Context-based lossless and near-lossless compression ... - IEEE Xplore

IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE, VOL. 3, NO. 3, SEPTEMBER 1999

231

Context-Based Lossless and Near-Lossless Compression of EEG Signals Nasir Memon, Xuan Kong, Senior Member, IEEE, and Judit Cinkler

Abstract— In this paper, we study compression techniques for electroencephalograph (EEG) signals. A variety of lossless compression techniques, including compress, gzip, bzip, shorten, and several predictive coding methods, are investigated and compared. The methods range from simple dictionary-based approaches to more sophisticated context modeling techniques. It is seen that compression ratios obtained by lossless compression are limited even with sophisticated context-based bias cancellation and activity-based conditional coding. Though lossy compression can yield significantly higher compression ratios while potentially preserving diagnostic accuracy, it is not usually employed due to legal concerns. Hence, we investigate a near-lossless compression technique that gives quantitative bounds on the errors introduced during compression. It is observed that such a technique gives significantly higher compression ratios (up to 3-bit/sample saving with less than 1% error). Compression results are reported for EEG’s recorded under various clinical conditions. Index Terms—Context modeling, data compression, EEG compression, entropy coding, error models, lossless compression, near-lossless compression.

I. INTRODUCTION

E

LECTROENCEPHALOGRAPHY (EEG) is a recording of the brain’s electric activities. Since its discovery by Burger [3], many research activities have centered on how to automatically extract useful information about the brain’s conditions based on the distinct characteristics of EEG signals. Many applications require acquisition, storage, and automatic processing of EEG during an extended period of time. For example, 24-h monitoring of a multiple-channel EEG is needed for epilepsy patients. The frequency range of a normal adult EEG lies between 0.1–100 Hz; thus, a minimum sampling rate of 200 Hz is needed. At the quantization level of 16 bit/sample, a 10-channel EEG for a 24-h period would amount to 346 Mb. Effective compression techniques are desired in order to efficiently store or transmit the huge amount of EEG data. However, liability consideration often prevents lossy data compression techniques from being used. An excellent survey of the performance of lossless EEG compression techniques can be found in [1]. In Section II of this paper, we present a new context-based lossless compression technique for EEG Manuscript received May 7, 1998; revised December 7, 1998 and February 4, 1999. The work of N. Memon was suported by the National Science Foundation under Grant NCR 9703969. The work of X. Kong was supported by the National Institutes of Health under Grant NS34145. N. Memon is with the Department of Computer Science, Polytechnic University, Brooklyn, NY 11201 USA. X. Kong and J. Cinkler are with the Department of Electrical Engineering, Northern Illinois University, DeKalb, IL 60115 USA. Publisher Item Identifier S 1089-7771(99)07091-0.

that makes use of some recent developments in lossless image coding [9]. Though the technique is computationally efficient, it provides significantly improved compression over comparable predictive and dictionary-based techniques. While lossless data compression techniques allow a perfect reconstruction of the original waveform, they do not yield very high compression ratios. Reconstructed EEG’s often serve the following two purposes: 1) visual inspection by human experts and 2) automatic analysis via signal processing algorithms. This suggests that any lossy compression technique might be acceptable as long as it can provide a reconstruction with no interference to the above operations. Hence, in Section III, we look at near-lossless compression techniques that give quantitative guarantees about the type and amount of distortion introduced. Based on these guarantees, a neurologist can be assured that the information of interest extracted from the compressed EEG will either not be affected or will be affected only within a bounded range of error. Such techniques lead to a significant increase in compression, thereby giving more efficient utilization of the precious bandwidth. We provide compression results for the proposed techniques and compare these results with some well-known techniques available in the public domain. The EEG data used for our study were provided by Dr. N. Thakor of The Johns Hopkins University, Baltimore, MD. The EEG’s were recorded from piglets experiencing hypoxic brain injury. Compression is carried out for 22 data segments extracted from different stages of a carefully designed brain injury protocol [6]. Utilization of EEG under a wide range of clinically significant conditions allows one to obtain a more complete picture of the performance of the EEG compression algorithms. Changes in brain injury protocol may alter the exact compression results, but the general trend should remain the same. The following is a brief description of the experimental protocol used. One-week-old piglets (3–4.5 kg) were anesthetized, kept on a warm blanket, and covered with a plastic bag to maintain their normal physiological temperature. After a 120min post-surgery stabilization period, hypoxia was induced for a 30-min period by ventilating the animals using a gas mixture with a fractional inspired O level of 0.1. This was followed by 5 min of room air. The animals’ airways were then occluded for 7 min to produce asphyxia. CPR was then initiated following ventilation with 100% oxygen. Anesthesia was maintained throughout the first four hours after resuscitation. EEG samples (sampled at 200 Hz) from various stages of the experiment were pieced together to give a representative view of the following important events: baseline

