Table 1: Coecients a K of Best Low-order Complex Wavelet Lter

k Re[a k ] Im[a k ] Total length = 14, a k = a 10k 6 length, orthogonality, and image quality. [10] D. Le Gall and A. Tabatabai, \Subband coding of digital images using symmetric short kernel lters and arithmetic coding techniques," Proc. ICASSP '88, pp. 761-5, 1988. 5 comparison, the 9-tap/7-tap biorthogonal lter discussed in [2] had a mean PSNR of 29.67 dB and the well-known 5-tap/3-tap biorthogonal lter [10] gave a mean PSNR of 29.13 dB. Table 1: Coecients a k of best low-order complex wavelet lter Note that the energy in the imaginary part of the coecients is signicantly smaller than that in the real part. Discarding the imaginary part will clearly lead to a lter that is no longer perfect reconstruction, raising the question of where most of the loss occurs when this and similar lters are used in image coding. Applying this lter on the same image set and simply performing analysis followed by synthesis with no quantization or other coding of the transformed data gave a mean PSNR of 47.10 dB. Since a well designed compression scheme gives PSNR's much lower than this even for low compression ratios, it is clear that for coding applications the loss of the perfect reconstruction property is not signicant. While PSNR is far from a perfect measure of image quality, we found that the visual quality of the images correlated well with the PSNR gures given above. PSNR dierences of several tenths of a dB are generally not signicant, and we found that the image quality after compression with the complex lter of Table 1 was equivalent to that obtained with the real lters mentioned above with which we compared it. While the previously known best real lters and the best complex lters all give excellent compression, the complex lters have the property of combining advantages (symmetry, simpler boundary conditions, etc.) that can not be obtained simultaneously with real orthogonal or biorthogonal lters. In addition to the immediate practical benets this allows, it suggests interesting theoretical possibilities in terms of complex lter design. IV. Conclusions When complex solutions are admitted, orthogonal wavelet bases can be designed that possess the symmetry characteristics that are important for image coding applications. By seeking solutions in which most of the lter energy is concentrated in the real part of the lter coecients, we have identied lter banks in which the imaginary …