reduces the space complexity and save the memory space, but
for each combinational case (i, j, k), we need more time to
verdict whether this case has existed, which sacrifices the run-
time efficiency.
However, runtime is also very important, especially for real-
time processing, even equal to space assigned, so the principal
indexes are divided (Figure. 1. c) into four or more indexes,
which is equivalent to increase of the number of the principal
indexes. This improved method helps to reduce runtime (Table
1), but to some extent, increases the storage. Then this solution
finally finds the balance between time and space of this
algorithm.
In order to test the efficiency of algorithm, the experiments are
done to calculate three-dimensional joint entropy of TM images.
The test data is TM multispectral images (six bands, except for
thermal band 6) in Wuhan, China and the size is 1024x1024
pixels. Then total number of the combinatorial cases is 20.The
comparisons of the runtime between index and improved index
methods are partly listed in Table 1.
Seq Bands Joint Runtime(s)
; Selected Entropy Index Improved
1 3,4,5 15.1920 163.313 49.531
5 4,5,7 14.4935 54.265 22.610
10 15,7 14.0925 82.000 25.547
15 23:9 13.1894 55.844 30.859
20 1,23 11.3511 1.718 1.360
Table 1. comparisons of the runtime for calculation of three-
dimensional joint entropy
[t's showed in Table 1 that the greater the value of joint entropy
is, the more the time cost by index or improved index solution
generally is. The maximal runtime of index is 163.313s and the
minimal is 1.718s; while the maximum of improved index is
49.531s, and the minimum is 1.360s. The result indicates that
the new solution works more efficient than the conventional
index method and has better stability.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
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Figure 1. Solutions to joint entropy. a. represents the definition
algorithm and the total number of units is 2**. b. shows the
index solution. c. shows the improved index solution.
In order to get the expansibility of this new solution, we extend
it to the calculation of multidimensional joint entropy the
method and apply the method to the aerial hyper spectral image
data (512x512 pixels) acquired near Poyang Lake, China with
totally 30 bands. The computing results show the efficiency of
six-dimensional joint entropy and they are partly listed in Table
2.
Seq. Bands Selected — Joint Entropy Runtime(s)
1 5,6,7,9,10,12 17.8912 26.828
5 6.7,3.9.10.12 17.8868 20.110
10 6,7.9,10,11,12 17.8822 19.750
20 5,8,9,10,11,12 17.8688 24.984
28 86,789 10 17.8426 25.500
Table 2. Calculation of six-dimensional joint entropy
From this experiment, the runtime of improved solution to
calculate multidimensional joint entropy is acceptable. As we
know, three optimum bands are not enough for hyper spectral
band selection, and there is not only one best band triplet
(Alejandra, 2003). Then according to practical needs, the new
solution to multidimensional joint entropy could help divide
groups to analyse the hyper spectral image data.
3. APPLICATION OF JOINT ENTROPY
3.1 Optimum Band Selection Based on Information
Content
Optimum band selection is first used to colour synthesizing for
visual interpretation, so three-band selection is usually used.
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