## 4 Efficient Band Ordering
### Correlation-Based Band Ordering  
An important measure of dependence between two random distributions is Pearson's Correlation [22]. This measure is obtained by dividing the covariance of two variables by the product of their standard deviation, the formula is defined as:  
\[\rho_{x,y}\!=\!\frac{\sigma_{xy}}{\sigma_{x}\sigma_{y}}\]  
where \(\sigma_{xy}\) is the covariance between X, Y (the two variables).  
\(\sigma_{x}\) and \(\sigma_{y}\) are respectively the standard deviations for X and Y. The Pearson correlation coefficients have values in the range [\(-\)1, 1].  
When \(\rho_{xy}>0\), the variables are directly correlated. The variables are not correlated when \(\rho_{xy}=0\) and if \(\rho_{xy}<0\), the variables are inversely correlated.  
Figures 12, 14, 16 report the graphs of the Pearson's correlation for each pair of continuous bands for all the scenes respectively of Lunar Lake, Moffett Field and Low Altitude.  
The X axis refers to the \(x\)-th pair of bands in the ordering, where the pair is defined as (\(x\)-th band, (\(x+1\))-th band) (for example, without band ordering, the value 1 of the X axis refers to the pair of bands (1, 2), the value 2 refers to the pair of bands (2, 3), and so on) and the Y axis indicates the corresponding value of the Pearson's correlation between the pair of bands on X axis.  
Figure 13 reports the average correlation of all scenes of the Lunar Lake. As in Figure 13.  
Figures 15 and 17 report the average correlation of all scenes respectively for Moffett Field and Low Altitude.  
Figure 11: A correct band ordering given by DFS on MST M.  
Figure 14: Correlation for continuous band for each scene of Moffett Field.  
Figure 12: Correlation for continuous band for each scene of Lunar Lake.  
Figure 13: Average scene correlation for continuous band for Lunar Lake.  
Figure 16: Correlation for continuous band for each scene of Low Altitude.  
Figure 17: Average scene correlation for continuous band for Low Altitude.  
Figure 15: Average scene correlation for continuous band for Moffett Field.  
By looking at the above figures we can see that there are many points where there is a discontinuity between continuous bands.  
For example, consider the minimum peak in the graph of Figure 17, regarding the pair of bands (160, 161) (values 160 on the X-axis). The correlation of the pair of bands (160, 161) is approximately 0.102007273, this means that there is very low correlation between those two bands, and this means that the inter-band predictor shall fail its predictions, because there is not enough inter-band correlation.  
Figure 18 reports the graph of the correlation for continuous band (represented by the green line) and the correlation after the execution of the proposed band ordering (the red dotted line in the graph) for the Scene 03 of Moffett Field.  
**Figure 18.** The graph reports the trend of the correlation for continuous band (in green) and the trend of the correlation after the execution of the proposed band ordering (in red).  
From the figure it can be seen that there are less points of discontinuity after the execution of band ordering.  
Without band ordering, [PERSON]'s correlation assumes the minimal value of approximately 0.101975512 (very low correlation).  
After band ordering, [PERSON]'s correlation assumes the value of approximately of 0.415955776 (that means that the bands are correlated).