Databases Reference
In-Depth Information
In the next several sections, we will briefly examine the construction of wavelets and
describe how we can obtain a decomposition of a signal using multiresolution analysis. We
will then describe some of the currently popular schemes for image compression. If you are
primarily interested at this time in implementation of wavelet-based compression schemes,
you should skip the next few sections and go directly to Section
15.5
.
In the last two chapters, we have described several ways of decomposing signals. Why do
we need another one? To answer this question, let's begin with our standard tool for analysis,
the Fourier transform. Given a function
f
(
t
)
, we can find the Fourier transform
F
(ω)
as
∞
e
−
j
ω
t
dt
F
(ω)
=
f
(
t
)
−∞
Integration is an averaging operation; therefore, the analysis we obtain, using the Fourier
transform, is in some sense an “average” analysis, where the averaging interval is all of time.
Thus, by looking at a particular Fourier transform, we can say, for example, that there is a large
component of frequency 10 kHz in a signal, but we cannot tell when in time this component
occurred. In other words, Fourier analysis provides excellent localization in frequency and
none in time. The converse is true for the time function
f
, which provides exact information
about the value of the function at each instant of time but does not directly provide spectral
information. It should be noted that both
f
(
t
)
represent the same function, and
all the information is present in each representation. However, each representation makes
different kinds of information easily accessible.
If we have a very nonstationary signal, like the one shown in Figure
15.1
,wewouldliketo
knownot only the frequency components but when in time the particular frequency components
occurred. One way to obtain this information is via the
short-term Fourier transform
(STFT).
With the STFT, we break the time signal
f
(
t
)
and
F
(ω)
into pieces of length
T
and apply Fourier analysis
to each piece. This way we can say, for example, that a component at 10 kHz occurred in
the third piece—that is, between time 2
T
and time 3
T
. Thus, we obtain an analysis that is a
function of both time and frequency. If we simply chopped the function into pieces, we could
get distortion in the form of boundary effects (see Problem 1 at the end of this chapter). In order
to reduce the boundary effects, we
window
each piece before we take the Fourier transform.
If the window shape is given by
g
(
t
)
(
t
)
, the STFT is formally given by
∞
g
∗
(
e
j
ω
t
dt
(ω, τ )
=
(
)
−
τ)
(1)
F
f
t
t
−∞
t
0
2
t
0
F I GU R E 15 . 1
A nonstationary signal.
