Wavelet Transforms for Biomedical Signal Processing Tutorial

CONTENTSChapter 1Introduction3Chapter 2Biomedical signal processing in time domain62.1ECG62.1.1Compression62.1.2 Denoising82.1.3QRS detection112.2EEG132.3Spike detection...14Chapter 3Biomedical imaging processing153.1Biomedical image compression153.2 Biomedical image enhancement and edge detection..163.3Biomedical image registration.17Chapter 4Conclusion19REFERENCE

Chapter 1 IntroductionFor biomedical signals, most of the statistical characteristics of these signals are non-stationary. In particular, the analysis of biological signals should exhibit good resolution in both time domain and frequency domain. Several time-frequency analysis methods such as the short-time Fourier transform (STFT), Wigner-Ville Distribution function (WDF), Hilbert-Huang transform (HHT), etc, were proposed to represent the signals in both time and frequency domains at the same time. The problem with the STFT is that using a large window size may improve frequency resolution, but the assumption of stationary within the window may be compromised; whereas using a small window leads to poor frequency resolution. WDF offers high clarity in both time and frequency domains, but suffers from cross-term problem. For most biomedical signals, WDF is not suitable either because they have multiple components or because their phase terms are higher than second order. In addition, time-frequency analysis methods An alternative way to analyze the non-stationary biomedical signals is the wavelet transform, which has variable time-frequency resolution over the time-frequency plane. The analysis phase of the wavelet transform decomposes a signal into elementary building blocks or frequency bands that are localized in both space (time) and frequency (scale). This differentiates a wavelet transform from a Fourier transform. The window size (scale) used in wavelet transform is chosen to be short at high frequencies and long at low frequencies (to pick up all the abrupt changes), providing good time resolution at high frequency and good frequency resolution at low frequencies. Because of this localization property, wavelets are very good in isolating singularities and irregular structures in signals. The main advantage of wavelet transform over other time-frequency analysis is little storage space. The dimension and size of the output signal is about the same as the input signal, which gives wavelet transform very powerful potential in image processing. Because of the above reasons, the wavelet transform has become a popular technique in feature detection, noise reduction, signal compression, and image and video processing. With the rapid development of computer and improvement of fast algorithm of wavelet transform, the discrete wavelet transform (DWT) has been widely applied in biomedical signal processing and played an important role in clinical diagnosis and therapy of doctor and teaching and researching, such as magnetic resonance imaging (MRI), computerized tomography (CT), radiography, electrocardiogram (ECG), and electroencephalography (EEG) etc. This tutorial introduces several applications of the wavelet transform in biomedical signal processing. Chapter 2 focuses on 1-D wavelet transform applied in biomedical signal processing. Chapter 3 focuses on 2-D wavelet transform applied in biomedical image processing. Chapter 4 is the conclusion.

Chapter 2 Biomedical signal processing in time domainIn this chapter, I will present three types of applications of the DWT which are compression, filtering, and feature detection in biomedical signals. When measuring biomedical signals, the sensing device is desired to have information stored or transmitted with high quality and low redundancy; this scheme involves two blocks (filtering and compression) to filter out redundant signal and compress the signal. In practice, feature detection of the biomedical signals is required for clinical diagnosis such as QRS detection for ECG signals and spike detection. The following part of this chapter narrates the DWT applications in ECG in section 2.1, in EEG in section 2.2, in heart sounds in section 2.3, in ultrasounds in section 2.4, and in spike detection in section 2.5 respectively.2.1 ECG2.1.1 CompressionBy converting the signal into its DWT coefficients and then removing all except those containing the most pertinent signal information, the resulting transform is much smaller in size, which provides a good way of compressing a signal. Performing an inverse transform on the remaining components recreates a signal that very nearly matches the original. This concept has been widely adopted for effective signal compression and applied in ECG data. Data compression seeks to reduce the number of bits of information required to store or transmit digitized signals without significant loss of signal quality. In practice, ECG signals are collected both over long periods of time and at high resolution. Thus, effective ECG compression techniques are required to enlarge storage capacity and improve methods of ECG data transmission over internet lines. Crowe et al. in 1992 (Crowe, Gibson et al. 1992) used the ECG and heart rate variability data to demonstrate that the DWT is well suited for the compression and reconstruction of ECG data. To compress data, we have to find the redundancy in the information and eliminate it. The basic idea of compression model is showed in Fig. 1. The thresholding block is used to remove the redundancy in the signal. The encoding block is used to improve the Compression Relation (CR).

