Biomedical Engineering Reference
In-Depth Information
overviews the theoretical fundamentals of wavelet theory and related multiscale
representations. As an example, the implementation of an overcomplete dyadic
wavelet transform will be illustrated. Section 6.3 includes a general introduc-
tion of image denoising and enhancement techniques using wavelet analysis.
Sections 6.4 and 6.5 summarize the basic principles and research works in lit-
erature for wavelet analysis applied to image segmentation and registration.
6.2 Wavelet Transform and
Multiscale Analysis
One of the most fundamental problems in signal processing is to find a suitable
representation of the data that will facilitate an analysis procedure. One way to
achieve this goal is to use transformation, or decomposition of the signal on a
set of basis functions prior to processing in the transform domain. Transform
theory has played a key role in image processing for a number of years, and
it continues to be a topic of interest in theoretical as well as applied work in
this field. Image transforms are used widely in many image processing fields,
including image enhancement, restoration, encoding, and description [12].
Historically, the Fourier transform has dominated linear time-invariant signal
processing. The associated basis functions are complex sinusoidal waves
e
i
ω
t
that correspond to the eigenvectors of a linear time-invariant operator. A signal
f
(
t
) defined in the temporal domain and its Fourier transform
f
(
ω
), defined in
the frequency domain, have the following relationships [12, 13]:
+∞
f
(
ω
)
=
f
(
t
)
e
−
i
ω
t
dt
,
(6.1)
−∞
+∞
1
2
π
f
(
ω
)
e
i
ω
t
d
w.
f
(
t
)
=
(6.2)
−∞
Fourier transform characterizes a signal
f
(
t
) via its frequency components.
Since the support of the bases function
e
i
ω
t
covers the whole temporal domain
(i.e. infinite support),
f
(
ω
) depends on the values of
f
(
t
) for all times. This
makes the Fourier transform a global transform that cannot analyze local or
transient properties of the original signal
f
(
t
).
In order to capture frequency evolution of a nonstatic signal, the basis func-
tions should have compact support in both time and frequency domains. To
achieve this goal, a windowed Fourier transform (WFT) was first introduced
