The International Archives oj the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
2) Calculating the eigenvalues and eigenvectors according to 
the correlation matrix; 
6) Replacing the first principal component by the higher 
resolution band; 
3) Sorting the eigenvalues and eigenvectors; 
4) Calculating the principal components one by one according 
the PCA transform; 
pc = ®-xs L 
P an OJ) = P c V,iJ) +S VJ) 
(3) 
7) Obtaining the fusion results after inverse PCA transform 
5) Selecting the first principal component; 
xs H =n-pc\ where n = ®'‘ (4) 
xs" 
~«ll 
«12 
« 13 
« 14 
pan 
~«n 
(Ù\2 
«13 
«14 _ 
V 
+ 
XS2 
«21 
«22 
«23 
cd 24 
pc 2 
«21 
CO 22 
co 23 
«24 
pc 2 
xs" 
«31 
co 32 
OJ 33 
«34 
pc 3 
«31 
co 32 
co 33 
00 34 
pc3 
1 
% 
* a: 
1 
_« 41 
«42 
OJ 43 
«44. 
_ P C 4 _ 
_« 4 > 
co 42 
0) 43 
«44. 
PC 4 
«12 
«13 
«14 
«11 " 
«12 
«13 
«14 _ 
r -1 
’«11 ~ 
«22 
PC 2 
PC 2 
• (pc x + 8) + 
co 23 
«24 
PC3 
= 
«21 
•PC, + 
«22 
«23 
«24 
pc 3 
+ 
«2. 
co 32 
«33 
«34 
PC 4 
«31 
«32 
«33 
«34 
«31 
PC4 
_«42 
«43 
«44. 
_«41 _ 
_« 42 
«43 
«44. 
«41 _ 
«1. 
«12 
«13 
«14' 
PC\ 
«11" 
Xs/ 
«11' 
«21 
«22 
«23 
«24 
pc 2 
+ 
«2. 
■8 = 
xs 2 
+ 
«21 
«31 
«32 
«33 
«34 
pc 3 
«31 
xs/ 
«31 
«41 
«42 
«43 
«44. 
_P C \ _ 
.«41. 
1 
a 
1 
.«41. 
(5) 
Thus, the whole algorithm mainly consists of calculating the 
correlation matrix, forward PCA transform and inverse PCA 
transform. Through deduction the final fusion results are 
pan = c, • xs/ + c 2 ■ xs 2 + c 3 • xs/ + c 4 • xs/ + 8 
(7) 
xs 
(k,i,j) 
XS 
(k,i,j) 
CO, 
where S = pan-pc t (6) 
2) Computing the linear combination of blocks from 
multispectral bands in terms of coefficients. 
4.2 Modulation-based Fusion Technique 
The typical algorithms applying the modulation-based fusion 
technique include Brovey, SFIM and HPF fusion algorithms. 
To illustrate the deduction for the modulation-based fusion 
technique, following is the transformation steps taking 
Block-regression fusion algorithm presented by the authors as 
an example. 
XS 
The lower resolution multispectral band k is resampled to 
have the same size as the higher resolution panchromatic band 
P an after those bands are co-registrated: XSk rs P( xs k ) ? 
and after the resampling the implementatiom steps for 
Block-regression based fusion algorithm are as follows (Zhang 
and Yang,2006): 
1) Obtaining linear regression coefficients through multiple 
linear regression between the blocks from the panchromatic 
band and from the multispectral bands. ^ is equal to 4; 
syn = c x • xs/ + c 2 ■ xs/ + c 3 • xs/ + c 4 • xs/ 
(8) 
3) Finishing the fusion operation for every block using tt 
following expression: 
xs 1 / 
xs/ 
1 
% 
1 
xs/ 
xs/ 
xs/ 
pan 
xs/ 
syn+ 8 
XS/ 
xs/ 
— 
* — 
— 
xs/ 
syn 
xs 3 
syn 
xs L 3 
H 
L 
L 
XS 4 
L x5 4 J 
/xs 4 J 
L X5 4 j 
(1 + —) 
syn 
xs/ 
xs/ 
xs/ /syn 
XS2 
Ô 
XS2 
+ 
XS/ /syn 
xs/ 
syn 
xs/ 
xs/ /syn 
1 
<3 Tf 
% 
1 
.xs/ _ 
JCS4 /syn _ 
(9)