- Istanbul 2004 International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
llowing six Here, d; corresponds to the j-1 level approximate 2.2.3 A trous (Nonthogonal wavelet)
are
| generate
jez
Vk eZ
1, So that
sZ basis of
yf wavelet
W; is the
V; ,. Then,
sum.
n situation;
function is
(12)
1g function,
three 2-D
insform can
- 2yMf jte)
(13)
- 2y)f;(eN
(14)
- 2y)fj(e)
; 1 2 3 : .
image, and d p] ti are the horizontal,
vertical, and diagonal subimages, respectively.
2. Different wavelet used in the image fusion
22.1 Orthogonal wavelet
The dilations and translations of the scaling function
(6; (C) constitute a basis for V; and,
similarly, V jk Q2) for W; ,ifthe 9 j, (x) and
Wj k(x) are orthonormal, it includes the following
properties:
Vi LW; (17)
(410:4;1) E ys vis vir) Ee
l6. vi) = 0 (18)
The orthogonality property puts a strong limitation on
the construction of wavelets. For example, it is hard to
find any wavelets that are compactly supported,
symmetric, and orthogonal.
2.2.2 Biorthogonal wavelet
If the orthogonality condition is relaxed to
biorthogonality conditions, wavelets with some special
properties that are not possible with orthogonal
wavelets can be obtained. In the biorthogonal
transform, there are two multi-resolution analyses, a
primal and a dual:
Primal: V;, W;,Pjk y jk
Duk. VW; gi p. p.
The dilations and translations of the scaling function
~
Wik (x)} constitute a basis for V; and,
similarly, V jk (x) for W; ; the biorthogonallity
conditions imply:
"iw LW, a9
(4 70 = Ó C2 vii) = 65 9] ;
(77 9, 1) = 0 > on Vj ;) = 0 (20)
For the biorthogonal transform, perfect reconstruction
IS available. Orthogonal wavelets give orthogonal
matrices and unitary transforms; biorthogonal
wavelets give invertible matrices and perfect
reconstruction. For the biorthogonal wavelet filter, the
low pass and the high pass filters do not have the same
length. The low pass filter is always symmetric, while
the high pass filter could be either symmetric or anti-
symmetric.
A trous (with holes) is a kind of Nonorthogonal
wavelet which is different from orthogonal and
biorthogonal. It is a “stationary” or redundant
transform; i.e., decimation is not implemented during
the process of wavelet transform, while orthogonal
and biorthorgonal wavelet transform can be carried out
using either decimation or undecimation mode.
Compared with other fusion-based wavelet transform,
this method is relatively easy to implement. The
limitation is that it will use a lot of computer memory.
3. EXPERIMENTAL RESULTS AND
COMPARISON
Corresponding to the different wavelets, six kinds of
wavelet methods are implemented to test their fusion
results. Decimation and undecimation cases are
considered in the orthogonal and biorthorgonal
wavelet, respectively. They are orthogonal wavelet
fusion with decimation (called ORTH method),
orthogonal wavelet fusion without decimation (simply
called UORTH), biorthogonal wavelet fusion with
decimation (simply called BIOR), biorthogonal
wavelet fusion without decimation (simply called
UBIOR), wavelet fusion based on the A trous (simply
called ATRO), wavelet fusion based on wavelet and
IHS transformation (simply called WIHS)(Hong and
Zhang, 2003). The undecimation orthogonal wavelet is
used in the WIHS fusion method. The orthogonal and
biorthogonal wavelet coefficients are-listed in Table 1
and Table 2, respectively. A subset of IKONOS data
(512 pixels by 512 pixels ) is used to evaluate the
fusion algorithm. The fusion results are listed in
Figure 3~Figure 8. Figure 1 is the original IKONOS
panchromatic image, Figure 2 is the original IKONOS
multispectral image, Figure 3 is the fusion result of
orthogonal wavelet fusion with decimation, Figure 4 is
the fusion result of biorthogonal wavelet with
decimation, Figure 5 is the fusion result of orthogonal
wavelet without decimation, Figure 6 is the fusion
result of biorthogonal wavelet without decimation,
Figure 7 is the fusion result of A trous wavelet, Figure
8 is the fusion result of the IHS transformation
combined with wavelet.
From the point of visual comparison, ORTH result is
similar to BIOR result, UORTH result is similar to
UBIOR; while there exists apparent color distortion in
ORTH and BIOR, the degree of color distortion in
UORTH and UBIOR is lighter than that in ORTH and
BIOR; however, the spatial detail information in
ORTH and BIOR is more plentiful than that in
UORTH and UBIOR. Combining the spatial and color
together, the rank of the fusion result is WIHS, ATRO,
UORTH (UBIOR), ORTH (BIOR). The biorthogonal
and orthogonal difference cannot be differentiated
from the fusion result. The decimation and
undecimation can be differentiated from the fusion
result.