Fig. 1: Compression model for transmission channel and reception channelSeveral encoding algorithms can be used such as Huffman encoding and run-length encoding (RL). Huffman encoding defines the codebook according to the repetition of every data. It uses more bits in the no-frequent data and fewer bits for the data with higher occurrence. The steps for creating the code are: (1) sorting the data from high to low level of repetition, (2) Grouping in pairs of minor repetition, (3) repeating the second step until all data have been combined, and (4) drawing the Huffman tree with branches of two nodes, where data sets with higher levels of repetition are located to the left of the tree and the lowest level on the right, and assigning 1 to the data of the left and a 0 to the right. Huffman code is read from top to bottom of the tree. The Run-Length encoding (RL) is done by the selection of a value that is repeated many times in a row to be represented through the number of times of the repetition RL encoding. The length of the new data decreases when the quantity of zeros increases. Ballesteros et al. in 2010 (Ballesteros, Moreno et al. 2010) developed a compression model for ECG based on DWT and RL encoding. In practice, the real time processing the DWT in filtering and compression of biomedical signals is conceived on FPGAs because of lower time of response compared to implementations on software.2.1.2 DenoisingThe morphology of ECG signal has been used for recognizing much variability's of heart activity, so it is very important to get the parameters of ECG signal clear without noises and artifacts in order to support clinical decision making. To address this issue, DWT allows effective noise reduction. In contrast to continuous wavelet transform, DWT is a fast algorithm for machine computation. Like the Fast Fourier Transform (FFT), DWT is linear operation that operates on a data vector, transforming it into a numerically different vector of the same length. In addition, DWT is invertible and orthogonal. Instead of using sines and cosines as the basic functions in FFT, the basic functions in the DWT are hierarchical set of wavelet functions that satisfy certain mathematical criteria and are all translations and scaling of each other. The DWT splits the signal into two components, each of half the original length, with one containing the low-frequency or smooth information and the other the high-frequency or difference information. The process is performed again on the smooth component, breaking it up into low-low and high-low components and it is repeated several times. A remarkable feature of many useful wavelet transforms is that they obey a perfect reconstruction theorem. The block diagram of the decomposition and reconstruction of DWT is displayed in Fig.2.

Fig. 2: Filter bank tree of a) Decomposition and b) Reconstruction M. Alfaouri and K. Daqrouq in 2008 (Alfaouri and Daqrouq 2008) employed Daubechies wavelet to decompose the signal into five levels of wavelet transform and determined a threshold through a loop to find the value where minimum error is achieved between the detailed coefficients of thresholded noisy signal and the original. Their denoising method proceeds in the following steps, as illustrated in Fig. 3: (1) decomposing of the noisy and original signals using wavelet transform, (2) choosing and applying threshold value by finding minimum error of denoised and original wavelet subsignal (coefficients), and (3) reconstructing denoised signal using inverse DWT.

Fig. 3: The flow chart of denoising based on DWT and IDWTThe crucial issue of this approach is determination of an appropriate threshold value. The threshold T was determined as , where n is the number of samples in a subsignal, is the standard deviation of the noisy ECG signal, is the standard deviation of the jth detailed coefficients, and C is a constant. The wavelet detailed coefficients of the noisy ECG signal denoted as d(C(j,k)) is one if the wavelet transform coefficient C(j,k) is larger than or equal to the threshold value T; otherwise d(C(j,k)) is zero. The adaptive thresholding allows good quality of noise reduction. The in vivo ECG signal was used to evaluate their denoising method, illustrated in Fig. 4.

Fig. 4: ECG signal before and after denoising.2.1.3 QRS detectionFig.5 shows the typical ECG waveform in normal subjects.

