Page 273 - Digital Analysis of Remotely Sensed Imagery

Image Enhancement 235

where

V
r = r = ij (6.16)
V × V
ij ji
ii jj
V 214
.
r = 12 = = .
044
12
V × V 6 × 4
11 22
⎛ 100 044⎞
.
.
R = ⎜ ⎟
⎝ 044 100⎠
.
.
In other words, 44 percent of information is shared between bands
1 and 2 as illustrated in Fig. 6.22. The variance matrix after transforma-
tion must meet the following condition:
|V − k I| = 0 (6.17)
where I = identity matrix; k = eigen value matrix. It has the fol-
lowing form:
⎛λ 0 ... 0 ⎞
⎜ 1 λ ⎟
k = ⎜ 0 2 ... 0 ⎟
⎜ ... ... ... 0 ⎟
⎜ ⎝ 0 0 ... λ n ⎟ ⎟ ⎠

Plugging the V and I matrices into Eq. (6.17) yields the following
formula:

⎛ 6 2 14⎞ ⎛ 10⎞ 6 − λλ 214 0
−
.
..
⎜ ⎝ 214 4 ⎠ ⎟ − l ⎜ ⎝ 01⎠ ⎟ = 0 or 214 0 4 − λλ = 0
−
.
.
Or (6 − l)(4 − l) − 2.14 × 2.14 = 0
So l = 7.36 l = 2.76
1 2
The eigen value matrix k after the transformation is
.
.
k = ⎛ ⎜ ⎝ 736 000⎞ ⎟
000 264⎠
.
.
The following three points should be noted from the variance-
covariance matrix V and the eigen value matrix k:
• The total variance of the two spectral bands (6 + 4 = 10) before
the transformation is exactly the same as the total eigen values
(the main diagonal elements) (7.36 + 2.64 = 10) after the
transformation. This clearly demonstrates that PCA does not