2013
Brislawn, Christopher M.
Group-theoretic structure of linear phase multirate filter banks Journal Article
In: IEEE Transactions on Information Theory, vol. 59, no. 9, 2013, (LA-UR-12-20858).
Abstract | Links | BibTeX | Tags: Filter bank, free product, group, group lifting structure, JPEG 2000., Lifting, Linear phase filter, Polyphase matrix, semidirect product, Unique factorization, Wavelet
@article{Bris:13b:Group-theoretic-structure,
title = {Group-theoretic structure of linear phase multirate filter banks},
author = {Christopher M. Brislawn},
url = {https://datascience.dsscale.org/wp-content/uploads/2016/06/Group-TheoreticStructureOfLinearPhaseMultirateFilterBanks.pdf},
year = {2013},
date = {2013-08-06},
journal = {IEEE Transactions on Information Theory},
volume = {59},
number = {9},
abstract = {Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W = SW, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, CH, or, alternatively, into cosets of its scaled lifting group, SH. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a 'noncommutative vector space.'},
note = {LA-UR-12-20858},
keywords = {Filter bank, free product, group, group lifting structure, JPEG 2000., Lifting, Linear phase filter, Polyphase matrix, semidirect product, Unique factorization, Wavelet},
pubstate = {published},
tppubtype = {article}
}
Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W = SW, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, CH, or, alternatively, into cosets of its scaled lifting group, SH. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a 'noncommutative vector space.'
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.
Group-theoretic structure of linear phase multirate filter banks Journal Article
In: IEEE Transactions on Information Theory, vol. 59, no. 9, 2013, (LA-UR-12-20858).
@article{Bris:13b:Group-theoretic-structure,
title = {Group-theoretic structure of linear phase multirate filter banks},
author = {Christopher M. Brislawn},
url = {https://datascience.dsscale.org/wp-content/uploads/2016/06/Group-TheoreticStructureOfLinearPhaseMultirateFilterBanks.pdf},
year = {2013},
date = {2013-08-06},
journal = {IEEE Transactions on Information Theory},
volume = {59},
number = {9},
abstract = {Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W = SW, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, CH, or, alternatively, into cosets of its scaled lifting group, SH. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a 'noncommutative vector space.'},
note = {LA-UR-12-20858},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Unique lifting factorization results for group lifting structures are used to characterize the group-theoretic structure of two-channel linear phase FIR perfect reconstruction filter bank groups. For Dinvariant, order-increasing group lifting structures, it is shown that the associated lifting cascade group C is isomorphic to the free product of the upper and lower triangular lifting matrix groups. Under the same hypotheses, the associated scaled lifting group S is the semidirect product of C by the diagonal gain scaling matrix group D. These results apply to the group lifting structures for the two principal classes of linear phase perfect reconstruction filter banks, the whole- and half-sample symmetric classes. Since the unimodular whole-sample symmetric class forms a group, W, that is in fact equal to its own scaled lifting group, W = SW, the results of this paper characterize the group-theoretic structure of W up to isomorphism. Although the half-sample symmetric class H does not form a group, it can be partitioned into cosets of its lifting cascade group, CH, or, alternatively, into cosets of its scaled lifting group, SH. Homomorphic comparisons reveal that scaled lifting groups covered by the results in this paper have a structure analogous to a 'noncommutative vector space.'
2.
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.