  The polyphase-with-advance matrix representations of whole-sample symmetric
(WS) unimodular filter banks form a multiplicative matrix Laurent polynomial
group. Elements of this group can always be factored into lifting matrices with
half-sample symmetric (HS) off-diagonal lifting filters; such linear phase
lifting factorizations are specified in the ISO/IEC JPEG 2000 image coding
standard. Half-sample symmetric unimodular filter banks do not form a group,
but such filter banks can be partially factored into a cascade of whole-sample
antisymmetric (WA) lifting matrices starting from a concentric, equal-length HS
base filter bank. An algebraic framework called a group lifting structure has
been introduced to formalize the group-theoretic aspects of matrix lifting
factorizations. Despite their pronounced differences, it has been shown that
the group lifting structures for both the WS and HS classes satisfy a polyphase
order-increasing property that implies uniqueness ("modulo rescaling") of
irreducible group lifting factorizations in both group lifting structures.
These unique factorization results can in turn be used to characterize the
group-theoretic structure of the groups generated by the WS and HS group
lifting structures.
