Quantization of adaptive 2D wavelet decompositions

Classical linear wavelet representations of images have the drawback that they are not well suited to represent edge information. To overcome this problem, nonlinear multiresolution decompositions are being designed that can take into account the characteristics of the input signal/image. In our previous work [(G. Piella et al., July 2002), (H.J.A.M. Heijmans et al., 2002)] we have introduced an adaptive lifting framework that does not require bookkeeping but has the property that it processes edges and homogeneous regions in an image in a different fashion. The current paper discusses the effects of quantization in such adaptive wavelet decomposition. We provide conditions for recovering the original decisions at the synthesis and for relating the reconstruction error to the quantization error. Such an analysis is essential for the application of these adaptive decompositions in image compression algorithms.