A general scheme for finding the static rate–distortion optimized ordering for the bits of the coefficients of all subbands of an N-level dyadic biorthogonal DWT

Abstract

The expected distortion decrease per bit attained by conveying the significance and refinement information of transform coefficients is derived taking into consideration their Probability Distribution Function (PDF) and their entropy. A general scheme for finding the static rate–distortion optimized ordering for the bits of the coefficients for all subbands of a generic -level dyadic biorthogonal Discrete Wavelet Transform (DWT) is given by weighting their distortion decrease per bit using the gain of each decomposition subband. Specific formulation for the Exponential Power Distribution (EPD) family is given and closed formulae are derived for the special cases of the Uniform and Laplace distributions. It is shown that under certain circumstances some refinement information of larger magnitude coefficients should be conveyed before the significant information of the current ones. The results can be applied by both conventional context modeling entropy coding algorithms or by set-partitioning algorithms which use multiple lists to keep track of significant and refinement coefficients. A very fast set-partitioning compression algorithm (Depth Embedded Block Tree — DEBT) has been developed based on variable-depth blocks and trees which uses the results presented here and achieves excellent compression ratios along with many other desired properties, e.g., embedded rate–distortion optimized lossy or lossless stream, quality or resolution scalability, and region of interest, among others. Tests show that the rate–distortion curves of most images show a significant improvement when using the weighting derived by the appropriate Exponential Power Distribution fitting in comparison to using a priori standard Uniform or Laplace Probability Distribution Function when using a non orthogonal Discrete Wavelet Transform.