1089–7771/99$10.00  1999 IEEE

232


(control period before any insult), hypoxia, asphyxia, and 4-h recovery. The different compression techniques compared in this paper were evaluated for their effectiveness under various brain conditions based on the above comprehensive set of EEG segments. II. LOSSLESS COMPRESSION Dictionary-based coding is perhaps the most commonly used technique for lossless compression. Specifically, most publicly available lossless compression software is based on some variant of the Lempel–Ziv family of dictionary-based techniques, LZ77 [16] and LZ78 [17]. The reason for the popularity of dictionary-based techniques is the excellent compression performance provided, despite the low computational complexity. Although more sophisticated lossless compression techniques are known, they are usually not employed in practical applications because the improvement in compression obtained is often not considered worth the extra computation and/or memory requirements imposed by such techniques. For an excellent treatise on dictionary-based compression techniques the reader is referred to [2]. Unfortunately, dictionary-based techniques do not work well when applied to EEG data. This is because such techniques exploit the frequent reoccurrence of certain exact patterns found in the data. For example, in textual data, there are certain sequences of letters that occur frequently. However, with EEG data, due to the inherent nondeterministic nature of the EEG signal, there are no such frequently reoccurring exact patterns. Hence, alternative techniques need to be designed for lossless EEG compression. Many techniques for the lossless compression of EEG data have indeed been proposed in the literature. For an excellent recent survey of such techniques, the reader is referred to [1]. Among the various techniques compared in this survey, it was clearly demonstrated that predictive techniques performed as well as much more sophisticated and computationally involved techniques, which are based on transforms and vector quantization. Hence, in this paper, we restrict our attention to predictive coding. We show that it is possible to obtain compression performance significantly better than the more complex predictive techniques proposed in the literature. The improvement in compression is obtained by using techniques recently developed for lossless image compression, namely, context-based bias cancellation and improved error modeling techniques. The former is a simple but effective technique for removing any local trend in prediction errors and was first used in CALIC [15], a state-of-the-art lossless image coder. Improved error modeling is achieved by using a conditional entropy coding technique commonly used in lossless image compression that separates prediction errors into classes with different expected variances based on a local activity measure and employs a different entropy coder for each class. In the rest of this section, we show how these two techniques that have been used successfully for lossless image coding can be adapted to provide improved lossless EEG compression. However, we first briefly give an overview of predictive coding.

A. Predictive Coding Predictive coding has been commonly employed for waveform compression. For example, most of the earlier speech and audio coders were based on a predictive approach. In predictive coding, the transmitter (and receiver) predict the value of the current sample based on the samples that have already been transmitted (received). If the predicted value is , then only the prediction error needs to be transmitted to allow a perfect reconstruction of the current . sample through Various predictive models have been proposed and utilized for different types of data. A popular approach for modeling an EEG signal is to treat it as a time series and to describe it concisely as an autoregressive (AR) process [8] (1) where is the order of the AR process and is the prediction (residue error). error for There are many ways to estimate the parameters of the AR model. Levinson–Durbin’s method is used to determine the AR model parameters for each segment of EEG (6000 points corresponding to 60 s). Once the parameters are selected, the prediction errors are obtained via (2) is a quantization function to round its variable to the where is the original EEG sample in integer nearest integer and and will yield the format. At the receiver, combining . original One important factor in AR modeling is its model order which is data-dependent. Separate analysis [8] of the same . In Fig. 1, data set revealed that the optimum order was we show compression results for several different . The bit rate is calculated based on the entropy of the prediction error . It was seen that very little improvement is achieved for orders higher than six. Results for two other simple predictors, which involve only difference operations, are also included in the figure. These two predictors are first-order predictor and slope predictor, defined as First Order Predictor: Slope Predictor:

(3) (4)

