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orthonormal eigenvectors, X l , X 2 , A , X d , of 
C t corresponding to the first d largest eigenvalues as the 
optimal projection axes set. 
For a given image sample A , let 
Y k = AX k , k = 1, 2, A ,d. (3-7) 
Then, we obtain a family of projected feature vectors, 
F[, Y 2 , A , Y d , which are called the principal component 
vectors of the sample image A . 
The principal component vectors obtained are used to form an 
mxd matrix F — [YJ, Y 2 , A , Y d ] , which is called 
the feature matrix or feature image of the image sample A . 
After a transformation by 2DPCA, a feature matrix (image) is 
obtained for each image. Then, M images has M feature 
matrices (images) F t = [Yj (,) , Y 2 l) , A , ] , 
i = 1, 2, A , M . 
An image can be reconstructed by the principal component 
vectors and the feature matrices obtained by 2DPCA. This is 
the procedure of inverse transformation of 2DPCA. 
Suppose the orthonormal eigenvectors corresponding to the 
first d largest eigenvectors of the image covariance matrix 
C t are X x , X 2 , A , X d . After the image samples are 
projected onto these axes, the resulting principal component 
vectors are Y k = AX k , k = 1, 2, A ,d. 
Le, e = [V„r 2 ,A ,Y d ], P = [X„X 2 ,A ,X d l 
then 
Q = AP (3-8) 
Since X x , X 2 , A , X d are orthonormal, from (3-9), it is 
easy to obtain the reconstructed image of sample A . 
A = QP T =j^Y k X T k (3-9) 
k=1 
Let A k = Y k X k {k = 1, 2, A , d) , which is of the same 
size as image A , and represent the reconstructed subimage of 
A . That is, image A can be approximately reconstructed by 
adding up the first d subimages. In particular, when the 
selected number of principal component vectors d = Yl(Yl 
is the total number of eigenvectors of C t ), we have A = A , 
i.e., the image is completely reconstructed by its principal 
component vectors without any loss of information. 
3.2 2DPCA-based Algorithm 
We know that, in PCA-based method, the histogram of the 
panchromatic (high spatial resolution) image must be matched 
with that of the first principal component image, and the first 
principal component will be replaced with the matched image. 
Accordingly, the fused image can be obtained by reconstructing 
the images through inverse PCA transformation. New strategy, 
however, must be found and applied to 2DPCA-based 
algorithm, instead of applying the strategy in PCA-based 
technique. The main reasons lie in: Above all, there some 
differences between PCA and 2DPCA, and the feature matrices 
of the multispetral images can not be regarded as real images 
because of many pixel values of those images are negative. 
Moreover, there is not an unambiguous the first principal image 
in 2DPCA-based method, due to the difference between PCA 
and 2DPCA techniques. In the former, the multispetral images 
are regarded as a whole in the analysis and reconstruction 
processes, but it is oppositional in the latter. Therefore, we 
proposed a new 2DPCA-based algorithm on remotely sensed 
image fusion, after analyzing the objective of image fusion and 
the characteristic of PCA and 2DPCA. In a word, 
2DPCA-based algorithm is a quite different technique in 
contrast to PCA-based method. The main steps of this new 
technique will be listed as follows. 
(1) Image registration will be applied between the 
panchromatic (high spatial resolution) image and the 
multispectral (low spatial resolution) images, and the 
multispectral images will be resampled so that their cell scale 
equals to that of the panchromatic image. 
(2) The optimal projection axes, P = \X { , X 2 , A , X d ], 
will be evaluated by eignvalue decomposition the image 
covariance matrix C t . 
(3) The histogram of the panchromatic image will be matched 
with that of the M multispectral images respectively, instead 
of the first principal component image. 
(4) It will be obtained that M feature images (matrices) of 
the M matched panchromatic images after they are 
projected on the optimal projection axes. 
(5) The first principal component of each feature image of the 
M multispectral images will be replaced with the first 
principal component of each feature image of the matched 
images corresponding to the multispectral images. 
(6) The fused images will be obtained after the inverse 
transformation of the 2DPCA. 
4. EXPERIMENTAL RESULTS AND ANALYSIS 
4.1 Experimental Results 
The original multispectral images are Landsat ETM+ Band 1, 2, 
3, 4, 5, 7, whose spatial resolution is 28.5 meters, and the 
panchromatic image is Band 8, whose spatial resolution is 
14.25 meters. The scale size of the preprocessing multispectral 
and panchromatic images is 512x512 pixels, whose cell 
size of that is 14.25 meters.The preprocessing of the 
experimental images is completed under ERDAS Imagine 8.7 
platform. The experiments are completed on PC computer 
whose configurations are: CPU Pentium IV 3.06GHz, RAM 
1.0GB, and the programs are programmed on Matlab 7.0 
platform. The experimental results are shown in Figure 1.