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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
The principles of band selection (Beauchemin et al, 2001) are 
followed as: 
‚Whole information content of bands; 
Correlation among different channels; 
«Class separability. 
Some statistical methods at different aspects such as OIF 
(Optimum Index Factor), SI (Sheffield Index) and CI (Crippen 
Index) and so on, are employed for optimum band selection. 
OIF (Chavez et al., 1982) is introduced to select a three-band 
combination having high variances and low pair-wise 
correlation. It mainly emphasizes the differences between bands. 
SI (Sheffiled, 1995; Beauchemin et al., 2001) is proposed based 
on the image covariance matrix of multispectral bands. Another 
optimum index CI (Crippen, 1989) is based on image 
correlation matrix and it minimizes the effect of redundant 
image content. All methods referred above follow the first two 
principles to select best triplet that has the highest information 
content and the lowest correlation. 
Joint entropy is a statistical mean of the probabilities of the 
grey-value combinations and could also used to be an 
assessment index of information content in remote sensing 
images. Joint entropy as the general criterion of the information 
content could be employed in application to optimum band 
selection. The formula is seen in section 2 (2) and the 
comparisons of these methods are showed in the experiments 
(Table 3). 
32 Quality Assessment of Fused Imagery 
Quality assessment of image fusion is relatively new issue in 
recent years. A good fusion method lies on improving the 
spatial resolution of multispectral images as well as preserving 
their spectral characteristics. Various statistical methods are 
proposed to evaluate the quality of fused images. For example, 
average, entropy, variance, standard deviation, average gradient, 
correlation coefficient (Jia 2001; Li, 2000; Wang et al., 2002) 
of images have been employed for evaluation. 
The fused image can be evaluated both spectrally and spatially. 
The quality of spatial information could be judged by lucidity 
and local contrast of images and there are many methods for 
describing lucidity and contrast. For example, average gradient 
and variance are proposed to assess the details and variations in 
each channel of the merged image. And entropy is used to 
evaluate at the aspect of spatial information content. However, 
since there are redundancies between different channels of one 
fused image and entropy and average gradient can only be used 
to evaluate a single channel, the simple addition of entropies or 
gradients cannot represent the whole information of the fused 
image. Joint entropy eliminates the redundant information 
between channels and can solve this problem efficiently. 
4. EXPERIMENTS AND RESULTS 
The following experiments include two sections: joint entropy 
applied to optimum band selection and applied to quality 
assessment of merged images. By experiments, the comparisons 
between joint entropy and other methods are discussed. 
41 Optimum Band Selection 
The test data are the same as that of the above experiment in 
Table 1. SI, CI, OIF are also used to compare with joint entropy 
(JE), and the sequence results are showed in Table 3. 
  
  
Seq. Triplets JE SI CI OIF 
I 3.4.5 15.192 4 $ 4 
2 1,4,5 14.968 6 7 7 
3 3,4,7 14.963 1 2 2 
4 1,4,7 14.836 3 3 5 
5 4,5,7 14.493 19 19 19 
6 2,4,5 14.440 5 4 9 
7 3.5.7 14.321 14 17 13 
8 2,4,7 14.274 2 1 6 
9 1,3,4 14.234 7 6 1 
10 1.5.7 14.093 18 18 15 
11 1.3.5 14.036 8 8 10 
12 2.37 13.609 16 15 14 
13 1,3,7 15.567 9 9 16 
14 2,3,4 13.334 13 13 8 
15 2,55 13.189 15 14 11 
16 1,2,4 13.163 10 10 3 
17 1.2.5 13.025 11 11 12 
18 12.7 12.711 12 12 18 
19 2 d 12.610 17 16 17 
20 1.2.3 11.351 20 20 20 
  
Table 3. Comparisons between JE and different methods in 
application to optimum band selection 
As is showed above, all these methods obtain optimum band 
combinations, such as 543, 743, 742, 741. Whether judged by 
visual effect or by local contrast, they're all good triplets. This 
demonstrates that joint entropy could be applied to band 
selection. At the same time, the reasons for the ranking 
differences are the similarity of variances among different 
bands after adjustment of contrast. The similarity of variances 
causes OIF and SI only related to correlations between bands. 
OIF, SI, CI obtain the same sequences which might be a little 
different from reality, such as 431, the best triplet selected by 
OIF is not a good result. However, based on information 
content, joint entropy uses the probabilities of possible grey 
combinations instead of the variances and correlations, by 
which the optimum band selection could be judged more 
efficiently and this is why joint entropy is better than other 
statistical methods. 
4.20 Experiment on Quality Assessment of fused images 
In order to evaluate the quality of fused images, the experiment 
is based on IKONOS 4m multispectral (MS) and Im 
panchromatic images, which are taken from Beijing, China in 
1999. Fusion methods such as IHS (Intensity, Hue, Saturation), 
PCA (Principal Component Analysis) and á trous wavelet 
transform (Pohl et al. 1998, Jia, 2001) are employed to obtain 
the fused images. Entropy, gradient and joint entropy are 
applied as quality assessment criteria and the results are listed 
in Table 4.