From Fig. 1, we see that sixth-order AR predictor has the best compression result. The simple first-order predictor has a performance within 10% of the best compression results. In the remainder of the paper, we restrict ourselves to only the firstorder predictor and the sixth-order AR predictor, and we show how a predictor-based compression algorithm performance can be enhanced by utilizing some recent developments in lossless image coding [9]. B. Context-Based Bias Cancellation In predictive waveform coding, it is well known that prediction residuals generally conform to a zero-mean Laplacian

MEMON et al.: CONTEXT-BASED LOSSLESS AND NEAR-LOSSLESS COMPRESSION OF EEG SIGNALS

Fig. 1. Bits per EEG sample obtained by using various linear predictors. The original EEG data rate is 16 bit/sample. The experiment stages are baseline (B), hypoxia (H), asphyxia (A), and recovery (R). The numbers following the stage identifier is the number of minutes into that stage.

233

Fig. 3. EEG compression results in bits per sample for two linear predictors with and without bias cancellation (BC).

then estimating the conditional expectations within , by using the corresponding sample means . Since the is the most likely prediction error for a conditional mean given context , we can correct the bias in the prediction and adjusting the prediction to by feeding back . In order not to overadjust the predictor , is estimated in practice a new prediction error . This in turn leads to an improved rather than based on error predictor predictor for

Fig. 2. A realistic distribution for prediction errors (thicker line) which is in fact a weighted combination of nine different Laplacians (thinner lines).

(symmetric exponential) distribution. However, when conditioned on a sufficiently large number of contexts,1 error samples reveal systematic biases and are not zero-mean. The observed Laplacian distribution without conditioning on contexts is a composition of many distributions of different means and different variances, as shown in Fig. 2. Conditioning of the prediction error to its context provides a means to separate these distributions, and the systematic biases can then be removed by a process of bias-cancellation, employed in CALIC [14]. Conceptually, bias cancellation can be viewed as a two-stage adaptive prediction scheme via: 1) prediction of the current sample based on the past values ( represents the predictor) and 2) prediction error correction conditioned on the contexts and . the subsequent error feedback In practice, prediction error correction (bias cancellation) and is carried out by assigning each sample to a context 1 An

example of the context is the pattern of the previous prediction errors.

where is the sample mean of conditioned on context . Since only a single parameter is estimated within each context, a large number of contexts can be employed. In fact, is from zero, the more effective is the the more biased bias cancellation process. EEG compression contexts were formed by taking differfor ences between adjacent samples to obtain the quintuple . These differences effectively capture the local shape of the waveform immediately preceding the current sample . However, if is a 16-bit integer, then the total number of contexts is 2 , which is astronomically large. To reduce the number of the contexts, we quantize them into a smaller number of equivalent were quantized into two levels contexts. Specifically, the as follows: using a threshold if if

.

(5)

contexts. In our experiments, This gives a total of . In general, one can we used a threshold value use more than two quantization levels. Results show that the number of the quantization level does not change the compression results significantly [4]. Bias cancellation was then performed based on the accumulated sample means within each context. Fig. 3 gives results after bias cancellation was added to the two predictive coding techniques: the first-order predictor and the sixth-order AR predictor. It is seen that bias cancellation always results in a decrease in entropy of

234


the prediction errors, which would lead to a decrease in the coded bit rates obtained after entropy coding. It should also be noted that the improvement obtained by bias cancellation is significantly larger for the first-order predictor than for the sixth-order AR predictor. This is due to the fact that the prediction errors obtained after the first-order prediction are not white and contain redundancy due to the ineffectiveness of the prediction process. With a sixth-order AR model, prediction is more effective and the residual redundancy is much lower. Hence, bias cancellation is not as effective for AR model-based prediction errors. In light of these facts, bias cancellation can be viewed as a simple but effective way of improving prediction, because it does not require computing of optimal prediction coefficients and can be carried out with little additional complexity.

