Ordering for Embedded Coding of Wavelet Image Data Based on Arbitrary Scalar Quantization Schemes

Many modern wavelet quantization schemes specify wavelet coefficient step sizes as continuous functions of an input step-size selection criterion; rate control is achieved by selecting an appropriate set of step sizes. In embedded wavelet coders, however, rate control is achieved simply by truncating the coded bit stream at the desired rate. The order in which wavelet data are coded implicitly controls quantization step sizes applied to create the reconstructed image. Since these step sizes are effectively discontinuous, piecewise-constant functions of rate, this paper examines the problem of designing a coding order for such a coder, guided by a quantization scheme where step sizes evolve continuously with rate. In particular, it formulates an optimization problem that minimizes the average relative difference between the piecewise-constant implicit step sizes associated with a layered coding strategy and the smooth step sizes given by a quantization scheme. The solution to this problem implies a coding order. Elegant, near-optimal solutions are presented to optimize step sizes over a variety of regions of rates, either continuous or discrete. This method can be used to create layers of coded data using any scalar quantization scheme combined with any wavelet bit-plane coder. It is illustrated using a variety of state-of-the-art coders and quantization schemes. In addition, the proposed method is verified with objective and subjective testing.