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To appear in IEEE Transactions on Signal Processing, 1999
Robust Source Coding of Images for Very Noisy Channels Lisa M. Marvel, Ali S. Khayrallah, and Charles G. Boncelet, Jr. EDICS: SP 2.7.8 Abstract Robust source coding provides the compression and noise mitigation necessary for image transmission over noisy channels. Here two methods are compared, DPCM and PTCQ, both incorporating linear and nonlinear lters. Findings show that although PTCQ has better rate-distortion properties, the DPCM scheme provides better performance in noisy channels. Additionally, better images are achieved with the nonlinear lter.
I. Introduction
In environments where the channel is prone to errors, a productive image transmission system must provide error resiliency. However, typical error correcting codes tend to break down when bit error rates are high, unless very low rate codes are employed. Unfortunately these very low rate codes utilize much of the available bandwidth. Sayood and Borkenhagen presented approach that exploits the residual redundancy in the output of the imperfect source coder to assist in correcting channel errors at the decoder [1]. This scheme yields good performance, however the e ectiveness relies upon a priori knowledge of the channel A portion of this work was presented at the 1996 IEEE International Conference on Image Processing, Lausanne, Switzerland L. Marvel is with the Army Research Laboratory, Aberdeen Proving Ground, MD, e-mail:
[email protected], phone: 410-278-6508 fax: 410-278-2934. This research was funded by the Army Research Laboratory under Cooperative Agreement No. DAAL01-96-2-0002, ATIRP Federated Laboratory A. Khayrallah is with Ericsson, Inc., Advanced Development and Research Group, Research Triangle Park, NC. C. Boncelet, Jr. is with the Department of Electrical Engineering, University of Delaware, Newark, DE.
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characteristics. This dependency could be detrimental for a time-varying channel such as that of a wireless system. Therefore, we pursue a method that will perform robust source coding, as a technique which provides both compression and noise mitigation without channel coding or knowledge of the channel characteristics. Speci cally, the utilization of two source coding schemes, Predictive Trellis Coded Quantization (PTCQ) and DPCM, incorporating linear and nonlinear prediction lters is investigated. DPCM and trellis coding are incorporated within state-of-the-art image compression algorithms such as JPEG-v6 and JPEG-2000. Use of the nonlinear lter is e ective in limiting the the e ects of noise and minimizing the propagation of error. This research is motivated by the recent work of Khayrallah [2] and the original paper on PTCQ applied to speech signals [3]. In this paper our goal is two fold, we evaluate the incorporation of the optimal linear and nonlinear lters in two compression algorithms, DPCM and PTCQ to demonstrate resiliency to noise of both the algorithms and the lters. In addition, the performance e ects of various PTCQ parameters, such as bit rate and the number of trellis states, will be exhibited. II. DPCM
Predictive coding is one of the many methods of image compression. The principle of predictive coding with images is to eliminate the inter-pixel redundancy that exists between adjacent pixels. The redundancy is removed by taking the di erence between the predicted value of the pixel and its actual value. This di erence is the prediction error, or the residual, which constitutes the new information in the pixel. The residual is then quantized and encoded as the compressed image [4].
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III. PTCQ
PTCQ is a combination of Trellis Coded Quantization (TCQ) and predictive coding. It is known that for a memoryless source, TCQ outperforms the optimal scalar quantizer and comes within 0.21 dB of the distortion rate lower bound [5]. TCQ exploits the duality between modulation for digital communication systems and source coding while following the general principles of Trellis Coded Modulation (TCM) signal expansion described by Ungerboeck [6]{[7]. When Ungerboeck designed the trellis for TCM, convolutional encoders with feedback were used. For TCQ using a feedback encoder causes diculties when bit errors occur during transmission. If the TCQ encoder output sequence is sent over a noisy channel, a single bit error can result in the TCQ decoder diverging inde nitely from the intended trellis path. This is considered a catastrophic failure and should be avoided. To remedy this catastrophic failure, a convolutional encoder without feedback can be used to specify the branch selection [8][9]. Fortunately, for every convolutional encoder with feedback, there exists a feedback-free encoder, for which any given input bit can a ect no more that 1+log2 (N ) outputs, where N is de ned as the number of trellis states [5]. A detailed description of the PTCQ algorithm can be found in the seminal PTCQ paper by Marcellin and Fischer [3]. IV. Prediction Filters
Within the schemes of interest, DPCM and PTCQ, two prediction lters are evaluated: the optimal linear predictor [4] and the nonlinear Ll lter [10]. A. Optimal Linear Prediction Filter
Linear prediction lters produce a prediction for a future value based on a linear combination of previous values. For images, the predicted value of the next pixel is the linear
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combination of the previous pixels encountered during the raster scan. The lter is optimal in the sense of minimizing the mean-squared prediction error. The coecients of the optimal linear lter are computed based on the autocorrelation of the three surrounding pixel values [4]. B. Nonlinear Ll Prediction Filter
The nonlinear lter utilized is a special case of the Ll lter. The invocation of the lter is shown in (1). A nonlinear prediction lter help limit the in uence on prediction values by outliers, extreme pixel values, which result from channel errors. The causal nonlinear lter coecients, km, are selected based on the ranking of the elements within a ve-point window, u^m; (m = 1; : : : ; 5).
