A. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo :
“Lossless image compression using integer to integer wavelet transforms ,”
pp. 596–599
in
1st international conference on image processing
(Santa Barbara, CA, 26–29 October 1997 ),
vol. 1 .
IEEE (Piscataway, NJ ),
1997 .
incollection

Abstract
People
BibTeX
@incollection {key71393781,
AUTHOR = {Calderbank, A. R. and Daubechies, I.
and Sweldens, W. and Yeo, B.-L.},
TITLE = {Lossless image compression using integer
to integer wavelet transforms},
BOOKTITLE = {1st international conference on image
processing},
VOLUME = {1},
PUBLISHER = {IEEE},
ADDRESS = {Piscataway, NJ},
YEAR = {1997},
PAGES = {596--599},
DOI = {10.1109/ICIP.1997.647983},
NOTE = {(Santa Barbara, CA, 26--29 October 1997).},
ISBN = {9780818681837},
}
A. R. Calderbank, I. Daubechies, W. Sweldens, and B.-L. Yeo :
“Wavelet transforms that map integers to integers ,”
Appl. Comput. Harmon. Anal.
5 : 3
(July 1998 ),
pp. 332–369 .
MR
1632537
Zbl
0941.42017
article

Abstract
People
BibTeX

Invertible wavelet transforms that map integers to integers have important applications in lossless coding. In this paper we present two approaches to build integer to integer wavelet transforms. The first approach is to adapt the precoder of Laroia et al. , which is used in information transmission; we combine it with expansion factors for the high and low pass band in subband filtering. The second approach builds upon the idea of factoring wavelet transforms into so-called lifting steps. This allows the construction of an integer version of every wavelet transform. Finally, we use these approaches in a lossless image coder and compare the results to those given in the literature.

@article {key1632537m,
AUTHOR = {Calderbank, A. R. and Daubechies, Ingrid
and Sweldens, Wim and Yeo, Boon-Lock},
TITLE = {Wavelet transforms that map integers
to integers},
JOURNAL = {Appl. Comput. Harmon. Anal.},
FJOURNAL = {Applied and Computational Harmonic Analysis},
VOLUME = {5},
NUMBER = {3},
MONTH = {July},
YEAR = {1998},
PAGES = {332--369},
DOI = {10.1006/acha.1997.0238},
NOTE = {MR:1632537. Zbl:0941.42017.},
ISSN = {1063-5203},
}
I. Daubechies and W. Sweldens :
“Factoring wavelet transforms into lifting steps ,”
J. Fourier Anal. Appl.
4 : 3
(1998 ),
pp. 247–269 .
This was later published (without an abstract) in Wavelets in the geosciences (2000) .
MR
1650921
Zbl
0913.42027
article

Abstract
People
BibTeX

This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists, and expressed by the formula
\[ \mathrm{SL}\bigl(n;\mathbf{R}[z,z^{-1}]\bigr)=E\bigl(n;\mathbf{R}[z,z^{-1}]\bigr) ;\]
it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.

@article {key1650921m,
AUTHOR = {Daubechies, Ingrid and Sweldens, Wim},
TITLE = {Factoring wavelet transforms into lifting
steps},
JOURNAL = {J. Fourier Anal. Appl.},
FJOURNAL = {The Journal of Fourier Analysis and
Applications},
VOLUME = {4},
NUMBER = {3},
YEAR = {1998},
PAGES = {247--269},
DOI = {10.1007/BF02476026},
NOTE = {This was later published (without an
abstract) in \textit{Wavelets in the
geosciences} (2000). MR:1650921. Zbl:0913.42027.},
ISSN = {1069-5869},
}
I. Daubechies, I. Guskov, and W. Sweldens :
“Regularity of irregular subdivision ,”
Constr. Approx.
15 : 3
(1999 ),
pp. 381–426 .
MR
1687779
Zbl
0957.42022
article

Abstract
People
BibTeX

We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four-point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always \( C^1 \) ; under slightly stronger restrictions we show that the limit function is almost \( C^2 \) , the same regularity as in the regularly spaced case.

@article {key1687779m,
AUTHOR = {Daubechies, I. and Guskov, I. and Sweldens,
W.},
TITLE = {Regularity of irregular subdivision},
JOURNAL = {Constr. Approx.},
FJOURNAL = {Constructive Approximation. An International
Journal for Approximations and Expansions},
VOLUME = {15},
NUMBER = {3},
YEAR = {1999},
PAGES = {381--426},
DOI = {10.1007/s003659900114},
NOTE = {MR:1687779. Zbl:0957.42022.},
ISSN = {0176-4276},
}
I. Daubechies, I. Guskov, P. Schröder, and W. Sweldens :
“Wavelets on irregular point sets ,”
R. Soc. Lond. Philos. Trans. Ser. A, Math. Phys. Eng. Sci.
357 : 1760
(1999 ),
pp. 2397–2413 .
MR
1721247
Zbl
0945.42019
article

Abstract
People
BibTeX

In this article we review techniques for building and analysing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular, we focus on subdivision schemes and commutation rules. Several examples are included.

