The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
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essentially a wavelet transform in the spectral domain of the 
input image as opposed to the spatial domain as most often 
discussed in the literatures. As for the statistical fusion 
approach, a criteria-based solution is proposed. This is an 
enhancement and modification to the published c?-p approach 
(Gungor and Shan,2005, 2006). The new method starts from 
designing the desired properties for the fused images, and then 
uses them as criteria to solve a system of equations to determine 
the pixel values for the fused imagery. This way the resultant 
fusion outcome should have such desired properties and are 
optimal in terms of these predefined criteria. Through this 
approach, we develop a novel image fusion framework that is 
user and application driven such that the properties of the fused 
image are known beforehand. It can produce the fusion 
outcome with desired properties based on user’s need, whereas 
the fusion outcome from other existing methods have unknown 
properties and must be evaluated in a case-by-case basis. The 
proposed image fusion techniques are tested by fusing 
QuickBird panchromatic image. Fusion results are evaluated 
along with discussions on the properties of the proposed fusion 
methods. 
2. GENERALIZED RGB-IHS TRANSFORM 
To generalize the IHS transform to support n multispectral 
bands, we first consider the traditional RGB to IHS transform, 
where there are only three bands involved. Harrison and 
Jupp,(1990) define an auxiliary Cartesian coordinate 
system / , V x , V 2 and the transform matrix T 3 between these two 
spaces. 
Figure 1. Relation between RGB and IHS Spaces 
In the I , V x , V 2 system, / stands for the intensity and is 
defined as the line which connects the origin of the RGB colour 
space ‘O’ to the point ‘White’ (see Figure la). Using the 
coordinates of the vertices of ARBG in Figure lb, the 
coordinates of ‘P’ (the centre of ARBG) and ‘A’ (the middle 
point of line GR ) can be determined as /*(—,—,—) and 
3 3 3 
. The orthonormal vector defining the direction of I 
can then be written as y? = (—J=-,—J=r,-j=r). Here, the subscript 
V3 V3 V3 
“1” indicates that / is the first vector and the superscript “3” 
denotes the dimensionality. The other two vectors V x and V 2 
are defined on the ARBG plane. V x is defined in the direction 
of BA as perpendicular to the / axis, and V 2 is chosen as 
perpendicular to V x . The orthonormal vectors for V x and V 2 are 
obtained as yl = (~^= and y\ =(—^,—^,0) . Thus 
V6 V6 V6 y[2 V2 
the forward RGB to IHS transform is described as follows 
(Harrison and Jupp,1990) 
’ 1 1 1 
V3 VI V3 
v 
1 1 -2 
G 
r 1 
/2. 
V6 Vó yfó 
1 -1 
B 
TT vr V 
(1) 
To generalize the above relationship from 3-D to n-D, we first 
examine the 2-D case. The normalized intensity vector for 2-D 
7 1 1 
would be y x ={—j=,—=) . The other normalized vector 
\[2 ’ V2 
1 -1 
orthogonal to y x is y\ = (-2=r,—-2=). Thus, the 2-D transform 
V2 V2 
matrix is 
7; = 
1 
1 
1 -1 
V2 V2 
(2) 
For 3-D, the intensity axis is y x - (—j=,—|=r) . The third 
V3 ’V3 'fi 
1 -1 
vector y\ = (—=r,—t=-,0) can also be obtained directly from the 
V2 V2 
3-D transform matrix T 2 by simply adding a zero to y\ as the 
last element. Finally, the second vector y\ can be calculated as 
yl = (—which is orthonormal to y\ and y\ . Thus 
V 6 V 6 V 6 
the transform T 3 is (as given in Eq. 1) 
T,= 
1 
1 
1 
fi fi fi 
1 1 -2 
fi fi fi 
1 -1 
fi fi 
0 
(3) 
Using the intensity definition given for 3-D above, the first axis 
in 4-D would be y x = (—)=r,-^,—j=) . The two orthogonal 
V 4 V 4 V 4 V 4 
vectors y 3 and y\ can directly be copied from y\ and y\ by 
adding zeroes as the last elements as 
y* =(^=,-^,^,0), y\ =(—^=,^¿,0,0) . Finally, we form 
V6 V6 V6 V2 V2 
y 2 =(- 1 !=,- 1 L=-,—jL=,- 7 ^=r) to be orthonormal to y x , y 3 , 
V12 Vl2 V12 V12 
y\ .The transform T 4 in 4-D is