w^0 = k1 u^1 + k2 u^2 + k3 u^3 + k4 u^4 + k5 u^5 where u^1 = u^i?2;j u^2 = u^i?1;j?1 u^3 = u^i?1;j u^4 = u^i;j?2 u^5 = u^1;j?1:
(1)
The lter coecients for the minimum and maximum u^m's within the lter window are set to zero, while a linear lter is invoked for the three remaining elements in the window. There are
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= 10 possible lters, indicated by k , (k = 1; : : : ; 10). The coecients of the 10
lters are calculated by taking the autocorrelation of the three remaining pixels within the window for all possible con gurations. The linear lter is selected using the coecients of the lter that minimizes mean squared error. For example, if the minimum and maximum values of the elements within the ve-point window are u^1 and u^4, the corresponding lter coecients, 1 and 4, would be set to zero and the lter coecients for the remaining elements will be identical to those of the optimal linear lter.
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V. Performance
In order to demonstrate the e ectiveness of the nonlinear lter in a noisy environment, both the DPCM and PTCQ schemes will be utilized with identical output bit rates. The PTCQ scheme implemented here is the four-state trellis designed utilizing the feedback-free convolutional encoders. The notations for the encoding schemes incorporating the linear lter and the nonlinear lter will be designated l and Ll, respectively. The performance of each algorithm as a whole within a noisy environment will also be assessed. A. Linear lter vs. Nonlinear lter
To compare the performance of the linear lter and the nonlinear lter, three images were encoded and transmitted through a noisy channel. The noisy channel is simulated by the binary symmetric channel with three levels of error probability. The reconstructed Lena image after encoding via l-DPCM and transmission though a simulated noisy channel with error probability of 10?3 is shown in Figure 1. The resultant image utilizing the Ll-DPCM encoding scheme and an identical transmission noise level is shown in Figure 2. At this low noise level, the errors caused by the imperfect channel are more evident in the l-DPCM image as small streaks which are short in duration. However, errors in the reconstructed Ll-DPCM algorithms are barely noticeable. An image of the M2 Bradley Infantry Fighting Vehicle (M2BIFV) is transmitted over a channel with probability of error 10?2:5 using the PTCQ scheme with both lter con gurations. The l-PTCQ and Ll-PTCQ algorithm results are displayed in Figures 3 and 4, respectively. In this instance, the errors are very evident in the linear implementation. They appear as abrupt streaks at error locations with long plumes as the error propagates into the surrounding pixels. Conversely, the reconstructed image from the nonlinear lter implemen-
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tation does not contain these anomalies. In fact, the error propagation is severely limited due to the elimination of outliers during the nonlinear lter's execution. Here errors appear as impulses, which is much more acceptable to the human visual system than the streaks produced by the linear method. These impulsive errors can often be eliminated by simple nonlinear post ltering, e.g., as in [11]. Once more, the noise level is increased to 10?2 ; Figures 5 and 6 show the earth image transmitted through this noisy channel utilizing l-PTCQ and Ll-PTCQ, respectively. Again it is evident that the nonlinear lter limits the propagation of error when errors occur during transmission. It should be noted that the performance of the linear lter within both schemes, DPCM and PTCQ, is very similar. By evaluating both systems, it has been determined that both the prediction lters perform well in a noiseless environment. However, when the image is transmitted over a noisy channel, the nonlinear systems demonstrate the most noise immunity. B. DPCM vs. PTCQ
The primary di erence between DPCM and PTCQ schemes incorporating the same prediction lter is the method of quantization. As previously mentioned, TCQ outperforms optimal scalar quantization in a rate distortion sense. Therefore it is expected that the PTCQ scheme will achieve lower distortion than the DPCM implementation. By implementing the four-state PTCQ scheme operating at 3 bpp and comparing it to DPCM operating at the same rate, we found this to be true for low BER. In fact, l-PTCQ outperforms l-DPCM by 3:5 dB Power Signal to Noise Ratio (PSNR), and nonlinear LlPTCQ exceeds Ll-DPCM by 3:32 dB at low BER. However, as the BER is increased to a
