A New Compression Scheme for Color-Quantized Images

904 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 12, NO. 10, OCTOBER 2002 A New Compression Scheme for Color-Quantized Images Xin Chen, Sam Kwong, and Ju-fu Feng Abstract An efficient compression scheme for color-quantized images based on progressive coding of color information has been developed. Instead of sorting color indexes into a linear list structure, a binary-tree structure of color indexes is proposed. With this structure, the new algorithm can progressively recover an image from two colors to all of the colors contained in the original image, i.e., a lossless recovery is achieved. Experimental results showed that it can efficiently compress images in both lossy and lossless cases. Typically for color-quantized Lena image with 256 colors, the algorithm achieved 0.5 bpp below state-of-the-art lossless compression methods while preserving the efficient lossy compression. Such a compression scheme is very attractive to many applications that require the ability of fast browsing or progressive transmission, and necessary, to exactly recover the original image. Index Terms Color-quantized images, data compression, index sorting, progressive refinement. I. INTRODUCTION COLOR-QUANTIZED images are often used in low-cost devices that are restricted to a small number of colors displayed or printed simultaneously. A compression scheme for these images would thus require low computational complexity, especially at the decoding end. A large number of color-quantized images stored in image databases exist, such as geographic information systems, or on the World Wide Web. When loading these images from a database or browsing them online over a slow communication link, it would be more ideal to have progressive refinement of the encoded bit-stream to enable us to recognize the image before all the compressed data has finished transmission. Furthermore, lossless compression may be needed in some cases since it is unwise to discard any information that may be useful later. When all these practical demands are taken into consideration, it is desirable to have a compression scheme with low complexity, particularly in decoding, and to allow fast browsing and exact recovery of the image necessary. The correlation between index pixels appears to be lost in color-quantized images; however, these images still preserve visual smoothness and have high dependence in the neighboring pixels as natural color images. Therefore, most methods [1], [2] Manuscript received January 8, 2001; revised August 12, 2002. This work was supported by NKBRSF G1998030606 and by City University under Grant 7001181. This paper was presented in part at the IEEE Conference on Image Processing, Thessaloniki, Greece, October 2002. This paper was recommended by Associate Editor O.K. Al-Shaykh. X. Chen is with the National Key Laboratory on Machine Perception, Peking University, Beijing 100871, China and the Department of Computer Science, University of Calornia, Santa Barbara, Goleta, CA 93106 USA (e-mail: chxin@cs.ucsb.edu). S. Kwong is with the Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong, China. J. Feng is with the National Key Laboratory on Machine Perception, Peking University, China and the Center for Information Science, State Key Lab of Machine Perception, Peking University, Beijing 100871, China (e-mail: fjf@cis.pku.edu.cn). Digital Object Identier 10.1109/TCSVT.2002.804896 (a) (b) Fig. 1. Dferent index structures. (a) Linear list structure. (b) Binary-tree structure. adopt a color reindexing method to restore the missing correlation among indexes, and then compress the re-indexed image using some conventional techniques, such as dferential pulse code modulation (DPCM) and discrete cosine transform (DCT). Usually, color indexing is such that consecutive indexes are assigned to visually similar colors [1], [3]. If predictive lossless compression is applied, a better color correlation measure is proposed in [2] in order to minimize the resulting prediction errors between indexes. Various sorting methods are also investigated, such as the simulated annealing algorithm [1], the pairwise merge (PM) heuristic method [2], and a hybrid sorting method called YCP in [4]. Although dferent color correlation and sorting algorithms are applied, they all attempt to establish a linear list structure of color indexes. The concept of progressive color refinement is first introduced in [4] as resolution refinement in wavelet coding. Progressive coding of color information refers to the generation of an embedded bitstream so that the reception of code bits can stop at any point to achieve an efficient compression of the original image with a limited number of colors. In this paper, we will present a new compression scheme that allows progressive coding of color information. In Section II, a binary-tree correlation structure of color indexes is first introduced, which is more complex than any linear list structure used in previous works. In Section III, a context-based entropy coding method proposed for binary values will explore local redundancy for achieving high compression efficiency. Compression results are reported in Section