408 IEEE COMMUNICATIONS LETTERS, VOL. 4, NO. 12, DECEMBER 2000 Fuzzy Channel-Optimized Vector Quantization Wen-Jyi Hwang, Member, IEEE, Faa-Jeng Lin, Senior Member, IEEE, and Chin-Tsai Lin Abstract—A novel fuzzy clustering algorithm for the design of channel-optimized source coding systems is presented in this letter. The algorithm, termed fuzzy channel-optimized vector quantizer (FCOVQ) design algorithm, optimizes the vector quantizer (VQ) design using a fuzzy clustering process in which the index crossover probabilities imposed by a noisy channel are taken into account. The fuzzy clustering process effectively enhances the robustness of the performance of VQ to channel noise without reducing the quantization accuracy. Numerical results demonstrate that the FCOVQ algorithm outperforms existing VQ algorithms under noisy channel conditions for both Gauss–Markov sources and still image data. Index Terms—Fuzzy clustering, vector quantization. I. INTRODUCTION V ECTOR quantizers (VQ’s) [3] have been established as important source-coding techniques in many digital communication applications. The techniques effectively re- move redundancy in the source and retain useful information for subsequent transmission. In the presence of channel noise, this removal of redundancy may cause significant performance degradation. One way to achieve some degree of robustness to channel errors is to design a channel-optimized VQ (COVQ), which optimizes a VQ under a specific noisy channel condition. The common algorithm for COVQ design is the crisp COVQ (CCOVQ) algorithm [1], which is an extension of the -means algorithm for noisy channels. While enhancing the robustness of the performance of VQ, the algorithm reduces the number of encoding regions for source coding. In fact, the algorithm tends to remove encoding regions having higher probability of transmission errors for less sensitivity to channel noise. The reduction in encoding regions lowers the performance of the VQ. Another alternative for robust VQ design involves careful assignment of the binary index associated with each codeword of the VQ [6]. The index assignment (IA) algorithms offer uncompromised performance under clean channel conditions. Nevertheless, the performance of the IA algorithms may be lower than that of the CCOVQ algorithm at the prescribed channel noise [2]. In this letter, we present a novel approach for the COVQ design, which reduces the sensitivity of VQ to channel noise without removing the encoding regions. The algorithm, termed Manuscript received June 27, 2000. The associate editor coordinating the re- view of this letter and approving it for publication was Dr. P. Cosman. This work was supported by National Science Council of Taiwan under Grant NSC 89-2213-E-033-036. The authors are with the Department of Electrical Engineering, Chung Yuan Christian University, Chungli, 32023, Taiwan R.O.C. (e-mail: wh- wang@dec.ee.cycu.edu.tw). Publisher Item Identifier S 1089-7798(00)11519-4. fuzzy COVQ (FCOVQ), performs fuzzy clustering for robust performance under noisy channel conditions. A fuzzy clustering process is an overlapping partitioning procedure where no en- coding region becomes empty during the VQ training process. As a result, the FCOVQ algorithm will not reduce the quan- tization accuracy of a VQ. In addition, since training vectors usually have features belonging to different clusters, it is not natural to assign each training vector to only one encoding re- gion for codebook construction. Fuzzy clustering algorithms are therefore more effective for VQ design [4] as compared with their crisp counterparts. Nevertheless, the existing fuzzy clus- tering algorithms such as the fuzzy K-means (FKM) algorithm and its variations [4] are sensitive to channel noise because the effects of noisy channel conditions are not considered in these algorithms.The FCOVQ algorithm takes into account the index crossover probabilities imposed by a noisy channel during the course of fuzzy clustering, and therefore reduces its sensitivity to transmission errors. Moreover, similar to the CCOVQ algo- rithm, the FCOVQ algorithm can be used in conjunction with noisy channel relaxation (NCR) [2] to prevent the algorithm from falling into a poor local optimum. In addition, as compared with the deterministic annealing approach [5], the FCOVQ al- gorithm does not require cooling schedules and phase transition processes for clustering, and therefore has lower design com- plexity. Numerical results show that the VQ’s designed by the FCOVQ algorithm obtain substantial improvement in rate-dis- tortion performance over existing approaches. II. THE ALGORITHM A VQ is specified by a partition of input space into disjoint encoding regions . Given a source vector , an index is determined and transmitted. The index indicates the encoding region which contains (i.e., ). We assume the channel for the VQ is memoryless and noisy. Let be the probability of receiving index when index is transmitted. Suppose some index is received by the decoder. According to the received index, the decoder reproduces the corresponding codeword from its codebook Let be the set of training vec- tors for the VQ design. Given codewords , and a noisy channel having crossover probabilities , the average distortion of the VQ sub- ject to the noisy channel is obtained by (1) 1089–7798/01$10.00 © 2001 IEEE