Minimum memory implementations of the lifting scheme

All publications on the lifting scheme up to now [1,2] consider non-causal systems, where the assumption is that the whole input signal is buffered. This is problematic if we want to use lifting in a low memory scenario. In this paper we present an analysis for making a lifting implementation of a filter bank causal, while at the same time reducing the amount of delay (or memory) needed for the whole system. The amount of memory needed for the lifting implementation of any filter bank can be shown to be always smaller than the corresponding convolution implementation. The amount of memory savings is filter bank dependent, it ranges from no savings for the Haar transform to 40% for a 2 - 10 filter bank. The amount of savings depends on the number of lifting steps as well as the length of the lifting steps used. We will also elaborate on the use of boundary extensions on each lifting step instead of the whole signal. This leads to lower memory requirements as well as simpler implementations.