Image Processing Reference
In-Depth Information
11.2
Advanced Mathematical Tools for Data Fusion
Wavelet transform (WT) analysis appeared in the early 1980s at the junction of
signal processing, applied mathematics, and quantum mechanics to provide an
efficient mathematical framework for the study of non-stationary signals.
Combined with the multi-resolution analysis (MRA), introduced by Mallat (
1989
),
the WT was applied to the analysis and processing of images. In the field of Earth
observations, the first applications of these two mathematical tools (i.e. MRA and
WT) appeared in the early 1990s and focused on the analysis of remotely sensed
images in different fields of investigation, including geology, urban areas, and
oceanography, as well as on the processing of SAR imagery and data fusion for
the improvement of spatial resolution of images. Currently, the use of WT and
MRA is quite common for modeling and fusing data in the field of remote
sensing.
wavelet transform
produces a
time-frequency
representation of
signals and images
obtained from
decomposition over
a base generated by
dilations and
translations of a
single function
called the mother
wavelet
11.2.1
Wavelet Transform (WT)
The main property of the WT is to adapt the analysis
window to the phenomenon under study based on local
information. The WT leads to a time-frequency repre-
sentation. In the case of images, it leads to a scale-space
representation. As in Fourier transform, WT is equiva-
lent to a decomposition of the signal on the basis of
elementary functions: the wavelets. Wavelets are generated
by dilation and translation of a single function called the
mother wavelet:
xb
−
1
2
y
=
a
y
(11.2)
ab
a
where
a
and
b
are reals and
a π
0.
a
is called the dilation step and
b
the translation step.
Many mother wavelets exist. They are all oscillating functions, which are well
localized both in time and frequency. All the wavelets have common properties
such as regularity, oscillation and localization. They also need to satisfy an admis-
sibility condition (see Meyer (
1990
) or Daubechies (
1992
) for more details about
the properties of the wavelets).
Despite the common properties, each wavelet brings a single decomposition of
the image signal related to the used mother wavelet. In the one dimension case, the
continuous WT of a function
f
(
x
) is:
+∞
1
xb
−
∫
WT a b
(,)
=
〈
f
,
y
〉
=
f x
(
)
y
(
)
dx
(11.3)
f
ab
,
a
a
−∞
