2.3 One-Dimensional Wavelet Based ECG Compression Methods
2.3.5 Quantization Approaches for Wavelet Coefficients
The second step of the wavelet compression is quantization. The purpose of quantization is to reduce data entropy by compromising the precision of the data and to reduce the irrelevance in a signal [188, 189].
Reducing entropy allows more compression. The quantizer typically performs an operation such as trunca- tion or rounding to the nearest integer, thus creating an integer valued output. This process may introduce quantization errors, distortion or noise, to the signal. Thus, the original signal may not be recovered exactly after quantization. It is therefore very important to design a quantization strategy which selectively quan- tizes the wavelet coefficients and preserves the ECG quality. There is a tradeoff between signal quality and degree of quantization. It also has significant impact on the bit rate of the encoder. Thus, the performance is governed by the rate-distortion theory. A large quantization step size can produce high compression ratios with unacceptably large clinical feature distortion. Unfortunately, finer quantization leads to lower com- pression ratios. Then, the question is how to quantize the wavelet coefficients most efficiently. The design of the quantizer has a significant impact on the amount of compression obtained and loss incurred in a lossy compression scheme. Therefore, we present different quantization approaches applied to the wavelet coefficients of the ECG signals. In this section, we also present some of other simple quantization schemes which are reported for subband coding of images and then we describe the issues involved in the reported quantization schemes in order to present the motivation for the present research work.
The scalar quantizer is the simplest of all lossy compression schemes that is widely employed to code
the retained wavelet coefficients. In [126], each group of coefficients is quantized using the number of bits per coefficient,b, which is given by
b=1+
log2
1+ cl−cs 2×TOL
(2.10) whereclandcsdenotes the largest and the smallest coefficient within the group, TOL specifies the maximum allowable quantization error per coefficient, and b.c denotes the integer part. A value of TOL=0.008 is used, and the number of bits is restricted within 5≤b ≤16. In [129], uniform quantization with bin or step size of ∆ is used for each wavelet subband coefficients. The ∆ is found according in adaptive manner for a desired quantization MSE. In [131], a uniform scalar quantizer with 16-bit resolution is used because of its simplicity and ease of implementation. Results presented in [131] indicate that the number of quantization levels to be used has an impact on the quality of the compressed signal and the compression ratio. But the rate-distortion performance may be improved if the number of quantization levels of the quantizer is chosen based on the dynamic range of the significant coefficients vector. In [134], the retained coefficients are grouped and then encoded by assigning a fixed number of bits for each coefficient. A sign bit (‘1’ for negative coefficient and ‘0’ for a positive coefficient) is added to encode the sign of the retained coefficient. The retained coefficients are stored using 7 bits signed representation and then the compression performances are compared with the results of the SPIHT and the ASEC algorithms. Note that there is no bits assigned to the integer part of the coefficient because it is always zero due to the preprocessing steps that were applied to the ECG signal before applying the DWT. Disadvantages of this approach will be discussed in later with a set of experiments. In [137], the nonzero wavelet coefficients (NZWC) were transformed into in the range 0-255 using the 8 bits linear quantizer defined by
QNZWC=255× NZWC−NZWCmin
NZWCmax−NZWCmin (2.11)
where QNZWC denotes the quantized NZWC vector, NZWCmax and NZWCmin denotes the maximum and minimum value of the NZWC vector. Note that fixed quantization scheme is used for the coding of the coefficients of the time varying PQRST morphology. In general, for a given quantizer resolution, the quantization step∆is determined for each signal based on its dynamic range. The quantizer replaces each coefficient with ∆[NZWC/∆], referred as quantized NZWC (QNZWC). This is the simplest quantization rule which is often used in the wavelet based methods.
An excellent overview of scalar quantization scheme and its variants are discussed for coding of wavelet coefficients of the image [195]. In [195], the issues of different quantization schemes are discussed. We provide here the limitations of the schemes reported in [195], since there is no analytical framework of quantization for coding of wavelet coefficients of the ECG signals. The JPEG 2000 uses a dead-zone uniform scalar quantizer to wavelet coefficients resulting from the DWT of image samples [195]. The
midtread quantizer [195, 196] with step size∆yields quantization indexesq q=Q(c) =sign(c)
|c|
∆ +τ
(2.12) wherecdenotes a wavelet coefficient of the ECG signal block andτ is a parameter controlling the width of the central zero-zone. Whenτ=0.5, the quantizer is uniform, whileτ=0 corresponds to the case in which the zero-zone width is 2∆. The wavelet coefficients inside the interval (−∆,∆) are quantized to zero. Thus, this interval is referred as dead-zone (or zero-zone) [195]. The width of zero-zone interval is 2∆while all other intervals are of width∆. The inverse quantizer is given by
˜
c=Q−1(q) =
0, q=0
sign(q)(|q|+ξ)∆ q6=0 (2.13)
where parameterξ is in the range [0,1] (typicallyξ =0.5). An increase of∆implies that greater compres- sion can be achieved, but with low quality compressed signal. It is known that compressed quality and rates are controlled by the amount of quantization applied to each coefficient. The quantization step sizes are specified relative to the nominal dynamic range of the coefficients. Part II of the standard generalizes the zero-zone midtread quantizer to allow more flexible zero-zone selection while maintaining the fixed width
∆ for all other intervals [195]. In [200], the performance of the three quantizers such as uniform scalar quantizer, zero zone midtread quantizer and trellis coded quantization is tested for coding wavelet coeffi- cients of the image. Among these three quantizers, the performance of the USQ with control zero-zone is significant and its computational complexity is lower than the universal trellis coded quantizer [200]. For each threshold T, the best choice of parameter∆ in the rate-distortion sense can be determined by some searching algorithm. However, for natural image, the value of∆should be among 1.2T−1.8T, specifically
∆=1.55T, for good compression performance [200, 201]. In this way, only one parameter (the threshold T) should be adjusted to determine the specific quality or rate. Then, this quantizer is applied to the the wavelet coefficients of the ECG signal. In [200, 202, 203], the wavelet coefficients vector is quantized with a quantizer defined in (2.14), whereT is 0.5∆<T <∆.
dp=
((2p−1)δ,(2p+1)δ) p=−2,−3,−4, ....
(−3δ,−T) p=−1
(−T,T), p=0
(T,3δ), p=1
((2p−1)δ,(2p+1)δ), p=2,3,4, ....
(2.14)
Op=
0, p=0
p∆, p=±1,±2,±3, ... (2.15)
where, ∆ is the quantization step size,δ is the half of the step size,dp is the pth decision level, Op is the pth output level andT is the threshold of the zero-zone. If the threshold,T =δ, then the characteristics of
this quantizer is same as that of midtread quantizer. The width of the zero-zone is 2T. In [142], the nonzero wavelet coefficients of the threshold vector are quantized using fixed quantizer as mentioned before in (2.11) with 8 bits resolution. In [143], the nonzero wavelet coefficients (NZWC) of the thresholded vector are quantized as in (2.16)
QNZWC= (2b−1)× NZWC−NZWCmin
NZWCmax−NZWCmin+2b (2.16)
whereb is the quantizer resolution. The parameter bstands for the quantization resolution determined in such a way that the PRD of the compressed signal resulting from the quantized wavelet coefficients, denoted QPRD, meets the specified accuracy of QPRD. The minimal resolution(b+1)is chosen and attributed to the quantizer from the set of bits{7,8,9,10,11}. If a suitable distortion measure is defined, an effective optimal quantizer can be designed by a simple iterative algorithm. The advantages and disadvantages of the above quantizers are presented in section 2.6 with different sets of the experiments.