IV to illustrate the advantage of the binary-tree index structure. II. BINARY-TREE INDEX STRUCTURE A binary-tree structure, which is more complex than a linear list structure, can preserve more correlation between indexes. For example, there are three color indexes of,, and, as shown in Fig. 1, where index represents a color more similar to index than to index. It can be seen that a linear list structure cannot determine whether index or represents a color more similar to index because the distance between indexes from a linear list structure may not correctly imply the relevant distance between their color values; but the binary-tree index structure there can. From the above observations, a binary-tree index 1051-8215/02$17.00 2002 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 12, NO. 10, OCTOBER 2002 905 structure may provide an opportunity to obtain higher compression efficiency for color-quantized images. Assume that the color table of a color-quantized image is, where is the number of colors. Each index corresponds to a color value, where is the transpose operation and is defined as the occurrence number of in the image. We would like to build a binary-tree structure of color indexes by partitioning. During the partitioning process, each generated set is labeled, such that and, where and is the total number of indexes generated up to now, i.e., the number of subsets. is the representable color of, which is later defined as the weighted mean of colors in. The generating procedure of the binary-tree index structure we choose is based upon a binary-tree palette design method presented in [5], which will be incorporated into the following compression procedure. We also assume that the original index image is. Step 1: All the color indexes in an image first constitute one set as the oot node of the binary tree to be built. Then we initialize,, and a blank index image. Step 2: For each where, we first define Then the representable color and the covariance of can be calculated, respectively, by Step 3: Since is a real symmetric matrix of three by three, its eigenvector corresponding to the principal eigenvalue can be obtained with low computational cost using the Jacobi method [6]. Step 4: In order to output more important color information earlier and have similar colors cluster in the same set, the criteria used for splitting nodes and determining which nodes to split at each step is to minimize the total square error (TSE) It can also be observed that [5] the principal eigenvalue can be considered an approximate measure of the expected reduction in TSE node is split. Therefore, a specic node with the largest absolute eigenvalue of among all the leaf nodes is selected to be split. Step 5: Split the selected into two children nodes and in the following way: Step 6: Update each image pixel with the specic index in by otherwise where and denote index values at the location of and, respectively. Meanwhile, a binary value is defined by [see Fig. 1(b)] This is to say, the value 1 of indicates that the index value of was updated to be. Step 7: Set and, followed by. Now observe that a newly reconstructed image with finer quality is obtained due to one more color being contained in the color table. Step 8: If, calculate with and with, respectively, for two newly generated sets in the way described in Step 3 and then go to Step 4; or else, exit. Such a process can be repeated until each leaf node contains only one of the color indexes belonging to the original color table. When at the end, an exact lossless recovery of the original image is obtained with a reindexed color table. We should point out that exact recovery here refers to the same color value for each pixel rather than index value. Image sequences produced during the above process are then the successive finer approximations to the original image. In addition, all the operations for building a binary-tree index structure are only made over the image color table. Therefore, building such a structure can be implemented with operations, which makes the proposed algorithm practical. Typically, for the Lena image with, it takes about 60 ms on a P3-667 machine. Observe from the binary-tree structure that each color index now has a unique and variable-length binary sequence associated with it, which is formed by combining bits from the root to the leaf node, e.g., index in Fig. 1(b) is represented by. Reading a binary sequence from left to right, we know that the first bit represents the most important part of the color information, i.e., in which set this color belongs there are only two sets clustered, while the subsequent bits will further refine this color value. Because no binary sequence representation of an index is the prefix sequence of any other index, all the index values can be correctly recognized without any ambiguity when concatenating binary representations of these indexes into one sequence. Therefore, a simple and natural way we can consider to encode an image is to output its binary-tree index structure, followed by a binary sequence representation for each pixel in the image. This method can surely compress an image; however, it is very inefficient. A comparison is performed in the next section. III. CONTEXT-BASED ENTROPY CODING In our algorithm implementation, we apply an efficient bitencoding scheme, shown in Fig. 2. It can be seen that three kinds of information need to be encoded after each splitting: the leaf

