978-1-4799-0902-5/13/$31.00 ©2013 IEEE 333

Second Generation Wavelets: Advantages in Cardiosignal Processing

Ana Gavrovska1, Goran Zajić2, Irini Reljin3, Vesna Bogdanović4, Branimir Reljin5

Abstract – In this paper, the cardiosignal analysis using second generation wavelets and lifting structure based on clinically-relevant extracted information is described. The information extraction can be performed by different spectral (such as multifractal) and spectrogram (time-frequency) representations. Here, the novelties are related mainly to the knowledge-controlled analysis and noise reduction in phonocardiograms. We suggest adaptive (signal-dependent) wavelet-based processing able to simultaneously preserve relevant high-frequency details and reduce noise.

Keywords – Phonocardiogram, signal decomposition, lifting, signal representation, wavelets, multifractals.

I. INTRODUCTION

Telecardiological system enables the transfer of medical information regarding cardiovascular state between two distant locations. It usually involves cardiological data exchange from the signal acquisition location, i.e., from the patient, to the receiver side where the health care provider (HCP) analyzes transferred data [1]. The cardiosignals used in primary health care (PHC), such as electrocardiograms (ECGs) and phonocardiograms (PCGs), can be analyzed locally (at the same location where acquisition is performed), or at a distant location (within the telemedical system). In the last two decades, wavelet transform (WT) has become a popular signal processing tool [2]. There are two major uses of WT and generally multiresolution transforms: compression and (feature) analysis [3]. The latter one usually includes segmentation and restoration tasks, such as denoising.

Lifting scheme is a ladder structure which enables efficient implementation of discrete wavelet transform (DWT) [4-7]. Wavelets constructed using this structure are called second generation wavelets. In this paper, we point out several advantages and possibilities of the lifting in cardiosignal processing, particularly in PCG signal pre-processing.

1Ana Gavrovska and 3Irini Reljin are with the School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia, E-mail: anaga777@gmail.com, irinitms@gmail.com

2Goran Zajić is with the ICT College of Vocational Studies, Zdravka Čelara 16, 11000 Belgrade, Serbia, E-mail: gzajic@gmail.com

4Vesna Bogdanović is with the Health Center "Zvezdara", Olge Jovanović 11, 11000 Belgrade, Serbia, E-mail: vesna.bogdanovic54@gmail.com

5Branimir Reljin is with the Innovation Center of the School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, Serbia, E-mail: reljinb@etf.rs

In Section II a brief description of the lifting and its advantages in signal processing are described. Knowledge-based control of noise reduction by lifting is introduced in Section III, where the importance of signal-dependent WT and feature extraction are emphasized. Furthermore, novelties regarding PCG analysis and context modelling via spectral and spectrogram representations are discussed. Some of the simulation results in adaptive pre-processing are presented in Section IV. Finally, conclusions are given in Section V.

II. SECOND GENERATION WAVELETS AND LIFTING STRUCTURE

A. Signal decomposition using wavelets and polyphase construction

Wavelet filter banks are widely used for cardiosignal analysis. The block scheme of signal analysis (decomposition) and synthesis (reconstruction) using DWT is depicted in Fig.1. On the input side the signal x is passed through the low-pass (LP) and high-pass (HP) filter pair denoted as H0 and H1 respectively, and decimated (downsampled by 2), producing signals 0y and 1y . On the reconstruction side an opposite processes are applied: signals 0y and 1y are upsampled by 2 and passed through the LP-HP filter pair G0 and G1 producing outputs 0x and 1x . If the filter pairs (H0,H1) and (G0,G1) are adjusted properly, reconstructed output 10 xxx += will be equal to the input signal x . Certainly, this case is of no interest here. Usually, signals 0y and 1y are changed before reconstruction, producing new values 0y and 1y . By appropriate choice of system parameters the new signals ( 0y and 1y ) will contribute to desired final result x .

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Fig. 1. Analysis and synthesis wavelet filter bank.

Each FIR (finite impulse response) filter can be defined via Laurent polynomial. Using Euclidian algorithm for computing the greatest common divisor of two Laurent polynomials can be written in a matrix form based on polynomial divisions throughout the algorithm (such divisions are not unique) [8].

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Matrix filter representation can be used in polyphase construction. If two polyphase components have length twice shorter than signal's length: ( ) ( ) ( )nxnxnx even 20 == and

( ) ( ) ( )120 +== nxnxnx odd , the input signal can be presented

as ( ) ( ) ( )212 zXzzXzX oe−+= , where 0XX e = and

1XXo = are z-transforms of evenx and oddx , respectively. The polyphase decomposition is defined as

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20 zzHzHzH P += and can be represented in 2x2

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Fig. 2. The polyphase construction - analysis and synthesis filter

bank.

B. Lifting structure and its advantages

The lifting structure represents suitable framework for perfect reconstruction filter bank realization and construction of the second generation wavelets without the use of translations and dilatations as in standard definition of WT [4]. The basic idea behind lifting is not to rely on the Fourier transform, but to use the correlation between neighboring samples and work within physical domain. The lifting structure is consisted of two types of elementary lifting steps (ELSs): primal (or update, U) step and dual (prediction, P) step. The filter pairs can be modified using ELSs.