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level greater than 10?2:5 , the method of least distortion becomes the DPCM scheme. For these increased noise levels, l-DPCM performs 1:22 ? 2:0 dB PSNR better than l-PTCQ, and Ll-DPCM exceeds Ll-PTCQ in performance by 0:6 ? 1:27 dB PSNR. This performance is demonstrated for both lter implementations in Figures 7 and 8. This phenomenon is attributed to the construction of PTCQ, where the the output for each pixel is broken into two portions: the branch bit and the scalar quantization bits. The branch bit speci es which branch to take within the trellis, and the scalar quantization bit selects the quantization point within the partitioned scalar quantizer. If an error occurs in the branch bit, it can a ect no more than 1 + log2 (N ) inputs where N is the number of trellis states. If a bit error occurs in the scalar quantization bits, the result of this error is no more severe than a scalar quantization error in the DPCM scheme. The property that allows PTCQ to exceed DPCM at low BER, namely the trellis structure, provides a hindrance when the BER is increased. C. Variations in the PTCQ Con guration
Variations in the PTCQ con guration, namely bit rate and the number of trellis states, further exhibit that the best con guration for the noisy channel is the two-state PTCQ with rate 3{4 bpp. The two-state system provides the best performance due to the bounded error when a branch bit error occurs. As previously indicated, this bounded error is dependent on the number of states within the trellis. Therefore the trellis with the smaller number of states would prevail over those with increased number of trellis states. When the noise was increased to a very high level, greater than 10?1:5 , it was determined that the better performing con guration is the two-state PTCQ algorithm operating at a bit rate of 2 bpp. This result is based on the simple fact that each pixel value is based on fewer bits, and as
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the noise level is increased, many more bits are a ected. Subsequently, the least number of bits per pixel to be corrupted, the better the system performance at these very high noise levels. VI. Conclusions
We have presented two simple-to-implement methods of robust source coding which provide compression and error resiliency without channel coding. PTCQ has been reviewed, and it has been demonstrated that when feedback-free convolutional encoders are employed, error propagation is limited. For both the DPCM and PTCQ systems, the Ll lter provides the nonlinearity necessary to conceal the e ects of noise and minimize the propagation of error. At low BER, PTCQ outperforms DPCM by approximately 3:3 ? 3:5 dB PSNR. However, at BER greater than 10?2:5, the DPCM scheme should be utilized to achieve a gain of 0:6 ? 2:0 dB PSNR over PTCQ. In addition, variations in the PTCQ con guration parameters such as rate and number of trellis states were discussed. References
[1] K. Sayood and J.C. Borkenhagen. Use of residual redundancy in the design of joint source/channel coders. IEEE Transactions on Communications, 39(6):838{846, June 1991. [2] A. S. Khayrallah. Nonlinear lters in joint source channel coding of images. Presented at the IEEE International Symposium on Information Theory, Whistler, CA, September 1995. [3] M. W. Marcellin and T. R. Fischer. Predictive trellis coded quantization of speech. IEEE Transactions on Acoustics, Speech and Signal Processing, 38(1):46{55, January 1990. [4] R. C. Gonzalez and R. E. Woods. Digital Image Processing. Addison-Wesley Publishing, New York, NY, 1992. [5] M. W. Marcellin and T. R. Fischer. Trellis coded quantization of memoryless and Gauss-Markov sources. IEEE Transactions on Communications, 38(1):82{93, January 1990. [6] G. Ungerboeck. Channel coding with multilevel/phase signals. IEEE Transactions on Information Theory, 28(1):55{67, January 1982. [7] G. Ungerboeck. Trellis-coded modulation with redundant signal sets, parts 1 and 2. IEEE Communications Magazine, 25(2):5{21, February 1987. [8] M. Wang and T. R. Fischer. Trellis-coded quantization designed for noisy channels. IEEE Transactions on Information Theory, 40(6):1792{1802, November 1994. [9] P. Sriram and M. W. Marcellin. Performance of adaptive prediction algorithms for trellis coded quantization of speech. IEEE Transactions on Communications, 42(2/3/4):1512{1517, February/March/April 1994. [10] F. Palmieri and C. G. Boncelet, Jr. Ll- lters - a new class of order statistic lters. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37(5):691{701, May 1989. [11] R. Hardie and C. G. Boncelet, Jr. Lum lters: A class of rank-order-based lters for smoothing and sharpening. IEEE Transactions on Signal Processing, 41(3):1061{1076, March 1993.
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List of Figures
1 Lena l-DPCM, error 10?3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2 Lena Ll-DPCM, error 10?3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3 M2BIFV l-PTCQ, error 10?2:5 : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 4 M2BIFV Ll-PTCQ, error 10?2:5 : : : : : : : : : : : : : : : : : : : : : : : : : : 11 5 Earth l-PTCQ, error 10?2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 6 Earth Ll-PTCQ, error 10?2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 7 Performance of l-Filter Implementation : : : : : : : : : : : : : : : : : : : : : : 13 8 Performance of Ll-Filter Implementation : : : : : : : : : : : : : : : : : : : : : 13
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