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

−∞