Input Samples
Transformed parameters
Bit Stream Symbol
Stream
Prediction, Transform Model fitting, etc.
Scalar Quantizer (SQ) Vector Quantizer (VQ)
Fixed length Variable length Transformation
Decorrelates Samples
Quantization
Information Loss Occurs Here
Entropy Encoder
Efficient Lossless Representation of Symbol Steam
Figure 1.2: Basic components in a data compression system.
bols. Quantization is the process of replacing the continuous amplitude samples with approximate discrete values which are taken from a finite set of allowed values [36, 38]. The goal of quantization is to quantize the transformed coefficients or parameters so that the entropy of the resulting distribution of bin indexes is small enough and the symbols can be entropy coded at some target low bit rate. The quantization process results in a discrete amplitude signal which is different from the continuous amplitude signal by the quanti- zation error or noise. Information loss occurs in the quantization stage. When each set of parameters (or a sequence of signal values) is quantized separately, the process is known as scalar quantization (SQ). When the set of parameters is quantized jointly as a single vector, the process is known as vector quantization (VQ) or block quantization [39, 40]. Some of the scalar quantizers [36] are: uniform quantizer (midtread quantizer and midriser quantizer), dead zone quantizer, nonuniform quantizer, logarithmic quantizer, adap- tive quantizer and differential quantizer. The vector quantization (VQ) is an efficient data compression method for speech signals and images [39–41]. In VQ, the optimal encoder operates on a nearest neigh- bor or minimum distortion fashion and determines the closest codeword in its collection by an exhaustive search. A constrained search can speed up the encoding but may not guarantee to find the overall nearest neighbor in the codebook. The choice of distortion measure permits us to quantify the performance of a VQ in a manner that can be computed and used in analysis and design optimization. The performance of VQ depends crucially on the dimension, size of the codebook, the choice of code vectors in the codebook and the distortion criterion [40–42].
Entropy Coding: After quantization of data into a finite set of values, it can be encoded using an entropy coder to provide additional compression [37]. The entropy or statistical coder encodes a given set of symbols with the minimum number of bits required to represent them. If the original signal is transformed and quan- tized, then the application of lossless coding techniques [35] such as Huffman, Lempel-Ziv-Welch (LZW) or arithmetic can provide further compression without any additional loss of information. An efficient data compression system can be designed using methodologies which exploit irrelevancy and redundancy of the cardiovascular signals.
1.3.1 Signal Irrelevancy and Redundancy
For heart disease analysis, electrocardiography data is continuously recorded for 12-72 hours to monitor ischemia, ventricular and supra-ventricular dysrhythmias, conduction abnormalities, QT interval and heart rate variability [5]. The ECG signals are typically sampled at different sampling frequencies ranging from 100 Hz to 1000 Hz. For good signal quality, typically 8 to 16 bits of precision is used for samples. Hence, the amount of ECG data that has to be stored or transmitted depends upon the sampling frequency, the number of quantization levels, the number of leads and the recording time. Hence, ECG record management system and telecardiology application requires reliable and efficient compression techniques for storage and continuous transmission purposes [27]. The standard features of an ECG signal are the P wave, the QRS complex and the T wave. Additionally, small U wave (followed by the T wave) is present occasionally.
These local waves are separated by baseline or isoelectric regions. The local waves are characterized by the feature parameters such as amplitude, duration and shape. Alterations of these features are considered as diagnostic criteria for cardiac abnormalities. In medical practice, ECG signals are recorded for longer time for diagnosis of cardiovascular diseases. It can be observed that there is a concatenation of nearly similar cardiac events or periods which yields beat to beat correlation while looking at the long-term ECG signal. Digital sampling of ECG signals is generally performed by sampling at uniform intervals with rates high enough to record the fastest or short signal components [43]. Hence, many redundant samples are recorded in slow wave regions and baseline regions or isoelectric regions. Multi-channel ECG signals have three types of correlations [5, 6]: the intra-beat, the inter-beat and the inter-channel/lead. The intra- beat correlation represents the correlation between the successive samples in an ECG cycle. The inter- beat correlation represents the correlation between successive beats in a single-channel ECG signal. The correlation that exists between the signals from different channels is termed as inter-channel correlation.
