Environmental Engineering Reference
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a multidimensional filter bank, as shown in Figure 3.9). This approach is one of the
most frequently used in image processing [46].
a
columns
0
h
12
↓
j
rows
0
h
21
↓
d
h
j
h
columns
1
12
↓
a
−
j1
v
j
d
columns
0
h
12
↓
h
rows
1
21
↓
d
j
d
Column
decimation
h
columns
1
12
↓
Row decimation
Figure 3.9
A 2-D separable solution for applying DWT to an image. Adapted from Liu
et al.
[37]
0. Note that in texture anal-
ysis, only gray scale or univariate images are used. Therefore,
a
0
could be an image
captured using a monochrome camera or an RGB image converted into gray levels.
Each row of this image is filtered using the low-pass (
h
0
) and the high-pass (
h
1
)
filters, followed by column decimation keeping only one column out of two for the
next step. The resulting low and high frequency information along the horizontal
direction is then filtered vertically (
i.e.
, along each column) using again both filters.
A final row decimation step removes one row out of two. The output of this algo-
rithm consists of four new images containing the wavelet coefficients. The low-pass
information in both directions yields an approximation image,
a
j
+
1
or
a
1
, capturing
low frequency information. The horizontal low-pass and vertical high-pass extracts
vertical edges in the image, thus capturing the horizontal details (
i.e.
high frequency
information) into a detail image
d
1
. On the other hand, horizontal edges (or vertical
details) are characterized by the high pass horizontal and low pass vertical informa-
tion and are observed in detail image
d
1
. Finally, the high frequency information
in both directions corresponds to diagonal details,
d
1
. The approximation and de-
tail images corresponding to the first level of decomposition now have a four times
smaller resolution due to the column and row decimation (
i.e.
, the number of hor-
izontal and vertical pixels are divided by 2). Such image resolution will be further
divided by 4 at each level of decomposition. This is how the dyadic shifting and
scaling of the discrete wavelet is implemented within the filter bank solution. Fur-
ther decomposition of the textural information at higher levels (or towards lower
frequencies) can be performed using approximation image
a
j
The algorithm begins with the original image,
a
j
,
j
=
1
as the input of the
+
filter bank (see dashed line in Figure 3.9).
The link between the decomposition level and the resolution of the textural infor-
mation is as follows, assuming that each image has its own distribution of textural
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