On the Group-Theoretic Structure of Lifted Filter Banks

Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

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 JPEG2000 image coding standard. Half-sample symmetric unimodular filter banks do not form a group, but such filter banks can be partiallyfactored 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 structurehas 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 propertythat 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.

Keywords

Lifting Filter bank Linear phase filter Group theory Group lifting structure JPEG 2000 Wavelet Polyphase matrix Unique factorization Matrix polynomial 

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Copyright information

© Birkhäuser Boston 2013

Authors and Affiliations

  1. 1.Los Alamos National LaboratoryLos AlamosUSA

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