C. Activity-Based Conditional Coding The last step of predictive coding is entropy coding of prediction errors. In [1], prediction residuals are treated as an independent identically distributed (i.i.d.) source and are encoded using a Huffman code. Unfortunately, even after applying the most sophisticated prediction techniques, the residual sequence generally has ample structure which violates the i.i.d. assumption. Hence, in order to encode prediction errors efficiently, a model that captures the structure that remains after prediction is needed. This step is often referred to as error modeling [5]. The error modeling techniques employed by most lossless compression schemes proposed in the literature can be captured within the context modeling framework described in [10] and applied in [5] and [12]. In this approach, the prediction error at each sample is encoded with respect to a conditioning state or context, which is computed from the values of previously encoded neighboring samples. Viewed in this framework, the role of an error model is essentially to provide estimates of the conditional probability of the prediction error, given the context in which it occurs. This can be done by estimating the probability density function (pdf) by maintaining counts of sample occurrences within each conditioning state [12], or by estimating the parameters (variance, for example) of an assumed pdf (Laplacian, for example) as in [5]. We adopt the former approach here. One approach would be to condition the encoding of the prediction error on neighboring samples or their differences as done in the bias cancellation. However, even after quantization, the large number of conditioning states leads to two problems: the use of prohibitively large memory space for error modeling, and the lack of sufficient samples in each conditioning state during adaptive coding in order to make reliable probability estimations. Bias cancellation requires only a single parameter, the expected prediction error, to be estimated for each conditioning. For conditional coding, however, the entire pdf of the prediction error needs to be estimated for each conditioning state. Hence, the number of conditioning states needs to be kept very small. In view of this fact, we employ an activity-based conditional model, where an activity measure of the local neighborhood is computed and then quantized to a small number of levels, each

Fig. 4. EEG compression results in bits per sample for two predictive coding techniques. Bias cancellation (BC) and conditional coding (CC) improve the EEG compression results.

level representing a distinct conditioning state for entropy coding. Specifically, the conditioning state for the current sample is computed by classifying the activity of its local region based on prediction errors made at previous samples. Conditioning the prediction error on a local activity level leads to separation of prediction errors into classes of different variances. Thus, entropy coding of errors using estimated conditional probabilities improves coding efficiency. The conditioning model essentially keeps the different distributions separate in order to maintain smaller conditional entropies and to achieve improved compression. The conditioning state at a given data sample is defined as where is an level nonuniform scalar quantizer and is the weighted average of the previous prediction errors [14]. For EEG compression, the value of is computed as follows:

(6) In Fig. 4, we show results obtained with the sixth-order AR predictor, with and without conditional coding. A value of was used. It is seen that conditional coding results in a significant decrease in bit rates. D. Comparison with Other Techniques In order to compare the performance of the proposed lossless compression algorithms, Figs. 5 and 6 plot results of the same data sets with some publicly available techniques for lossless compression. Fig. 5 shows the results with the widely used compress, gzip, and lesser known bzip techniques. Bzip is a block-sorting technique that compresses files using the Burrows–Wheeler–Fenwick block-sorting transform. Bzip generally compresses better than the more conventional LZ77/LZ78-based compressors like compress and gzip. However, it is computationally more intensive than compress and gzip, requiring significantly more memory and twice the


235

the highest compression ratio, but even with a significantly lower linear prediction order the compressed file sizes are not significantly higher. Again, both proposed techniques perform better than the shorten with a predictor order up to 12. III. NEAR-LOSSLESS COMPRESSION

Fig. 5. Comparison between the proposed algorithm and three LZ77/LZ78-based algorithms. The entropy coding results show that the proposed methods have a better EEG compression performance.

In medical applications, lossless data compression techniques are preferred because they allow perfect reconstruction of the original waveform. However, lossless compression may not yield the desired bit rate. In many EEG monitoring applications, the reconstructed EEG’s often serve the following two purposes: 1) visual inspection by human experts and 2) automatic analysis via signal processing algorithms. Many automatic EEG analysis algorithms are based on the AR modeling assumption. Therefore, it is important to guarantee that the AR model parameters obtained from the reconstructed EEG are the same as the parameters obtained from the original EEG. This requirement can be satisfied with the AR-based predictive coding techniques, as the AR parameters will be available at the receiver. To ensure an accurate EEG reading by a human expert, the reconstructed EEG waveform must closely resemble the original on a point-by-point basis. This can be achieved by specifying a constraint such as (7) value of zero guarantees lossless reconstruction. A small leads to a lossy reconstruction, but the quality of the reconstruction can be tightly controlled to avoid any appreciable visual degradation of the EEG waveform. Because perfect reconstruction is not required, an optimal predictive error sequence can be selected to reduce its entropy so that a more efficient compression can be achieved. A near-lossless compression criterion similar to (7) was proposed by Ke and Marcellin [7] for image compression. Based on this criterion, they designed a context-based trellis-searched delta pulse code modulation (DPCM) technique that minimizes the entropy of the quantized prediction error sequence. The near-lossless EEG compression algorithm described here is based on the one proposed by Ke and Marcellin, in the sense that it is a context-based trellis-searched scheme and it satisfies the same near-lossless error criterion. We use an AR model of order six for the prediction step. However, unlike the previous section where lossless compression was provided, here the decoder and the encoder prediction must be based on the previous reconstructed samples instead of original samples . Instead of coding the prediction error

