Image Processing Reference

In-Depth Information

xb

a

−

where
Y
a,b
is defined as in Eq.
11.2
and

y

(

)

is the complex conjugated of
Y
.

WT
f
(
a,b
) represents the information content of
f
(
x
) at scale
a
and location
b
. For

fixed
a
and
b
,
WT
f
(
a,b
) is called the wavelet coefficient.

The application of the WT for each scale and each location of a signal provides

a local representation of this signal. The process can be reversed and the original

signal reconstructed exactly (without any loss) from the wavelet coefficients

according to the following equation:

+∞+∞

1

dadb

∫∫

f x

()

=

WT a b

(,)

y

(

x

)

(11.4)

f

ab

,

2

C

a

y

--

∞∞

where
C
Y
is the admissibility condition of the mother wavelet. This equation can be

interpreted in two ways:

•
f
(
x
) can be reconstructed exactly if one knows its wavelet transform.

•
f
(
x
) can also be thought of as a superimposition of wavelets.

These two points of view lead to different applications of the WT: signal processing

and signal analysis.

11.2.2

Multi-resolution Analysis (MRA)

The concept of MRA introduced by Mallat (
1989
) is

derived from the Laplacian pyramids (Burt and Adelson

1983
). In this approach, the size of a pixel is defined as a

resolution of reference and is used as a basis to measure

local variations in the image. Note that the resolution is

related to the inverse of the scale used by cartographers.

Hence, the larger the resolution of an image, the smaller the size (or the character-

istic length or characteristic scale) of visible objects on the image scene.

Figure
11.1
shows a description of MRA and more generally of pyramidal algo-

rithms. MRA computes successive, coarser and coarser approximations of the same

original image. The base of the pyramid corresponds to the original image. Climbing

the pyramid, the different steps represent the successive approximations of the

image. The theoretical limit of these algorithms is the top of the pyramid, which cor-

responds to a unique pixel. The difference of information existing between two suc-

cessive approximations of the same image is described by the wavelet coefficients.

The application of the Mallat's algorithm to images is well known for the wavelet

community. Another algorithm of interest has been proposed by Dutilleux (
1989
),

the so-called “à trous” algorithm. In this algorithm, only a scale function is used. The

approximation of the original image is obtained by filtering the original image, the

the larger the

resolution of an

image, the

smaller the size of

visible objects