@article {key1721247m,
AUTHOR = {Daubechies, Ingrid and Guskov, Igor
and Schr\"oder, Peter and Sweldens,
Wim},
TITLE = {Wavelets on irregular point sets},
JOURNAL = {R. Soc. Lond. Philos. Trans. Ser. A,
Math. Phys. Eng. Sci.},
FJOURNAL = {The Royal Society of London. Philosophical
Transactions. Series A. Mathematical,
Physical and Engineering Sciences},
VOLUME = {357},
NUMBER = {1760},
YEAR = {1999},
PAGES = {2397--2413},
DOI = {10.1098/rsta.1999.0439},
NOTE = {MR:1721247. Zbl:0945.42019.},
ISSN = {1364-503X},
}
I. Daubechies and W. Sweldens :
“Factoring wavelet transforms into lifting steps ,”
pp. 131–157
in
Wavelets in the geosciences: Collection of the lecture notes of the school of wavelets in the geosciences
(Delft, Netherlands, 4–9 October 1998 ).
Edited by R. Klees and R. Haagmans .
Lectures in Earth Sciences 90 .
Springer (Berlin ),
2000 .
This was earlier published (with an abstract) in J. Fourier Anal. Appl. 4 :3 (1998) .
Zbl
0963.65154
incollection

People
BibTeX
@incollection {key0963.65154z,
AUTHOR = {Daubechies, Ingrid and Sweldens, Wim},
TITLE = {Factoring wavelet transforms into lifting
steps},
BOOKTITLE = {Wavelets in the geosciences: {C}ollection
of the lecture notes of the school of
wavelets in the geosciences},
EDITOR = {Klees, Roland and Haagmans, Roger},
SERIES = {Lectures in Earth Sciences},
NUMBER = {90},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2000},
PAGES = {131--157},
DOI = {10.1007/BFb0011095},
NOTE = {(Delft, Netherlands, 4--9 October 1998).
This was earlier published (with an
abstract) in \textit{J. Fourier Anal.
Appl.} \textbf{4}:3 (1998). Zbl:0963.65154.},
ISBN = {9783540669517},
}
I. Daubechies, I. Guskov, and W. Sweldens :
“Commutation for irregular subdivision ,”
Constr. Approx.
17 : 4
(2001 ),
pp. 479–514 .
MR
1845265
Zbl
0994.42019
article

Abstract
People
BibTeX

We present a generalization of the commutation formula to irregular subdivision schemes and wavelets. We show how, in the noninterpolating case, the divided differences need to be adapted to the subdivision scheme. As an example we include the construction of an entire family of biorthogonal compactly supported irregular knot B-spline wavelets starting from Lagrangian interpolation.

@article {key1845265m,
AUTHOR = {Daubechies, Ingrid and Guskov, Igor
and Sweldens, Wim},
TITLE = {Commutation for irregular subdivision},
JOURNAL = {Constr. Approx.},
FJOURNAL = {Constructive Approximation. An International
Journal for Approximations and Expansions},
VOLUME = {17},
NUMBER = {4},
YEAR = {2001},
PAGES = {479--514},
DOI = {10.1007/s00365-001-0001-0},
NOTE = {MR:1845265. Zbl:0994.42019.},
ISSN = {0176-4276},
}
I. Daubechies, O. Runborg, and W. Sweldens :
“Normal multiresolution approximation of curves ,”
Constr. Approx.
20 : 3
(2004 ),
pp. 399–463 .
MR
2057535
Zbl
1051.42025
article

Abstract
People
BibTeX

A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.

@article {key2057535m,
AUTHOR = {Daubechies, Ingrid and Runborg, Olof
and Sweldens, Wim},
TITLE = {Normal multiresolution approximation
of curves},
JOURNAL = {Constr. Approx.},
FJOURNAL = {Constructive Approximation. An International
Journal for Approximations and Expansions},
VOLUME = {20},
NUMBER = {3},
YEAR = {2004},
PAGES = {399--463},
DOI = {10.1007/s00365-003-0543-4},
NOTE = {MR:2057535. Zbl:1051.42025.},
ISSN = {0176-4276},
}