Fig. 5: Normal ECG waveformThe QRS complex is the dominant feature of the ECG signal and accurate detection of QRS is of vital importance in number of clinical instruments. The problem of automation of this process is quite challenging because the morphologies of normal as well as abnormal QRS complexes may differ widely and the presence of noise from many sources make this problem more complicated. In addition, other sections of ECG (P and T waves) can hinder the detection of QRS complexes and often result in error in classification. In general, the commercially used equipment that detect QRS complex require band-pass filtering and temporal filtering (time windowing) of the signal. However, the choice of appropriate bandwidth is a tradeoff between noise and high frequency details and the duration of the sliding window is a tradeoff between false and missed detections. Further, the bandwidth of the signal and duration of the QRS complex are dynamic varying and fixed values of either are not suitable for QRS complex detection. In contrast to conventional techniques, the wavelet transform provides a new dimension to signal processing and event detection with higher degree of flexibility and adaptability. To determine the choice of wavelet, properties of the QRS were examined. There are three properties of the ECG that are useful for detection of the QRS complex: the slope, shape, and location of QRS complex. The shape of the signal is maintained if the phase shift is linear. Thus one requirement of the wavelet is that it should have a symmetrical function. Such wavelets are non-orthogonal. Time localization is important because the ECG events are transient. A number of wavelet-based techniques have been proposed to detect these features using different mother wavelets. For one, spline wavelets have properties satisfying the two requirements discussed above. The order of spline wavelets is a tradeoff factor between frequency resolution computational time-consumption. Commonly the cubic spline wavelet is assumed to have a high enough order for this application. Other mother wavelets such as Daubechies and rst-order derivative of the Gaussian function can also be use for the characterization of ECG waveform (Sahambi et al., 1997a;b). The basic scheme of QRS detection combines DWT and thresholding. Li et al. in 1995 proposed a method based on DWT and thresholded the modulus maxima larger than a threshold obtained from the pre-processing of preselected initial beats, this threshold can be updated during the analysis to obtain a better performance (Li et al., 1995). 2.2 EEG EEG signals are considered not to be deterministic and they have no special characteristics like ECG signals. In addition, when the Fourier transform is applied to successive segments of an EEG signal, the obtained spectra are observed to be time varying. This indicates that the EEG signal is also non-stationary. The spectral analysis based on the Fourier transform classical method assumes the signal to be stationary, and ignores any time-varying spectral content of the signal within a window. Both frequency and time-domain characteristics of EEG are very important in clinical diagnosis and studies. An EEG signal consists of several frequency bands, which are called and bands, and their corresponding bandwidths are 04, 48, 812, and above 12 Hz respectively. Akin in 2002 evaluated the wavelet transform and FFT methods in the analysis of EEG signals and concluded that WT is more suitable in EEG analysis then FFT as the EEG signals are non-stationary (Akin 2002).2.3 Spike detectionFor a neuroelectric event, there are sometimes small-scale transient events such as focal epileptogenic spikes occur in the signal. The flexible resolution in localization property makes the wavelet transform ideally suited to detect the time of occurrence and the location of such focal spikes. Several studies have explored the utility of wavelets for EEG spike identication (Kalayci, Ozdamar, & Erdol, 1994; Schiff, Heller,Weinstein, & Milton, 1994). Nenadic and Burdick in 2005 employed the continuous wavelet transforms with basic detection theory to develop a new unsupervised method for robustly detecting and localizing spikes in noisy neural recordings (Nenadic and Burdick 2005). Their methodology proceeds as the following steps: (1) performing multi-scale decomposition of the signal using an appropriate wavelet basis, (2) separating the signal and noise at each scale, (3) based on results from previous 2 steps, performing Bayesian hypothesis testing at different scales to assess the presence of spikes, (4) combining the decisions at different scales, and (5) estimating the arrival times of individual spikes. The fundamental issue of the spike detection is the design of a mother wavelet that is suitable for the signal of interest, which is the shape of the spike. Chapter 3 Biomedical imaging processing3.1 Biomedical image compressionAs medical images have higher resolution, it takes more storage space. To meet the demand for high-speed transmission of image in efficient image storage and remote treatment, the efficient image compression is essential. Recently some new and very promising method merge in the field of image compression algorithm based on wavelet transform, such as wavelet packet transform, multi-wavelet transform, the combination of wavelet transform with fractal, and so on. Thus, wavelet theory has great potential in medical image compression. Wavelet transform-based image data compression in general involves the following successive steps: (1) Selection of the best wavelet shape according to objective measures such as the peak signal-to-noise ratio (PSNR), the percent retained energy, the percent-rate of distortion (PRO), the correlation coefficient between the original and reconstructed images (), and the normalized mean-squared error (NMSE), (2) thresholding of the transform coefficients and forcing small coefficients to a zero value, (3) efficient vector quantization of the retained transform coefficients, (4) efficient encoding of transform coefficients. A.S. Tolba in 2002 (Tolba 2002) proposed the good design parameters for a data compression scheme applied to medical images of different imaging modalities using wavelet packets (WP). The proposed technique provides better image representation in the sense of lower entropy or minimum distortion by selecting the optimal filter bank and reconstruction basis for best compression rate. 3.2 Biomedical image enhancement and edge detectionMedical images generally have poor contrast of object with surrounding and vague edge. For breast images, the contrast among soft tissue of breast is little and the position and form of lump are difficult to distinguish. Thus, in order to convenient doctor to diagnosis, we need to enhance those image properties that are useful for clinical and difficult enhance to distinguish under ordinary conditions. According to the properties of its multi-scale, direction and local characteristic, the image edge features can be obtained by determining the local maxima of wavelet coefficients.3.3 Biomedical image registrationMedical image registration is a pre-processing step in object identification and object classification. Traditional image registration methods identify points on one image and match these points on the other image, obtaining translation and rotation parameters. Raj Sharman et al. (Sharman, Tyler et al. 2000) proposed a fast and accurate and automatic method to register medical image using Wavelet Modulus Maxima. The purpose of using Wavelet Modulus Maxima is obtain fewer points in an image corresponding to sharp varying edges and effectively denoised an image by thresholding the Wavelet Modulus Maxima coefficients. The registration method proceeds as follow: 1. Find Wavelet Modulus Maxima Image2. Find convex hull3. Find principal axis using principal component4. Find rotation parameters and rotate the imageA wavelet transform provides information that essentially allows us to isolate sharp variations in the gray level, which are essentially edges or object boundaries. The Wavelet Modulus Maxima needs a wavelet that is a derivative of its scaling function. Therefore the wavelet used in their method is a quadratic spline wavelet and the cubic spline function is its scaling function. Fewer vanishing moments were chosen because of computational efficiency. A Modulus Maxima occurs at a point (a0, x0), if (a0, x0) is a local maximum of the modulus and the position of Modulus Maxima points yields the location of edges or corners in the signal. In most cases Modulus Maxima points corresponding to noise in the signal usually have a smaller absolute value. Therefore, they can accurately and effectively detect sharp varying edges and corners from the Modulus Maxima image after thresholding. For image registration, it is essential to use features that are invariant between the images. For biomedical images such as CT and MRI, the shape of the head stays the same over reasonable periods of time. Therefore, they calculated the convex hull from the Modulus Maxima image derived in previous step to get the shape of the skull and make the registration insensitive to internal changes in the brain. Once the convex hull was computed, the principal component analysis was employed to center each of the images in the image frame and determine the translation and rotation parameters. ValidationFig. 10 showed the performance of registering MRI images. They first generated a misaligned image illustrated in Fig. 10(b) from the original image in Fig. 10(a) by rotation of 30and translation of a few pixels. Fig. 10(c) is the difference between the Figs. 10(a) and (b). Fig. 10(d) is obtained from Fig. 10(a) and Fig. 10(e) is obtained from Fig. 10(b) after our image registration procedure on the images in Figs. 10(a) and (b). Fig. 10(f) is obtained by taking the pixel by pixel difference of the images in Figs. 10(d) and (e). The correlation coefficient between the registered images in Figs. 10(d) and (e) is 0.98487, reflecting accurate registration for gray level MRI images.

