IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 3, NO. 3, MAY 1994 zyxwvuts 307 zyx i i i ! I ! I j I i i ~ ~ I , I i i ~ I I ! ! zyxwvutsrqponmlkjihgfedcbaZ A results for our coder both with and without entropy coding.* For the results in [7], we used the reported bit rate that also includes the bits required for encoding the codebook. The plot indicates that only one coder [7], which employs lattice vector quantization, outperforms the space-frequency localized coder that uses scalar quantization. We attribute the good performance of our coder to the space-frequency partition scheme. Further improvements can be expected using vector quantization in conjunction with this scheme. The subband coder of [2] employs scalar quantization along with Huffman coding. It performs comparably to the space-frequency localized coder without entropy coding. The good performance of the coder in [2] is perhaps not surprising because both frequency and spatial information are incorporated into the design. Finally, we point out that entropy coding is not used in [6]. Furthermore, the characteristics of the human visual system has been taken into account in both [6] and zyxwvutsrq [9]. V. SUMMARY We suggested a new image compression method based on a space- frequency partition where the product of the spatial coverage and the frequency coverage is a constant for all blocks. This means that high-frequency blocks have good spatial resolution, whereas low- frequency blocks have good frequency resolution. Each block is optimally (in the mean square sense) assigned an individual bit rate and is encoded accordingly using scalar quantization. It fully exploits the localization property of wavelets in both space and frequency and, hence, results in improvements in performance relative to traditional wavelet and subband schemes. It is found using the image “Lena” that this coder, although employing only scalar quantizers, outperforms most reported subband or wavelet based schemes [2], [6]-[lo] in terms of the PSNR versus rate. One recently reported scheme [7], however, has better performance than that of our coder. The coder in [7] employs lattice vector quantization, whereas the scheme in this paper only uses scalar quantization. It is expected that the performance of the space- frequency localized coder can be improved when vector quantization is employed. REFERENCES [I] P. A. Wintz, “Transform picture coding,” zyxwvutsrqpo Proc. IEEE, vol. 60, pp. 809-820, July 1972. [2] J. W. Woods and S. D. O’Neil, “Subband coding of images,” IEEE Trans.Acoust. Speech Signal Processing, vol. ASSP-34, pp. 1278-1288, Oct. 1986. 131 S. G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Parr. Anal. Machine Intell., vol. 11, pp. 674-693, July 1989. [4] Y. Meyer, “Orthonormal wavelets,” in Proc. Inr. Con$ Wavelets: Time- Frequency Methods Phase Space (J. M. Combes, A. Grossmann, and P. Tchamitchian, Eds.) (Marseille, France), Dec. 1987, pp. 21-37. [5] P. W. Wong, “Wavelet decomposition of harmonizable random pro- cesses,’’ IEEE Trans. Inform. Theory, vol. 39, pp. 7-18, Jan. 1993. 161 T. D. Lookabaugh and M. G. Perkins, “Application of the Princen- Bradley filter bank to speech and image compression,” IEEE Trans. Acoust. Speech Signal Processing, vol. 38, pp. 1914-1926, Nov. 1990. [7] M. Antonini, M. Barlaud, and P. Mathieu, “Image coding using lattice vector quantization of wavelet coefficients,” in Proc. ICASSP (Toronto, Canada), May 1991, pp. 2273-2276. 2Here, without entropy coding means that the quantized wavelet coefficients in each block are not entropy coded. The bits that describe the quantizer configuration, i.e., those that make up R,, are always entropy coded as described in Section 111. [8] M. J. T. Smith and S. L. Eddins, “Analysis/synthesis techniques for subband image coding,” IEEE Trans. Acoust. Speech Signal Processing, vol. 38, pp. 1446-1456, Aug. 1990. [9] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Processing, vol. 1, pp. 205-220, Apr. 1992. [IO] R. P. Rao and W. A. Pearlman, “Alphabet- and entropy-constrained vector quantization of image pyramids,” Opt. Eng., vol. 30, pp. 865-872, July 1991. Ill E. A. Riskin, “Optimal bit allocation via the generalized BFOS al- gorithm,” IEEE Trans. Inform. Theory, vol. 37, pp. 400-402, Mar. 1991. 121 M. Guglielmo, “Bit-assignment procedure for a block of uncorrelated random variables,” CSELTRapp. Technici., vol. 3, pp. 63-67, Dec. 1985. I31 I. Daubechies, “Orthonormal bases of compactly supported wavelets,” zy Commun. Pure Applied Math., vol. 41, pp. 909-996, Nov. 1988. Index Assignment for Progressive Transmission of Full-Search Vector Quantization Eve A. Riskin, Richard Ladner, Ren-Yuh Wang, and Les E. Atlas Absfruct-We study codeword index assignment to allow for progressive image transmission of fixed rate full-search vector quantization (VQ). We develop three new methods of assigning indices to a vector quantization codebook and formulate these assignments as labels of nodes of a full-search progressive transmission tree. The tree is used to design intermediate codewords for the decoder so that full-search VQ has a successive approximation character. The binary representation for the path through the tree represents the progressive transmission code. The methods of designing the tree that we apply are the generalized Lloyd algorithm, minimum cost perfect matching from optimization theory, and a method of principal component partitioning. Our empirical results show that the final method gives intermediate signal-to-noise ratios (SNR’s) that are close to those obtained with tree-structured vector quantization, yet they have higher final SNR’s. I. INTRODUCTION Data compression techniques have become important as large amounts of data need to be stored or transmitted through computer networks and telephone lines. Vector quantization (VQ) is a lossy compression technique that has been used extensively in speech and image compression. It is the extension of scalar quantization to a higher dimensional space. The motivation behind VQ is that the memory or correlation that exists between neighboring samples of a signal can be better exploited by quantizing samples together rather than individually. Discussions of vector quantization can be found in [l], [7] among others. Manuscript received February 7, 1992; revised May 7, 1993. This work was supported by the Washington Technology Center, the University of Washington Graduate Research Fund, and NSF Grants No. CCR-9108314 and MIP-9110508. This work was presented in part at the 1993 International Symposium on Information Theory, San Antonio, TX, USA, January 1993. The associate editor coordinating the review of this paper and approving it for publication was Prof. William Pearlman. E. A. Riskin, L. E. Atlas, and R.-Y. Wang are with the Department of Electrical Engineering, University of Washington, Seattle, WA ,USA 98195. R. Ladner is with the Department of Computer Science and Engineering, University of Washington, Seattle, WA, USA 98195. IEEE Log Number 9216569. 1057-7149/94$04.00 zyxwvut 0 1994 IEEE