1.8 Organization of the Thesis
2.1.1 Wavelet transform as sparsifying basis in CS-based applications
Discrete Wavelet Transform (DWT) is an alternate way of signal representation. DWT has very good energy compaction and time-frequency localization property which helps grossly segment the clinical information present in ECG signals in different wavelet subbands (Figure 2.1). The gross segmen- tation of features in specific wavelet subbands gives more control over clinically relevant information and helps retain them in various signal processing tasks. This is the reason DWT has been a very attractive tool among the signal processing community for analyzing biomedical signals like ECG.
DWT has been extensively used for ECG data compression applications [85–88]. DWT helps confine the diagnostically important ECG features within a few low frequency subbands. Therefore, a large number of high frequency wavelet subband coefficients (especially fromcD1,cD2, andcD3 subbands) can be zeroed without affecting the clinical features of ECG. This helps in data compression and signal denoising as well. However, despite their high data compression abilities, the DWT-based techniques are not found suitable for resource-limited applications, such as WBAN-enabled ambulatory ECG monitoring. This is because of different processes involved in transform-based data encoders such as noise-cancellation, filtering, thresholding, etc., which increase the circuitry complexity and results in a slower, bulkier and power-inefficient data-encoder [1, 39]. WBAN-enabled ECG telemonitoring involves miniaturized sensors driven by power-limited batteries. So, in these applications, transform- domain ECG compressors are not found suitable. CS-based embedded data compression techniques
2.1 Basics on Compressed Sensing
0 500 1000 1500 2000 2500 3000 3500 4000
−2000 0
2000 Time−domain ECG signal
0 500 1000 1500 2000 2500 3000 3500 4000
500 1000
1500 cA
6 Approximation subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−1000 0 1000
cD6 Detail subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−1000 0 1000
Amplitude
cD5 Detail subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−500 0
500 cD
4 Detail subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−500 0 500
cD3 Detail subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−100 0 100
cD2 Detail subband signal
0 500 1000 1500 2000 2500 3000 3500 4000
−50 0 50
Samples
cD1 Detail subband signal
Figure 2.1: Distribution of ECG information in various wavelet subbands
involve low-complex and energy-efficient data reduction procedure and emerged as an effective alter- native to wavelet-based ECG compression techniques for telemonitoring applications [1]. In CS-based
2. Compressed Sensing for ECG Signals- A Review
applications, sparsity feature of DWT is employed for data recovery task. DWT plays a vital role as the sparsifying transform (Ψ) in CS-based ECG data compression applications [1, 41, 45, 46].
In DWT domain, the signal is decomposed and represented in terms of finite-energy waves called wavelets. Here, wavelets are the basis functions in DWT representation. Wavelets can be defined as the families of functionsψa,b(t) generated by dilation and translation of a single mother waveletψand is given as follows [89, 90]:
ψa,b(t) = 1
√aψ t−b
a
(2.4) where a, b are the dilation (scaling) and translation parameters. The dilated wavelet function corresponds to the low frequency components, whereas the translated version corresponds to the high frequency components of the signal. Therefore, DWT allows a multi-resolution decomposition of the signal, where the signal is separated into finer parts called “details” and coarser representation called
“approximation” [91, 92]. A function f(t) can be represented as the linear combination of shifted and scaled version of the mother wavelet Ψ(t) as:
f(t) =X
a
X
b
wa,bψa,b(t) (2.5)
where ψa,b(t) is the shifted and scaled version of the mother wavelet Ψ(t) and wa,b are the wavelet coefficients. Hence, from (2.5), the wavelet coefficients wa,b of a signal f(t) are given by [93]:
wa,b=hf(t), ψa,b(t)i (2.6) Figure 2.2 shows an amplitude plot of wavelet coefficients in different ECG channels. Seven level wavelet decomposition is performed for each of the eight fundamental channels/leads using same Daubechies wavelets (db4). The clinically important ECG information are captured by only few significant wavelet coefficients in low frequency subbands of each lead. This results in a sparse repre- sentation of each individual lead signal and a joint sparse representation by all leads.
Multi-resolution analysis of DWT allows detailed study of subband level signals. It is obtained through filter bank implementation of DWT, where signal is passed through low-pass and high-pass filters. Low-pass filtered version of the signal is again filtered through a low-pass and high-pass filter pair in the next level of wavelet decomposition and this continues for further levels. If Φ(t) is a scaling function and Ψ is a wavelet function, approximation (cAn(b)) and detail (cDj(b)) subbands coefficients
2.1 Basics on Compressed Sensing
0 100 300 200
500 400 600
Lead I Lead II V1 V3 V2
V5 V4 V6
0 1 2 3 4
Wavelet coefficients Multichannel ECG
Amplitude
cD2 cD1
cD4 cD3
cD5
cA7+cD7+cD6
Figure 2.2: Amplitude plot of ECG wavelet coefficients in all eight independent channels in various subbands for 7 level wavelet decomposition
aftern-level decomposition can be defined as: cAn(b) =hf(t), φa,b(t)iandcDj(b) =hf(t), ψa,b(t)i, j = 1, ..., n. Taking the example of ECG signal, the nlevel wavelet decomposition of each lead of MECG gives n+ 1 number of subbands: cAn, cDn, cDn−1, ..., cD2, cD1. Here, the subbandcAn consists of the approximation coefficients and carries most of the low frequency information of the ECG signal. The finer details (or high frequency ECG information) are associated with n number of detail subbands cDj, j = 1 to n. This can also be verified from the Figure 2.1, where the distribution of ECG information is shown in various subbands.
In order to decide the number of levels of wavelet decomposition, frequency range of wavelet subbands should be known. The bandwidth ∆Fj ofj-th wavelet subband is given by [81]:
2−j−1.Fs≤∆Fj ≤2−j.Fs (2.7) whereFsis the sampling frequency of the signal. There is also a heuristic formula proposed by Fahoum
2. Compressed Sensing for ECG Signals- A Review
et al. [80] to decide the number of levels (n) and is given as:
n= [log2(Fs)−2.96] (2.8)