Every FIR filter pair that corresponds to DWT can be described using the finite number of successive ELSs [8]. Due to its efficiency, lifting-based approach is suggested for telemedicine applications [9]. Lifting structure enables fast decorrelation and simple inverse transform implementation. Furthermore, it enables in-place calculation of wavelet coefficients and working with irregularly sampled data. It represents a framework where modifications can be made in different ways (e.g. ELSs, redundance, etc.) [6,10]. One of the most interesting possibilities of filter banks implemented via lifting scheme is the one regarding adaptive realizations having in mind different constraints [6,11]. Adaptive realizations are usually based on decision maps, which define selection of parameters that should be used in lifting-based processing. The advantage of such approach is the ability to introduce signal-dependence into parameter selections and to develop ladder structure appropriate for particular purpose, such as cardiosignal processing.

III. LIFTING STRUCTURE CONTROL BY EXTRACTED CLINICAL INFORMATION

Raw cardiosignal is often pre-processed and used for further analysis and clinical information extraction (Fig.3). Clinically valuable components should not be affected by a noise reduction algorithm. So, considered cardiosignal or obtained wavelet coefficients may not be suitable for further analysis. Due to this, it is important to extract clinical information regarding the signal in order to control the processing. This is knowledge-based control of the lifting.

Wavelet (lifting) -based processing

Knowledge-based control

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Fig. 3. A knowledge-controlled lifting-based processing.

Depending on the analyzed cardiosignal, analysis usually starts with dominant intervals in time-frequency domain (for instance, heart sounds S1/S2s in PCG, or QRS complexes in ECG signal). Such intervals are often considered as references for further cardiosignal analysis usually found in low-frequency (LF) domain. Their amplitude extrema, like R peaks in ECG, represent another type of references. Since the lifting enables processing in time domain, an adaptive lifing can be an interesting solution for improving results in cardiosignal analysis. From the aspect of signal decomposition, adaptivity can be understood as choosing the right filter pair or wavelet type selected within the signal interval (decision map for ELSs, ELSD , or different maps for particular or each ELS, U/PD , Fig.4(a)). This is a space-adaptive transform [11]. Similar signal context-modeling can be applied for thresholding [12], where edges (or more generally, singularities) are often treated with special care (decision map ThrD , Fig.4(b)) The adaptivity can also be realized regarding each scale (scale-adaptive transform) [11].

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Fig. 4. Decision map implementation for (a) lifting structure and (b)

thresholding.

Since the most of challenges regarding noise reduction comes from relevant high-frequency (HF) components in cardiosignal (for instance from R peaks in ECG, as explained in [13]), their localization can be used for denosing. Adaptivity can be realized using the Euclidean distance from these singularities. Similarly as for a pulse signal (Fig.5(a)) [12], R peak or similar cardiac event found in high-frequency (HF) domain can be treated with special care. We point out

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that clinically important HF details can be found within as well as outside of dominant LF intervals. Due to this, we may consider two types of control in time domain: decision maps defined by signal singularities (Fig.5(a)) and decision maps defined by reference intervals (Fig.5(b)). A map does not enable necessary a binary choice ( 2=r , where r represent a number of choices - switchings) for particular parameter (e.g. wavelet, threshold) used in lifting-based processing [6].

r=2

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(b) Fig. 5. Decision maps in time domain: (a) defined by detection of singularities and (b)defined by interval detection within a signal.

Even interactive entries regarding the cardiosignal can be useful for introducing signal-dependence in the processing (Fig.3). Nevertheless, the automatically made decision (as well as automatic clinical information extraction) is much preferable for cardiosignal processing. The signal-dependent analysis for PCG is proposed in Fig.6.

Cardiosignal noise removal

LF components (reference) detection

Input cardiosignal

Time margins

Outputcardiosignal

HF components detection

Potential HF abnormality existence

Analysis system

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Lifting structure can be modified by different selection of dual vanishing moment number [6,10]. As there is no wavelet that can give consistently better results than the others, there is a recommendation that the number of vanishing moments should be higher for denoising purposes [15]. Higher numbers of the vanishing moments are related to signal reproduction using polynomials up to higher degree, as well as enlarging the number of coefficients used in a ELS.

Joint time-frequency (JTF) analysis is considered to be extremely useful in PCG analysis. It can be applied in LF component detection (heart sound detection) [14]. In our experiments, standard PCG denoising using DWT showed similar effects as ECG in [13]. Namely, even though mean squared error (MSE) calculated as a figure of merit for denoising may be satisfying (~ 410− ), as well as visual appearance of the denoised signal, a calculated difference (error) between target and denoised signal usually shows high correlation of the signal. This is often obvious in heart sounds (solid blue lines in Fig.7), where the amplitude values are

relatively high in comparison to other components in PCG. A denoising technique can affect clinically relevant HF components of low amplitude outside of the heart sounds. The existence of repeatable clinically relevant singularities can be detected by multifractal spectra (zooming, [16]) even if they are found at irregular distances and have low-amplitude. So, they can be revealed even if they are difficult to localize [17].

