3.2 Construction of Adaptive Subband Coding Scheme
3.2.3 Wavelet Thresholding and Threshold Selection
3.2.4.2 Background and Problem Statement
∆=1.55T for ECG signal compression. This choice has been empirically tested to give reasonably better performance. However, when the above condition is not met the result may not be optimal. It is well known that the parameters thresholdT and∆have a strong impact on compression and reconstruction error.
Thus, considerable gains in compression ratio could probably be achieved by using different optimized parameters set for different blocks. The limitations of the above quantizers in the point of ECG compression are illustrated with sets of experiments in Chapter 2.
Therefore, it is desired to have a larger zero-zone to set more high frequency coefficients to zero to achieve the necessary compression [203, 206]. In this case, the outer-zone width increases in magnitude as the zero- zone width increases. Thus, it may quantize the nonzero wavelet coefficients in the outer-zone coarsely and this may introduce significant features distortion if those nonzero coefficients are really from the relevant features of the ECG signal. When the second quantization procedure is opted it may not remove the small wavelet coefficients likely due to noise significantly to achieve good compression results.
The nearly uniform midtread quantizer (NUMQ) is reproduced here in brief. The NUMQ has 2L+1 zones, consists of one zero-zone plusL symmetric levels of zones on each side, with zero-zone width 2T and outer-zone width∆[198]. The decision levels of the quantizer are defined for a quantizer with 2L+1 zones that L levels of zones of equal quantization step size∆ on each side. The quantization rule of the quantizer [198] is defined as
y=
y−∞, x<−T−(L−1)∆
y−m, −T−m∆≤x<−T−(m−1)∆
y0, |x| ≤T
ym, T+ (m−1)∆<x≤T+m∆
y∞, x>T+ (L−1)∆
(3.4)
where 1≤m<L, x is the input value and {yi} denote the reconstructed or output values. The optimal reconstruction values are the centroids of the input values lying between two levels. In wavelet based image coding techniques, the modified midtread quantizer is widely used to achieve simultaneous denoising and compression [205, 206]. These techniques exploit the close similarity between the extended zero-zone midtread quantization and the soft thresholding structures, and approximate the denoising process using the lossy quantization [205]. In [205],BayesShrinkthresholdTB defines the zero-zone width and the quantized nonzero wavelet coefficients are found in minimum description length (MDL) rule over the number of quantization levelsLand step size∆. The compression performance of the technique in [205] depends upon the magnitude of the additive noise. With respect to the ECG signals, the above quantizer solution is most suitable if the threshold parameterTB is successfully defined for varying characteristics of the signal with different noises and artifacts. This design can be used to quantize higher detail subbands of the corrupted ECG signal with an apriori knowledge of the distribution to appropriately model these subband coefficient histograms and a noise threshold parameter TB estimated appropriately. Thus, it requires an apriori study of characteristics of the noises and artifacts to be present in the recorded ECG signals and modeling of the histograms of the subband coefficient. However, if high compression rate is demanded at the output of a variable rate coder with the quantizer design in [205], it provides a poor compression results when the power of the input corrupting noise is low (i.e., high SNR). This happens because of the smaller zero-zone width, defined by the noise power dependent threshold TB, and the rest of the nonzero coefficients to be then quantized coarsely with large distortion [206]. Therefore, it does not balance the tradeoff between the compression rate and distortion.
To overcome the above problem, the threshold parameter TB and the number of quantization levels required in the quantizer are adapted [206]. To provide a better explanation with the ECG signal com- pression aspects, we provide a brief summary of the quantizer design philosophy [206] and then describe the limitation of this quantizer design. The zones on the positive side are indexed asm=0,1, ...L. Let z−L, ...,z−1,z0,z1, ...,zL, ... denote the boundaries of the outer-zones with reconstruction or output values y−L, ....,z−1,z0,y1, ...,yL. In standard midtread quantizer, the zero-zone width is fixed by∆z= MAX(2L+1|WC|). Based on the modeling of the histograms, the adaptive subband coder (adaptation of the zero-zone and reconstruction levels of the quantizer) is reported in order to perform simultaneous denoising and compres- sion of the image. As per the quantizer design philosophy, the decision on the zero-zone width is made by comparing∆zwith noise threshold parameterTB. For design of the quantizer in [206], the zero-zone width is defined as
z0=
TB, if∆z≤TB
∆z, if∆z>TB. (3.5)
whereTB is the BayesShrink threshold parameter. The expression in (3.5) is slightly modified to perform zero-zone decision operation in closed form. The value ofz0 used to design the outer-zone width∆. For the positive side of the quantizer, the decision levels of the quantizer are defined as zm =z0+m∆ where∆=
MAX(|WC|)−z0
L . The decision levels for the negative side are calculated asz−m=−z0−m∆using the symmetry of the quantizer. This quantizer adapts its zero-zone and reconstruction levels according to the input noise level, the statistics of the noise-free signal and the compression rate required [206]. The results show that this scheme works better than the two stage schemes employed for simultaneous denoising and compression.
As we stated in the previous section, the compression rate depends upon the two parameters zero-zone width TB (thresholding) and the outer-zone width∆(quantization). When the input noise level is low or no noise is present in the input (i.e., at high compression rates,∆z>TB), the coder in [206] functions like an adaptive subband coder that performs quantization of the wavelet coefficients by adjusting the outer-zone width∆z
for a desired compression rate. In such a case, this adaptive subband coder may introduce a significant amount of quantization noise since the relatively larger width of the outer-zone∆ will be chosen in order to achieve a desired high compression rate when the input noise level is low. In this case, a significant improvement in compression rate and distortion could be achieved if the coder jointly selects the optimal values of the threshold (i.e., zero-zone width)TB in the allowable outside region and the outer-zone width
∆for quantizing nonzero wavelet coefficients. Thus, the approach with constraint NUMQ scheme provides an optimal bit allocation and simultaneous signal denoising and compression with optimal parameters set.
In ECG compression methods, the wavelet coefficients are thresholded iteratively until a user specified target value is reached with a preset error tolerance ε. As we demonstrated in the previous section, most of the signal energy is concentrated on a small number of wavelet coefficients while these coefficients are relatively large compared to the irrelevant coefficients, and less energy being in coefficients around zero in their distributions. The EPE/PWZC/RE criterion is chosen to select the threshold parameter T in most of
the reported works [134, 137, 211]. On the other hand, the methods set the insignificant coefficients to zero based on the thresholdT achieved with rate- or quality-driven adaptive threshold algorithm [142, 143]. In all the methods, significant coefficients should be quantized at a finer resolution so that the criterion can be satisfied at the output of the coder. In such a case, the quantizer design, in (3.5), may not be efficiently embedded in those well-defined compression methods.
The above survey and investigation by several experiments show that an optimal quantization procedure can significantly improve the coding performance and reduce the computational cost, and can satisfy the specific application requirements. Therefore, a complete study of the thresholding and quantization proce- dures are presented to provide a fast algorithm for optimal nearly uniform midtread quantizer design, and to achieve simultaneous denoising and compression with good rate-distortion performance. In this work, the adaptive subband coder is attempted to perform: signal denoising with the noise threshold parameter and finer resolution of nonzero wavelet coefficients; signal compression by finding the optimal choice of the zero-zone width T and outer-zone width ∆ in rate-distortion sense; and jointly signal denoising and compression with the optimal constraint noise threshold parameterT and outer-zone width∆. The choice of NUMQ design parameters is discussed next.