906 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 12, NO. 10, OCTOBER 2002 TABLE I LOSSLESS COMPRESSION VIA THREE DIFFERENT METHODS TO ENCODE b TABLE II BIT RATES (BPP) OF LOSSLESS COMPRESSION Fig. 2. Flowchart of coding procedure. TABLE III LOSSLESS COMPRESSION (LENA QUANTIZED BY [7]) Fig. 3. Context model. node which is selected for splitting, the two representable color values and of and, and the binary values for pixels with index in that will determine whether a pixel index would be updated by or not. For,, and, we simply output them in bytes; for the binary values a context-based entropy-coding method is proposed. Context modeling is a powerful technique that exploits higher order structures to obtain high compression gains. Here, a context is formed using several causal neighboring pixels of to account for local characteristics, as depicted in Fig. 3. They are numbered in the order of Hamming distance to the encoding pixel. In order to further reduce the number of contexts, each neighboring pixel with order in the latest reconstruction image is assigned one bit in the following way: otherwise where is representable color of the th neighboring pixel in that has been coded and is the Euclidian distance in RGB color space. We also observe that the number of pixels to be encoded at each step decreases gradually as increases on the whole. Thus, in practice, only the first neighboring pixels are chosen to combine a -bit context for dferent to avoid context dilution, where is set as varies from 8 down to 1 as increases. Each will be encoded conditionally depending on its causal context by the well-known arithmetic coder [10]. Experiments have shown that such an adaptive context model gives better bit rates as compared to a context with a fixed. Fig. 4. PSNR versus bit-rate curve of Lena image. TABLE IV MSE OF LUMINANCE AND CHROMINANCE COMPONENTS (LENA AND BOATS QUANTIZED BY [5]) In fact, our bit-encoding scheme is most similar to the bit-plane coding method employed by most wavelet-based image-compression algorithms. In the binary sequence associated with an index, bits on the left that represent more important color information were encoded earlier. Therefore, it can provide our algorithm with functions of progressively coding and displaying color-quantized images. On the other hand, pixels with the same value of tend to appear together because they indicate closer color groupings, which makes easy to compress. Moreover, a context was generated for each

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 12, NO. 10, OCTOBER 2002 907 (a) (b) (c) (d) (e) (f) Fig. 5. Compression results of Cmpndd images. (a) Original image. (b) Two colors (0.061 bpp/18.8 db). (c) Four colors (0.164bpp/25.1 db). (d) 16 colors (0.436 bpp/32.1 db). (e) 64 colors (0.787 bpp/40.0 db). (f) Wavelet coder (0.436 bpp/22.8 db). by trying to make use of all the color information that the previously encoded bit planes can provide, to signicantly improve its compression performance. This is demonstrated by a comparison experiment shown in Table I, where Literal coding means the simple coding method described at the end of the previous section, Order-1 coding means using the order 1 arithmetic coding algorithm to encode the binary sequence, and Context-based coding represents the method we just proposed. IV. EXPERIMENTAL RESULTS Ten popular color images, after being quantized to 256 colors by the algorithm in [7], were used as test images. When our algorithm was applied, the reconstruction image could be displayed progressively from two colors up to 256 colors, so that an exact recovery of the original color-quantized image was obtained at the end of decoding. Table II lists the lossless compression results from GIF, luminance-intensity-based JPEG2000 described in [11], and our proposed algorithm. All data here are file byte counts and include header information. It is observed that our proposed algorithm, on the average, reduces the bit rate by 25% when compared to the luminance-intensity-based JPEG2000. Typically, for the Lena image, more than 0.5 bpp lower bit rate was achieved than for the state-of-the-art lossless compression performance known to us [2] (see Table III). For lossy compression, on the other hand, its curve of PSNR versus bit rates is depicted in Fig. 4. Compared with the most efficient lossy

