2013
Brislawn, Christopher M.
On the group-theoretic structure of lifted filter banks Book Chapter
In: Andrews, Travis; Balan, Radu; Benedetto, John; Czaja, Wojciech; Okoudjou, Kasso (Ed.): Excursions in Harmonic Analysis, vol.~2, pp. 113-135, Birkh, Basel, 2013, (LA-UR-12-21217).
Abstract | Links | BibTeX | Tags: Filter bank, Group lift- ing structure, Group theory, group-theoretic structure, JPEG 2000, lifted filter banks, Lifting, Linear phase filter, Matrix polynomial, Polyphase matrix, Unique factorization, Wavelet
@inbook{Brislawn2013,
title = {On the group-theoretic structure of lifted filter banks},
author = {Christopher M. Brislawn},
editor = {Travis Andrews and Radu Balan and John Benedetto and Wojciech Czaja and Kasso Okoudjou},
url = {http://datascience.dsscale.org/wp-content/uploads/2016/06/OnTheGroup-TheoreticStructureOfLiftedFilterBanks.pdf},
doi = {10.1007/978-0-8176-8379-5_6},
year = {2013},
date = {2013-01-01},
booktitle = {Excursions in Harmonic Analysis, vol.~2},
pages = {113-135},
publisher = {Birkh},
address = {Basel},
series = {Applied and Numerical Harmonic Analysis},
abstract = {The polyphase-with-advance matrix representations of whole-sample symmetric (WS) unimodular filter banks form a multiplicative matrix Laurent poly- nomial group. Elements of this group can always be factored into lifting matrices with half-sample symmetric (HS) off-diagonal lifting filters; such linear phase lift- ing 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 al- gebraic 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.},
note = {LA-UR-12-21217},
keywords = {Filter bank, Group lift- ing structure, Group theory, group-theoretic structure, JPEG 2000, lifted filter banks, Lifting, Linear phase filter, Matrix polynomial, Polyphase matrix, Unique factorization, Wavelet},
pubstate = {published},
tppubtype = {inbook}
}
The polyphase-with-advance matrix representations of whole-sample symmetric (WS) unimodular filter banks form a multiplicative matrix Laurent poly- nomial group. Elements of this group can always be factored into lifting matrices with half-sample symmetric (HS) off-diagonal lifting filters; such linear phase lift- ing 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 al- gebraic 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.
: . .
1.
Brislawn, Christopher M.
On the group-theoretic structure of lifted filter banks Book Chapter
In: Andrews, Travis; Balan, Radu; Benedetto, John; Czaja, Wojciech; Okoudjou, Kasso (Ed.): Excursions in Harmonic Analysis, vol.~2, pp. 113-135, Birkh, Basel, 2013, (LA-UR-12-21217).
@inbook{Brislawn2013,
title = {On the group-theoretic structure of lifted filter banks},
author = {Christopher M. Brislawn},
editor = {Travis Andrews and Radu Balan and John Benedetto and Wojciech Czaja and Kasso Okoudjou},
url = {http://datascience.dsscale.org/wp-content/uploads/2016/06/OnTheGroup-TheoreticStructureOfLiftedFilterBanks.pdf},
doi = {10.1007/978-0-8176-8379-5_6},
year = {2013},
date = {2013-01-01},
booktitle = {Excursions in Harmonic Analysis, vol.~2},
pages = {113-135},
publisher = {Birkh},
address = {Basel},
series = {Applied and Numerical Harmonic Analysis},
abstract = {The polyphase-with-advance matrix representations of whole-sample symmetric (WS) unimodular filter banks form a multiplicative matrix Laurent poly- nomial group. Elements of this group can always be factored into lifting matrices with half-sample symmetric (HS) off-diagonal lifting filters; such linear phase lift- ing 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 al- gebraic 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.},
note = {LA-UR-12-21217},
keywords = {},
pubstate = {published},
tppubtype = {inbook}
}
The polyphase-with-advance matrix representations of whole-sample symmetric (WS) unimodular filter banks form a multiplicative matrix Laurent poly- nomial group. Elements of this group can always be factored into lifting matrices with half-sample symmetric (HS) off-diagonal lifting filters; such linear phase lift- ing 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 al- gebraic 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.