Contemporary Mathematics
Volume 345, 2004
How to construct wavelet frames on irregular grids and
arbitrary dilations in :!Rd
Akram Aldroubi, Carlos Cabrelli, and Ursula M. Molter
ABSTRACT. In this article, we present a method for constructing wavelet frames
of
£ 2(JRd)
of the type
{I
det
Aj
1
112,P(Ajx-
Xj,k) :
j E J,
k
E K} on irregular
lattices of the form X =
{xj,k
E
JRd : j
E
J, k
E
K}, and with an arbitrary
countable family of invertible d x d matrices
{Aj
E
GLd(JR) :
j
E
J}.
Pos-
sible applications include image and video compression, speech coding, image
and digital data transmission, image analysis, estimations and detection, and
seismology.
1. Introduction
In this article we present a general method for constructing well-localized
wavelet frames
{I
detAji 112 ~(Ajx-
Xj,k) :
j
E
J, k
E
K} of L 2 (~d) on arbi-
trary grids X
= {
Xj,k
E
~d
:
j
E
J, k
E
K}, and with arbitrary dilation matri-
ces { Aj }jEJ. The construction presented here is a special case of a more general
method for constructing time-frequency frame atoms in several variables discussed
in
[ACM03].
Although there has been considerable interest in trying to obtain
wavelet frame decompositions of the space L 2 (~d), on irregular grids and with un-
structured dilation matrices (see
[Bal97], [BCHL03], [Chr96], [Chr97], [CH97],
[CDH99], [FZ95], [Fei87], [FG89], [FWOl], [Gro91], [Gro93], [HK03], [OS92],
[RS95], [SZOO], [SZOl], [SZ02], [SZ03], [SZ03]),
most of the methods that have
been developed are small perturbations of wavelet frames on a regular grid and with
a fixed dilation matrix. In contrast, our approach presented in
[ACM03]
is not a
perturbation method and is very general, allowing quite general constructions. The
setting includes as particular cases, wavelet frames on irregular lattices and with
a set of dilations or transformations that do not have a group structure. For this
paper, we will be mainly concerned with an even more particular case consisting
of wavelet frames on irregular lattices and with an arbitrary but fixed expansive
matrix A (A is said to be expansive if
I.AI
1 for every eigenvalue
.A
of A). The
2000 Mathematics Subject Classification. 42C40.
Key words and phrases. Frames, irregular sampling, wavelet sets, wavelets.
The research of Akram Aldroubi is supported in part by NSF grant DMS-0103104, and by
DMS-0139740. The research of Carlos Cabrelli and Ursula Molter is partially supported by Grants:
PICT 03134, and CONICET, PIP456/98.
©
2004 American Mathematical Society
http://dx.doi.org/10.1090/conm/345/06237
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