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IV. SIMULATION

In order to test the influence of the wavelet-based lifting processing, and generally DWT, on clinically valuable HF components, we used two sets of recordings: PCG recordings from a group of healthy patients (Healthy set) and PCG recordings from a group of patients with premature mitral valve (PMV set). PCG signals have been acquired at Health Center “Zvezdara”, Belgrade, Serbia. The acquisition process has been explained in [17]. The reason for selecting a PCG from a patient with PMV is the fact that possible low-amplitude HF components (clicks) can be found within the signal. The possibility of their low amplitude increases the challenge of preserving clinically relevant HF details in comparison to details in fundamental heart sounds. The highest amplitude sample within a heart sound interval has been annotated in both sets (Healthy and PMV), as well as the samples corresponding to clicks (here, noted as selected singularities) in the PMV set.

In the simulation, we have added Gaussian noise to each acquired (target) signal. The test is performed using different noise standard deviation nσ (0, 0.001, 0.0025, 0.005, 0.0075, 0.01, 0.25). As a figures of merit for comparing different PCG denoising algorithms we have used the MSE and the distortion ratio (R) calculated as a ratio of amplitudes of selected singularities from target and denoised signals. For the comparison reasons, sym 8 as a wavelet considered to be an excellent solution regarding to MSE aspect and universal thresholding have been chosen.

We propose an adaptive update-predict lifting structure [11,10]. In the case of LF or HF detection (Fig.6), automatic selection of a vanishing moment number (linear to 9th order

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polynomial prediction) has been performed in order to minimize wavelet coefficients. Fixed 2-tap update filter has been applied.

Adaptive approach shows as a good solution to obtain sparse representation, preserve low MSE and keep clinically relevant singularities even outside of heart sounds (see Fig.8 and Fig.9).

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Fig. 8. (a) Noisy signal (black line) with annotated click. (b) DWT-sym8 denoising (magenta line) and (c) proposed denoising result (red

line) in comparison to noisy (black) and target signal (blue line).

Fig. 9. MSE (left) and distortion ratio R (right) for: sym8

and adaptive U-P lifting, respectively (in each row).

V. CONCLUSIONS

In this paper we have analyzed DWT application in cardiosignal, particularly phonocardiogram, signal processing, via lifting scheme. We want to point out that by denoising clinically relevant HF components (such as clicks) can be affected by the fixed choice of a wavelet function. The advatage of wavelet lifting is a possibility to develop adaptive solutions in order to control the filtering performed in time domain. Adaptive lifting approach has been proposed according to LF/HF detection.

ACKNOWLEDGEMENT

This work is partially funded by Serbian Ministry of Education, Science, and Technological Development through the project of Integrated and interdisciplinary research program, No. III44009. Authors are grateful to Health Center “Zvezdara”, Belgrade, Serbia, for providing medical data.

REFERENCES

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[2] A. Gavrovska, M. Paskaš, I. Reljin, D. Jevtić, D. Dujković, B. Reljin, “Review of selected techniques for cardiosignal analysis”, MD Medical Review, vol.2, no.4, pp.341-347, December 2010, re-printed April 2011.

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[8] I. Daubechies, W. Sweldens, “Factoring wavelet transforms into lifting steps”, J. Fourier Anal. Appl., vol. 4, no. 3, pp. 247-269, 1998.

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[12] S.G. Chang, B.Yu, M. Vetterli, “Spatially adaptive wavelet thresholding with context modeling for image denoising,” IEEE Trans. Img. Proc., vol. 9, pp. 1522–1531, September 2000.

[13] S.A. Chouakri, F. Bereksi-Reguig, S. Ahmaidi, O. Fokapu, “Wavelet denoising of the electrocardiogram signal based on the corrupted noise estimation”, Computers in Cardiology 2005; 32: 1021-1024.

[15] S.R. Messer, J. Agzarian, D. Abbott, “Optimal wavelet denoising for phonocardiograms”, Microelectronics Journal 32, pp. 931-941, 2001.

[14] A.M. Gavrovska, M.S. Slavković, M.P. Paskaš, D.M. Dujković, I.S. Reljin, “Joint time-frequency analysis of phonocardiograms”, in Proc. 11th Conference NEUREL 2012, pp.177-180, Belgrade, Serbia, September 20-22, 2012.

[16] S. Mallat, A Wavelet Tour of Signal Processing, The Sparse Way, Third edition, Academic Press, Elsevier Inc. 2009

[17] A. Gavrovska, G. Zajić, I. Reljin, B. Reljin, “Classification of Prolapsed Mitral Valve versus Healthy Heart from Phonocardiograms by Multifractal Analysis,” Computational and Mathematical Methods in Medicine, vol. 2013, Article ID 376152, 10 pages, 2013.

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