A

Fig. 6. Shorten 6, 12, and 6th AR with bias cancellation and conditional coding.

execution time. As can be seen from Fig. 5, the bzip compression algorithm performs the best among the LZ77/LZ78based algorithms for EEG compression. Overall, the proposed techniques yield significantly better performance than bzip. This is to be expected because bzip, gzip, and compress are string compression techniques that exploit the frequent reoccurrence of exact patterns and are more suited for text data. It should also be noted that the proposed technique based on the first-order predictor is significantly faster than bzip. We then compared our results with a publicly available waveform coder, shorten [11]. The shorten algorithm uses optimal linear prediction followed by Huffman coding of prediction residuals. The maximum order of the linear predictive filter can be specified. The default mode disables the use of linear prediction and a polynomial interpolation method is used instead. The use of the linear predictive filter generally results in improvement in compression ratio. In Fig. 6, the coded bit per sample rates are shown for the shorten compression scheme with linear prediction orders of 6 and 12. As expected, the highest linear prediction order 12 yields

(8) as in the case of lossless compression, one can code

as (9)

instead of is that a better The advantage of using compression may be achieved. The disadvantage, of course,

236

is that reconstructed sample


via (10)

will no longer be the same as . Therefore, it is necessary to develop an algorithm to properly select a member from to minimize the entropy of while maintaining the near-lossless criterion (7). A trellis was constructed describing all possible prediction that yield reconstructed values , error sequences satisfying the near-lossless requirement. The states in the trellis depend only on the reconstruction error. A cost based on the number of bits needed to code the errors was assigned to each previous states leading to it, state. Each state has and only the lowest cost state needs to be kept. Thus, at each states. The states are labeled instance there are by the allowable reconstruction for the present sample. The conditioning model utilized was the same as in the previous section. With the trellis and contexts defined, the Viterbi algorithm [13] was used to find a minimum entropy path. As the conditional probabilities within each context are not known, the algorithm was iterated to reduce the entropy. Initial conditional probabilities for contexts were chosen to have a fairly peaked probability distribution. After each iteration, new probabilities were calculated for the paths found. The procedure was repeated till the difference of the conditional entropies in two consecutive iterations was less than a preset threshold value. The EEG data was compressed using the near-lossless (to guarantee a sample-by-sample technique, with reconstruction error no greater than one), and the shorten scheme with linear prediction orders of 6 and 12. To make a fair comparison between the lossless shorten algorithm and the near-lossless algorithm, the EEG data was first quantized with a uniform scalar quantizer. The quantization steps were selected to guarantee the quantization errors to be within the . The quantized data was then compressed range of using the shorten algorithm so that the final uncompressed data had the same error bound as the near-lossless algorithm with the same setup. Fig. 7 summarizes the bits per sample results for the EEG test data set. The near-lossless algorithm, based on the sixth-order AR prediction, is superior to the shorten algorithm. at each stage grows Since the number of states exponentially with the AR prediction order , larger values pose a greater challenge to the computer memory needed. Since increasing the AR order itself does not significantly improve the compression results, a prediction order was used for the following comparison study. The EEG data were compressed using the near-lossless algorithm with , , , and . The results are plotted in Fig. 8. As expected, bit rates decrease as the allowable reconstruction error increases. Next, we compared the near-lossless compression algorithm described above with an extension of the lossless predictive coding technique described in the previous section. Extension of a lossless predictive coding technique to the case of near lossless compression requires prediction error quantization

Fig. 7. EEG compression results based on the near-lossless algorithm and the shorten algorithm. Inputs to the shorten algorithm have been properly prequantized so that both methods have the same reconstruction error bound.

