variation (WATV), 21–23, 25, 31

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Mourad, ECG Denoising Based on Total Variation Denoising and Wavelets, Synthesis Lectures on Biomedical Engineering, https://doi.org/10.1007/978-3-031-25267-9

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