Graphics Reference
In-Depth Information
8
Spline Wavelets: Construction, Implication,
and Uses
8.1 Introduction
At the beginning of the eighties while doing the seismic data analysis, J. Morlet
introduced wavelets as a tool for signal analysis. His success led A. Grossman
to make a detailed study of the wavelet transform [69]. Later on, Y. Meyer
pointed out that there was a connection between signal analysis methods and
existing powerful techniques in the mathematical study of singular integral op-
erators. Ingrid Daubechies, together with Grossman and Meyer [50], provided
first the construction of a special type of frames. Later on in 1988, Daubechies
[48] provided a major breakthrough with her construction of the families of
orthonormal wavelets with compact support. The remarkable papers of Mal-
lat [114, 115] and Daubechies [48] came out in 1988 and 1989. The subject,
along with its applications, then grew out in many diverse fields during the
last two decades.
To have an idea about various developments on wavelets, readers can go
first through an introduction to continuous wavelet transform in [156, 49].
Wavelet bases of Meyer, Battle [18] and Lemarie [103] can be easily realized
using orthonormal multirate filter banks. But the filters involved are not ratio-
nal and the corresponding wavelets cannot be computed exactly. Hence they
are limited from the signal processing viewpoint. Daubechies' compactly sup-
ported wavelets [48] are based on finite impulse response (FIR) filter banks.
Orthogonal filter banks and their relation to wavelet bases have been stud-
ied in [164, 165, 166]. Details about wavelets and various applications can be
found in topics [49, 166, 116, 14]. Other topics can also be consulted.
Different, well-known wavelets have been widely used in many problems.
Some are more ecient and more capable compared to others. Excepting
these remarkable wavelets, another class of wavelets that has gained attention,
interest, and importance (due to their simplicity in construction) is the class
of spline wavelets. These wavelets are found to secure a good place in signal
processing, as they have merit in implementations. They are also relatively
easy to understand and simple in their construction. The easiest of them uses
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