Fig. 8. EEG data compression results obtained for the near-lossless algorithm with different reconstruction error bound . = 0 indicates perfect reconstruction.

according to the specified pixel value tolerance. In order for the predictor at the receiver to track the predictor at the encoder, the reconstructed values of samples are used to generate the prediction at both the encoder and the receiver. It is well known that the optimal entropy coded quantizer is a uniform quantizer with the number of levels sufficient to cover the entire range of the quantizer input. Hence, we consider a procedure where the prediction error is quantized according to the following rule. At the encoder, a symbol is generated according to (11) denotes the integer part of the argument. This where symbol is encoded, and at the decoder the prediction error is reconstructed according to (12) The above quantization procedure was adopted in predictive for a sixth-order AR model coding techniques with


Fig. 9. EEG data compression results obtained from the near-lossless technique and the AR model-based predictive coding technique with error quantization.

predictor. Its EEG compression performance was compared in Fig. 9 with the near-lossless technique with the same . The results show that extension reconstruction error of the predictive coding technique to near-lossless compression by means of uniform quantization of prediction error yields a smaller entropy than the trellis-searched near-lossless technique described above. At first, this may seem surprising because the trellis search technique explicitly minimizes the entropy of prediction errors. However, it should be noted that, due to complexity concerns, bias cancellation cannot be performed in the trellis-based technique because it would lead to an intractable number of states in the trellis. Hence, we conclude that, for near-lossless compression, a simple uniform quantization of the prediction error is perhaps a good alternative to a sophisticated technique that attempts to minimize the entropy of prediction errors. IV. SUMMARY

AND

CONCLUSIONS

Lossless predictive coding techniques are seen to be very effective in removing the temporal redundancy in EEG signals recorded under various clinical conditions. The data compression rate of the predictive techniques can be significantly improved by using bias cancellation and activity-based conditional coding. The performance enhancement is greater for prediction errors with ample residual structures. This is evident in the comparison results shown in Figs. 3 and 4 where the improvement for a simple first-order predictor is more significant than for the sixth-order AR predictor. The simple first-order predictor cannot remove all the structures within the original EEG sequence, and the additional bias cancellation and conditional coding exploits this structure to cause a significant drop in bit rate. On the other hand, the sixth-order AR predictor provides a better prediction so the structure in the prediction error is less significant for the bias cancellation and context modeling schemes to take advantage of. The above facts imply that effective EEG compression can be done by a simple first-order predictor followed by bias cancellation and conditional coding. These results are

237

better in performance than simply using an optimal AR model and Huffman coding the prediction errors. Besides, computing optimal coefficients for the AR model is computationally intensive and cannot be done in many applications requiring real-time processing. Due to the limited compression provided by lossless compression, we then examined near-lossless compression. Since near-lossless compression does not guarantee a perfect reconstruction, the prediction errors do not have to be coded exactly. A trellis-based near-lossless compression technique can improve the coding efficiency of predictive coders by selecting an error sequence with a minimum entropy. One can consider this selection process as a means to optimally quantize the prediction errors. Due to the complexity of the trellis search, a joint optimization with sophisticated prediction and error modeling (i.e., bias cancellation and conditional coding) becomes computationally intractable. Hence, the simpler approach to achieve near-lossless compression via uniform quantization of the prediction errors as specified in (11) and (12) leads to similar performance. Our results suggest that it is more efficient to use uniform quantizer with a more elaborate error modeling technique than to use the optimal quantizer with no error modeling. Finally, we emphasize that the compression algorithms are evaluated for EEG’s recorded under a whole spectrum of brain conditions: from normal baseline, to initial injury, to severe injury, to recovery. Investigation of the compression algorithms based on a comprehensive database like this is critical since an effective EEG compression algorithm must be able to perform well for all stages of brain monitoring (including possible brain injuries). REFERENCES [1] G. Antoniol and P. Tonella, “EEG data compression techniques,” IEEE Trans. Biomed. Eng., vol. 44, pp. 105–114, 1997. [2] T. C. Bell, J. C. Cleary, and I. H. Witten, Text Compression. Englewood Cliffs, NJ: Prentice-Hall, Advanced Reference Series, 1990. [3] H. Berger, “Uber das Elektrenkephalogramm des Menschen,” Arch. Psychiat. Nervenkr., vol. 87, pp. 527–570, 1929. [4] J. Cinkler, “Near-lossless data compression,” M.Sc. thesis, Northern Illinois Univ., DeKalb, IL, 1998. [5] P. Howard and J. S. Vitter, “Error modeling for hierarchical lossless image compression,” in Proc. DCC’92., 1992, pp. 269–278. [6] V. Goel, A. Brambrink, A. Baykal, D. Hanley, and N. Thakor, “Dominant frequency analysis of EEG reveals brain’s response during injury and recovery,” IEEE Trans. Biomed. Eng., vol. 43, pp. 1083–1092, 1996. [7] L. Ke and M. W. Marcellin, “Near-lossless compression: Minimumentropy constrained-error DPCM,” IEEE Trans. Image Processing, vol. 7, pp. 225–228, 1998. [8] X. Kong, V. Goel, and N. Thakor, “Quantification of injury-related EEG signal changes using Itakura distance measure,” in Proc. Int. Conf. Acoustics, Speech and Signal Processing, 1995, pp. 2947–2950. [9] N. D. Memon and X. Wu, “Recent developments in lossless image compression,” The Computer J., vol. 40, nos. 2/3, pp. 127–136, 1997. [10] J. J. Rissanen and G. G. Langdon, “Universal modeling and coding,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 12–22, 1981. [11] T. Robinson, “SHORTEN: Simple lossless and near-lossless waveform compression,” Tech. Rep. CUED/F-INFENG/TR.156, Cambridge University, Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, UK. [Online]. Available FTP: ftp://svrftp.eng.cam.ac.uk/pub/comp.speech/sources/ [12] S. Todd, G. G. Langdon, and J. J. Rissanen, “Parameter reduction and context selection for compression of gray scale images,” IBM J. Res. Dev., vol. 29, pp. 188–193, 1985. [13] G. D. Forney, Jr., “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268–278, Mar. 1973.

