Tree coding of image subbands

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Abstract
Image subbands have previously been encoded with DPCM and PCM quantizers with very good results. In this paper we consider the encoding of image subbands with a tree code that is asymptotically optimal for Gaussian sources and the mean squared error (MSE) distortion measure. We first prove that optimal encoding of ideally filtered subbands of a Gaussian image source achieves the rate distortion bound for the MSE distortion measure. The optimal rate and distortion allocation among the subbands is a by-product of this proof. To help explain earlier empirical results on images and speech which show an unpredicted MSE advantage for encoding subbands, a bound is derived which shows that subband coding (SBC) is closer than full-band coding to the rate distortion bound for a finite length sequence. The aforementioned tree codes are then applied, with some compromises to true optimality, to encode the image subbands, both nonadaptively and adaptively. Since the tree codes are stochastic and the search of the code tree is selective, a relatively few reproduction symbols may have an associated squared error a hundred times larger than the target for the subband. Correcting these symbols through a postcoding procedure to be described improves the signal-to-noise ratio and visual quality significantly, with a marginal increase in total rate. The postcoding is extremely effective and typically provides a coding improvement of 1 dB or more with an overhead rate of less than 0.05 b per pixel. Our image coding results compare favorably with other coding techniques, as well as with the target distortion calculated above.

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