Construction of semi-recursive PR-filter banks via generalised lifting

Abstract

Two-channel filter banks with perfect reconstruction (PR) are used in different applications. Growing attention is paid to them, especially in relation to the discrete wavelet transformation. Typically, such filter banks consist of a combination of finite impulse response (FIR) filters plus downsampling and upsampling steps. In recent years it has been shown that these classical filter operations can be substituted by the so-called lifting schemes with a saving in calculation steps and time. In general, lifting consists of two steps, prediction and update. Unfortunately, filters with more than 5/3 taps show gaps between the filter coefficients (due to the lifting structure), if one wants to avoid additional (cascaded) lifting steps. It would be advantageous, however, to have longer impulse responses, since these allow more the so-called vanishing moments, which are desirable for signal approximation and especially in image compression. This contribution shows that it is possible to extend the conventional lifting scheme to filters with longer impulse responses without gaps, if filters with infinite impulse responses (IIR) are also permitted. Basically, semi-recursive lifting structures are proposed combining extended FIR filters in the analysis stage with IIR filters in the synthesis stage. It is further investigated, whether recursive filters in the analysis stage can improve the prediction step yielding higher energy compaction which is essential for image compression applications. All filter coefficients are derived and the properties of the filters analysed.