3420 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 12, DECEMBER 1998 TABLE III MSE FOR THE TI MULTIWAVELET BIVARIATE THRESHOLDING WITH DIFFERENT SIGNAL-TO-NOISE RATIO (SNR) TABLE IV MSE FOR DIFFERENT TI SINGLE WAVELETS THRESHOLDING AND TI MULTIWAVELET BIVARIATE THRESHOLDING the TI multiwavelet bivariate denoising works well when the noise level is high. The comparison between the TI multiwavelet bivariate denoising and TI single wavelet denoising is given in Table IV. The single wavelets that are used include Daubechies 4 (D4), Symmelet 8, Haar, and Coiflet 4. TI multiwavelet bivariate denoising obtains the superior performance over all single wavelet denoising. VI. CONCLUSION In this correrspondence, we discuss and implement signal denoising by using TI multiwavelets. Instead of applying univariate thresh- olding, we experiment with bivariate thresholding as pioneered by Downie and Silverman. Experimental results show that TI multi- wavelet denoising gives better results than the conventional TI single wavelet denoising. ACKNOWLEDGMENT The authors are grateful to T. R. Downie for helpful discussions and for providing the pseudo-code for robust covariance estimation. The authors would also like to thank the anonymous reviewers for their valuable suggestions and corrections. REFERENCES [1] R. R. Coifman and D. L. Donoho, “Translation invariant de-noising,” in Wavelets and Statistics, Springer Lecture Notes in Statistics 103. New York: Springer-Verlag, 1994, pp. 125–150. [2] D. L. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory, vol. 41, pp. 613–27, 1995. [3] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika, vol. 81, pp. 425–55, 1994. [4] D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Wavelet shrinkage: Asymptopia?” J. R. Stat. Soc. B, vol. 57, pp. 301–369, 1995. [5] T. R. Downie, “Wavelet methods in statistics,” Ph.D. dissertation, Univ. Bristol, Bristol, U.K., 1997. [6] T. R. Downie and B. W. Silverman, “The discrete multiple wavelet transform and thresholding methods,” IEEE Trans. Signal Processing, vol. 46, pp. 2558–2561, Sept 1998. [7] J. S. Geronimo, D. P. Hardin, and P. R. Massopust, “Fractal functions and wavelet expansions based on several scaling functions,” J. Approx. Theory, vol. 78, pp. 373–401, 1994. [8] P. J. Huber, Robust Statistics. New York: Wiley, 1981. [9] V. Strela, P. N. Heller, G. Strang, P. Topiwala, and C. Heil, “The application of multiwavelet filter banks to image processing,” Tech. Rep., Mass. Inst. Technol., Cambridge, 1995. [10] G. Strang and V. Strela, “Orthogonal multiwavelets with vanishing moments,” Opt. Eng., vol. 33, pp. 2104–2107, 1994. [11] , “Short wavelets and matrix dilation equations,” IEEE Trans. Signal Processing, vol. 43, pp. 108–115, Jan. 1995. [12] X. G. Xia, J. Geronimo, D. Hardin, and B. Suter, “Design of prefilters for discrete multiwavelet transforms,” IEEE Trans. Signal Processing, vol. 44, pp. 25–35, Jan. 1996. High Accuracy Multiwavelets with Short Supports Artur Sowa Abstract— We show how to construct multiwavelets from wavelets. The multiwavelets resulting from such constructions retain accuracy and regularity of the original wavelets and are all supported on a shorter interval than the original wavelets. In fact, we can arrange, at the expense of the number of channels, to obtain any accuracy obtainable for compactly supported wavelets and support in a prescribed neighborhood of the unit interval. Index Terms—Image coding, harmonic analysis, signal analysis, signal processing. I. INTRODUCTION A multiwavelet multiresolution analysis consists of a sequence of subspaces such that (1) where (2) (3) It is required that there be an -tuple of functions , which satisfy (4) Manuscript received November 26, 1996; revised March 20, 1998. This work was supported by DARPA under AFOSR Contract F 49620-931-0610. The associate editor coordinating the review of this paper and approving it for publication was Dr. Truong Q. Nguyen. The author is with the Department of Mathematics, Yale University, New Haven, CT 06520 USA. Publisher Item Identifier S 1053-587X(98)08703-0. 1053–587X/98$10.00  1998 IEEE