Discrete wavelet transforms that have an adaptive low pass filter

The design and performance analysis of wavelet transforms containing an adaptive low pass filter using the lifting framework is presented. Such transforms are useful in applications where it is undesirable to have smoothing artifacts in the low pass signal and it requires the signal singularities, sharp transitions and image edges to be left intact for further analysis and better visual quality of such low pass subbands. We show that the wavelet coefficients after the prediction lifting step can be used to compute a weighted mean gradient for each signal component of the low pass subband, magnitude of which can be used as a metric for spatially adaptive selection of low pass filters. The same selection can be realised in the synthesis without extra information being sent to the synthesis. Two examples using the 5/3 wavelets are shown and their lossless image coding and spatial scalability performances are analysed.