Group lifting structures are introduced to provide an algebraic framework forstudying lifting factorizations of two-channel perfect reconstructionfinite-impulse-response (FIR) filter banks. The lifting factorizationsgenerated by a group lifting structure are characterized by Abelian groups oflower and upper triangular lifting matrices, an Abelian group of unimodulargain scaling matrices, and a set of base filter banks. Examples of grouplifting structures are given for linear phase lifting factorizations of the twonontrivial classes of two-channel linear phase FIR filter banks, the whole- andhalf-sample symmetric classes, including both the reversible and irreversiblecases. This covers the lifting specifications for whole-sample symmetric filterbanks in Parts 1 and 2 of the ISO/IEC JPEG 2000 still image coding standard.The theory is used to address the uniqueness of lifting factorizations. With noconstraints on the lifting process, it is shown that lifting factorizations arehighly nonunique. When certain hypotheses developed in the paper are satisfied,however, lifting factorizations generated by a group lifting structure areshown to be unique. A companion paper applies the uniqueness results proven inthis paper to the linear phase group lifting structures for whole- andhalf-sample symmetric filter banks.
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