Fig. 10 The performance of registering MRI images.Chapter 4 ConclusionWavelet analysis in principle offers the researcher or clinician a superior alternative to standard Fourier analysis techniques. Fourier techniques are certainly adequate for some applications. However, wavelet analysis offers increased power to resolve transient and scale-specic events in neuroelectric data sets, to precisely filter neuroelectric waveforms for noise reduction, to efficiently store and transmit neuroelectric waveforms and images, and to observe and quantify their small-scale structure in time and space. For wavelet analysis to become an accepted analysis protocol for neuroscientists and clinicians, it will be necessary to demonstrate that it reveals important information about brain mechanisms or disease processes that is not readily obtained with other decomposition techniques. Although the growing experimental literature and theoretical considerations suggest that this is the case, rigorous comparative studies of wavelet techniques against alternative analysis protocols using comprehensive neuroelectric data sets have yet to be undertaken.ReferenceAkin, M. (2002). "Comparison of Wavelet Transform and FFT Methods for EEG." Journal of Medical Systems 26(3).Alfaouri, M. and K. Daqrouq (2008). "ECG Signal Denoising By Wavelet Transform Thresholding " American Journal of Applied Sciences 5(3): 276-281Ballesteros, D. M., D. M. Moreno, et al. (2010). Compression of Biomedical Signals on FPGA by DWT and Run-Length. Proceedings IEEE ANDESCON. Bogota, Colombia.Crowe, J. A., N. M. Gibson, et al. (1992). "Wavelet transform as a potential tool for ECG analysis and compression " J Biomed Eng 14(3): 268-272.Nenadic, Z. and J. W. Burdick (2005). "Spike Detection Using the Continuous Wavelet Transform." IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING 52(1).Kalayci, T., Ozdamar, O., and Erdol, N. 1994. The use of wavelet transform as a preprocessor for the neural network detection of EEG spikes. Proceedings of the IEEE Southeastcon 94 (pp. 13).Sahambi, J., Tandon, S. & Bhatt, R. (1997a). Quantitative analysis of errors due to power-line interference and base-line drift in detection of onsets and offsets in ecg using wavelets, Medical and Biological Engineering and Computing 35(6): 747751.Sahambi, J., Tandon, S. & Bhatt, R. (1997b). Using wavelet transforms for ecg characterization an on-line digital signal processing system, IEEE Engineering in Medicine and Biology Magazine 16(1): 7783.Schiff, S. J., Heller, J.,Weinstein, S. L., and Milton, J. 1994.Wavelet transforms and surrogate data for electroencephalographic spike and seizure detection. Optical Engineering, 33, 21622169.Sharman, R., J. M. Tyler, et al. (2000). "A fast and accurate method to register medical images using wavelet modulus maxima." Pattern Recognition Letters 21: 447-462.Tolba, A. S. (2002). "Wavelet Packet Compression of Medical Images." Digital Signal Processing 12(4): 441-470.

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