- 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.