response of six complex wavelets associated with the 2-D
complex wavelet transform is illustrated in Fig. 2. The
complex wavelet transform can discriminate between
features at positive and negative frequencies. Hence, there
are six subbands capturing features along the lines with
angles of
y
2
Y
, respectively.
3. Change detection algorithm
Let us consider two SAR images, X
1
¼fX
1
ði; jÞ; 1rirI;
1rjrJg and X
2
¼fX
1
ði; jÞ; 1rirI; 1rjrJg, with a size I J
each acquired on the same geographical area but at two
different time instances, t
1
and t
2
, respectively. Let us
assume that such images have been co-registered [14,15].
Let
O
¼fw
c
; w
u
g be the set of classes associated with
changed (denoted by w
c
) and unchanged (denoted by w
u
)
pixels on images X
1
and X
2
.
The decomposition of 2-D signals by DT-CWT produces
one complex-valued low-pass subband and six complex-
valued high-pass subbands at the each level of decom-
position. Let S level decomposition of 2-D signal X of size
I J produce the complex-valued high-pass subbands set
and magnitude of this set is denoted as L
X
s
¼
fl
X
s
ði; jÞj1rirI=2
s
; 1rirJ=2
s
g and H
X
y
;s
¼fh
X
s
ði; jÞj1rirI=2
s
;
1rirJ=2
s
g,
y
¼f715
3
; 745
3
; 775
3
g and s ¼ 1 ...S, respec-
tively. That is L
X
s
and H
X
y
;s
are real-valued 2-D signals
representing the magnitude of complex-valued low-pass
and high-pass subbands at a specific scale s, respectively.
The flowchart of the proposed multiscale change
detection algorithm is given in Fig. 3 and described as
follows. Because of the decimation operation in DT-CWT
decomposition, the size of subbands at the finest
resolution (i.e., s ¼ 1) is I=2 J=2. In order to create a
change detection map with the same size of that input
images, firstly, the two input images, X
1
and X
2
, are scaled
up by a factor of 2 in both dimensions. Then to enhance
low-intensity pixels, the ratio image is usually expressed
in a logarithmic scale, resulting in the absolute valued log-
ratio image X
r
X
r
¼ log
X
1
X
2
¼jlogX
1
logX
2
j; ð3Þ
where log stands for natural-logarithm. X
r
is decomposed
by the DT-CWT up to scale S. At each scale s, s ¼ 1; 2; ...; S,
one complex-valued low-pass and six complex-valued
high-pass subbands are generated, and the magnitude at
each pixel of these subbands are computed and
collectively denoted by L
X
r
s
and H
X
r
y
;s
, respectively,
where
y
2f715
3
; 745
3
; 775
3
g. In order to detect
changed pixels in high-pass subbands we define a new
statistics which is the average of magnitudes of high-pass
subbands, i.e.,
H
X
r
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
6
X
8
y
H
X
r
y
;s
s
: ð4Þ
Each one of the subbands H
X
r
y
;s
holds directional detail
information at scale s. The average of six subbands
embraces the overall detail information by combining
the detail information extracted from different directions
at scale s.
In Fig. 4, a set of two co-registered ERS-1 SAR images of
a rice plantation in Java Island, Indonesia, X
1
(Fig. 4(a))
and X
2
(Fig. 4(b)), are used to demonstrate the generation
of the log-ratio image and extraction of statistics using
DT-CWT decomposition. Figs. 4(d)–(i) depict the data
extracted from low-pass ðL
X
r
s
Þ and high-pass subbands
ðH
X
r
s
Þ of the DT-CWT decomposition of the log-ratio image
for different scales. It is clear from Fig. 4 that L
X
r
s
carries
the most discriminative data for the change detection,
meanwhile H
X
r
s
is sensitive to the high-frequency
differences (edges) between two SAR images.
ARTICLE IN PRESS
Fig. 1. Two-level 1-D dual-tree complex wavelet transform (DT-CWT).
Fig. 2. The real ðRÞ and imaginary ðIÞ parts of the impulse responses of the 2-D DT-CWT filters for the six directional subbands: (a) R
15
3
, (b) R
45
3
, (c) R
75
3
,
(d) R
þ75
3
, (e) R
þ45
3
, (f) R
þ15
3
, (g) I
15
3
, (h) I
45
3
, (i) I
75
3
, (j) I
þ75
3
, (k) I
þ45
3
, and (l) I
þ15
3
.
T. Celik / Signal Processing 90 (2010) 1471–1485 1473