Similarly, phonocardiography has correlations across the intra- and inter-heart sounds and also has long silence regions. The above signal irrelevancy and redundancy are exploited using different compression methodologies in ECG and PCG data compression methods. Note that the quality of the compressed signals is most important for diagnosis of heart diseases at the diagnostic center.
1.3.2 Coding Efficiency
Redundancy reduction ability is evaluated in terms of the following compression measures: the sample reduction ratio (SRR), the compression ratio (CR), the compressed data rate (CDR) and the decoding rate (DR). The sample reduction ratio (SRR) is defined as the ratio of the number of samples within a block to the number of retained or stored samples for compression [43]. This compression measure is utilized where the retained samples are coded at the same sample resolution. The amount of compression is often measured by the compression ratio (CR) which is defined as the ratio between the bit rate of the original signal and the bit rate of the compressed signal. The CDR (bits per second) is defined as the ratio of the product between the sampling rate (samples per second) and the amount of the compressed data (bits) to the number of samples within a block. The entropy in bits per sample is also used in many reported ECG compression methods. The coding efficiency can be measured easily. But measurement of clinical quality of the distorted signal is a challenging problem in many biomedical signal processing applications.
1.3.3 Distortion or Quality Measures
In literature many lossy compression methods are proposed. Lossy compression methods may distort the clinical information in the signal. The quality measures in the literature can be classified into two groups:
subjective and objective. Subjective evaluation is cumbersome as the physicians can be influenced by sev- eral critical factors including the knowledge, environmental conditions and motivation. Therefore a simple
mean square error (MSE) measure and its variants are commonly used for quality assessment [43]. The MSE is the L2 norm of the arithmetic difference between the original and the compressed signals. A squared error distortion measure is an assignment of cost, d(x,ex), of reproducing any original vector x as a reproduction vectorex. Given such a distortion measure, the performance of a compression method is quantified by an average distortion E[d(x,ex)]. These types of distortion measures are called as objec- tive distortion measures. A compression method will be good if it yields a small average distortion value.
Ideally a distortion measure should be manipulable to permit analysis and computable so that it can be evaluated and used in minimum distortion methods. It should be subjectively meaningful so that large or small value can correlate with bad and good subjective quality. The objective distortion measures are grouped into three categories viz global measure, local measure and similarity measure. For a given orig- inal discrete sequence x(n) = {x(1),x(2),x(3)...,x(N)} and a reconstructed or compressed sequence e
x(n) ={ex(1),ex(2),ex(3)...,ex(N)}the MSE is defined as MSE = 1
N
N n=1
∑
[x(n)−ex(n)]2 (1.1)
where N is the number of samples in the original vector. MSE distributes the error equally over all por- tions of the ECG signal. Every portion of the ECG cycle has a different diagnostic meaning and relevance.
It is a common practice to measure the performance of a compression method by normalized MSE. This corresponds to normalizing the average distortion by the average energy. The global error measures in- clude normalized MSE (NMSE), root MSE (RMSE), normalized RMSE (NRMSE), percentage root mean square difference (PRD) and signal to noise ratio (SNR). Some of the local error measures are maximum amplitude error (MAX) or peak error (PE), normalized MAX (NMAX) and standard error (StdErr). The similarity measures are normalized cross correlation (NCC) and the diagnostic distortion measures namely weighted PRD (PRD), weighted diagnostic distortion (WDD) and wavelet based weighted PRD (WWPRD).
Among these, the PRD is widely used in many compression methods. In general, the PRD is a normal- ized value which indicates the error between original and compressed signals [43]. It can be expressed as PRD= (RMSe/RMSv)×100, where RMSe and RMSv are the RMS values of the error and the ECG signal respectively. This corresponds to normalizing the average error by the RMS value of the signal. The mathematical expressions for other objective measures will be described in the next chapter. Since a highly distorted signal can be useless from a clinical point of view, a meaningful distortion measure is essential for local and global assessment. However, assessment of compressed signal quality is an open problem today.