908 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 12, NO. 10, OCTOBER 2002 TABLE V LOSSLESS COMPRESSION OF CMPNDD [8] TABLE VI LOSSLESS COMPRESSION COMPARED TO THE RE-INDEXING JPEG2000 METHOD IN [11] (BYTES) algorithm presented in [3], when bit rates are at 1.25 bpp, the reconstruction images have comparable mean squared errors of both luminance and chrominance components (see Table IV). Here, Lena and Boats were quantized into 256 colors using the algorithm in [5] so that a fair comparison could be made. Our algorithm is also suited for images containing various types of multimedia contents, such as the Cmpndd image of 246 colors in Fig. 5. Experiments show that although only a very low bit rate is transmitted, a good-quality image can be recovered. All characters can clearly be recognized when only 0.061 bpp is transmitted to decode into a two-color image, which is superior to the one produced by a wavelet coder, as shown in Fig. 5. Our lossless compression performance is also shown to be better than that of a predictive coding method [8], as listed in Table V. Table VI shows the compression results compared with the re-indexing JPEG2000 method in [11]. Signicant improvements of compression performance are also observed, except in image Andrene. V. CONCLUSION The advantage of our proposed algorithm is the progressive coding of color information so that it can recover the image with progressively finer image quality. This is done by building a binary-tree correlation structure of color indexes so that each index in the color table is assigned a unique and variable-length binary sequence. Such a tree structure can preserve more of the correlation that exists among color indexes than any other linear list structure. It is then explored by the proposed context-based entropy coder, as well as high dependence in the neighboring pixels, and leads to better performance for both lossy and lossless compression. Meanwhile, we code a binary sequence (instead of index number) associated with each index in the order of bit-planes so that more important part of color information can be output earlier. It can be seen that all these features are very attractive to many applications that require the ability of fast browsing or progressive transmission, and necessary, to exactly recover the original color-quantized image. At least two issues exist in our method which are worthy of further investigation. One is designing a faster entropy coder, for which we can consider using a fast binary entropy coder (e.g., Q-coder [9]) in place of the public domain implementation of arithmetic coder. The other is how to build a binary-tree index structure when color vectors are represented in any other color spaces than. REFERENCES [1] A. Hadenfelt and K. Sayood, Compression of color-mapped images, IEEE Trans. Geosci. Remote Sensing, vol. 32, pp. 534 541, May 1994. [2] N. Memon and A. Venkateswaran, On ordering color maps for lossless predictive coding, IEEE Trans. Image Processing, vol. 5, pp. 1522 1527, Nov. 1996. [3] A. Zaccarin and B. Liu, A novel approach for coding color quantized images, IEEE Trans. Image Processing, vol. 2, pp. 442 453, Oct. 1993. [4] U. Rauschenbach, Compression of palettized images with progressive coding of the color information, in Proc. SPIE Visual Communications and Image Processing, Perth, Australia, June 20 23, 2000. [5] M. Orchard and C. Bouman, Color quantization of images, IEEE Trans. Signal Processing, vol. 39, pp. 2677 2690, Dec. 1991. [6] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C: The Art of Scientic Computing, Second ed: Cambridge University Press, 1992. [7] S. Wan, S. Wong, and P. Prusinkiewicz, An algorithm for multidimensional data clustering, ACM Trans. Math. Software, vol. 14, no. 2, pp. 153 162, 1988. [8] P. J. Ausbeck, Jr., Context models for palette images, in Proc. IEEE Data Compression Conf., 1998, pp. 309 318. [9] W. Pennebaker, J. Mitchell, G. Langodn, and R. Arps, An overview of the basic principles of the Q-coder adaptive binary arithmetic coder, IBM J. Res. Devel., vol. 32, pp. 771 726, 1998. [10] I. Witten, R. Neal, and J. Cleary, Arithmetic coding for data compression, Commun. ACM, vol. 30, pp. 520 540, June 1987. [11] W. Zeng, J. Li, and S. Lei, An efficient color re-indexing scheme for palette-based compression, in Proc. IEEE Int. Conf. Image Processing, Vancouver, BC, Canada, Sept. 2000, pp. 476 479. Xin Chen received the Ph.D. degree in applied mathematics from Peking University, Beijing, China, in 2001. In 1999, he was a Research Assistant at City University of Hong Kong. He is now with the University of Calornia, Santa Barbara. His research interests include data compression, image processing, and bioinformatics. Sam Kwong received the B.Sc. degree from the State University of New York at Buffalo in 1983 and the M.A.Sc. degree from the Universtiy of Waterloo, Canada, in 1985, both in electrical engineering, respectively, and the Ph.D. degree in 1996 from the University of Hagen, Germany. From 1985 to 1987, he was a Diagnostic Engineer with the Control Data Canada, where he designed the diagnostic software to detect the manufacture faults of VLSI chips in the Cyber 430 machine. He later joined the Bell Northern Research Canada as a member of Scientic Staff. In 1990, he joined the City University of Hong Knog as a lecturer in the Department of Electronic Engineering. He is currently an associate Professor in the Department of Computer Science. Ju-fu Feng received the B.Sc. degree in 1989 and the Ph.D. degree in 1997, both in mathematics, from Peking University, Beijing, China. Since 1992, he has been with the Center for Information Science and State Key Lab of Machine Perception, Peking University. His current research interests are image processing, pattern recognition, machine learning and bioinformatics.