238


[14] X. Wu and N. Memon, “Context-based adaptive lossless image coding,” IEEE Trans. Commun., vol. 45, pp. 437–444, 1997. [15] X. Wu, “Efficient and effective lossless compression of continuoustone images via context selection and quantization,” IEEE Trans. Image Processing, vol. 6, pp. 656–664, 1997. [16] J. Ziv and A. Lempel, “A universal algorithm for data compression,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 337–343, 1977. , “Compression of individual sequences via variable-rate coding,” [17] IEEE Trans. Inform. Theory, vol. IT-24, pp. 530–536, 1978.

Nasir Memon received the B.E. degree in chemical engineering and the M.Sc. degree in mathematics from the Birla Institute of Technology, Pilani, India, in 1981 and 1982, respectively, and the M.S. and Ph.D. degrees, both in computer science, from the University of Nebraska in 1989 and 1992, respectively. He has held an Assistant Professor position at Arkansas State University and at Northern Illinois University. During 1997–1998, he was a Visiting Faculty Member at the Imaging Technology Department, Hewlett Packard Research Laboratories, Palo Alto, CA. He is currently an Associate Professor in the Computer Science Department, Polytechnic University, Brooklyn, NY. His research interests include data compression, data encryption, image processing, multimedia data security, and multimedia communication and computing. He has published more than 75 articles in journals and conference proceedings and holds two patents in image compression. From 1994 to 1996, he was actively involved in the formation of a new international standard on lossless image compression called JPEG-LS. He has been the principal investigator on several funded research projects. Prof. Memon received the National Science Foundation Career award in 1996.

Xuan Kong (S’87–M’90–SM’98) received the B.S. degree from Sichuan University in 1984, the M.S. degree from the University of Manitoba in 1986, and the M.S.E. and Ph.D. degrees from The Johns Hopkins University, Laurel Park, MD, in 1989 and 1991, all in electrical engineering. He is currently an Associate Professor in the Department of Electrical Engineering at Northern Illinois University, DeKalb. His research interests include adaptive signal processing and its applications in acoustic and biomedical signal processing. He is the co-author (with V. Solo) of the book, Adative Signal Processing Algorithms (Englewood Cliffs, NJ: Prentice-Hall, 1995). Dr. Kong is a member of Eta Kappa Nu.

Judit Cinkler received the B.S. and M.S. degrees in electrical engineering from Northern Illinois University, DeKalb, in 1995 and 1998, respectively. Currently, she is with the Fixed Products Design Center-Core Software, iDEN group, Motorola Inc., Schaumburg, IL. Her interests are in digital communications and data compression.