Databases Reference
In-Depth Information
(Note that these are the most commonly used choices, but they are not the only choices.) If this
set is
complete
, then
{
ψ
m
,
n
(
t
)
}
are called
affine
wavelets. The wavelet coefficients are given by
w
m
,
n
=
f
(
t
), ψ
m
,
n
(
t
)
(13)
a
m
/
2
0
a
0
t
=
f
(
t
)ψ(
−
nb
0
)
dt
(14)
The function
f
(
t
)
can be reconstructed from the wavelet coefficients by
f
(
t
)
=
w
m
,
n
ψ
m
,
n
(
t
)
(15)
m
n
Wavelets come in many shapes. We will look at some of the more popular ones later in
this chapter. One of the simplest wavelets is the Haar wavelet, which we will use to explore
the various aspects of wavelets. The Haar wavelet is given by
10
1
2
t
<
ψ(
t
)
=
(16)
1
−
1
2
t
<
1
By translating and scaling this mother wavelet, we can synthesize a variety of functions.
This version of the transform, where
f
is a continuous function while the transform
consists of discrete values, is a wavelet series analogous to the Fourier series. It is also called
the
discrete time wavelet transform
(DTWT). We have moved from the continuous wavelet
transform, where both the time function
f
(
t
)
w
a
,
b
were continuous functions
of their arguments, to the wavelet series, where the time function is continuous but the time-
scale wavelet representation is discrete. Given that in data compression we are generally
dealing with sampled functions that are discrete in time, we would like both the time and
frequency representations to be discrete. This is called the
discrete wavelet transform
(DWT).
However, beforewe get to that, let's look into one additional concept—multiresolution analysis.
(
)
t
and its transform
15.4 Multiresolution Analysis and the Scaling
Function
φ(
)
The idea behind multiresolution analysis is fairly simple. Let's define a function
that
we call a
scaling
function. We will later see that the scaling function is closely related to the
mother wavelet. By taking linear combinations of the scaling function and its translates, we
can generate a large number of functions
t
f
(
t
)
=
a
k
φ(
t
−
k
)
(17)
k
The scaling function has the property that a function that can be represented by the scaling
function can also be represented by the dilated versions of the scaling function.
