Page 275 - Digital Analysis of Remotely Sensed Imagery

Image Enhancement 237

Thus,
g ⎧ 2 + g 2 = 1
⎨ 11 2 12 2 (6.20)
⎩ g 21 + g 22 = 1

G is determined from the variance-covariance matrix and the
eigen values jointly, or

(V − l I)g = 0 where g = g ⎛ 11 ⎞
1 1 1 ⎜ g ⎝ 21 ⎠ ⎟

⎛ ⎛ 6 2 14⎞ ⎛ 10⎞⎞ g ⎛ ⎞
.
⎜ ⎜ ⎝ ⎝ 214 4 ⎠ ⎟ − 7.36 ⎜ ⎝ 01⎠⎠ g ⎝ 11 ⎟ = 0
⎟⎟ ⎜
21 ⎠
.
.
− ⎧ 136g + 214g = 0
.
or ⎨ 11 21 (6.21)
.
.
⎩ 214g 11 − 336g 21 = 0
From Eqs. (6.20) and (6.21), g = 0.8435 and g = 0.5372.
11 21
Similarly,
(V − l I)g = 0 where g = g ⎛ 12 ⎞
2 2 2 ⎜ g ⎝ 22 ⎠ ⎟

⎛ ⎛ 6 2 14⎞ ⎛ 10⎞⎞ ⎛ g ⎞
.
12
⎟⎟ ⎜
⎜ ⎜ ⎝ ⎝ 214 4 ⎠ ⎟ − 2.64 ⎜ ⎝ 01⎠⎠ ⎝ g ⎠ ⎟ = 0
.
22
⎧ 336g + 214g = 0
.
.
⎨ 12 22 (6.22)
.
.
⎩ 214g 12 + 136g 22 = 0
Solution of Eqs. (6.20) and (6.22) yields g = −0.5372 and g =
12 22
0.8435. Thus,
°
°
.
.
cos36 06
G = ⎛ ⎜ − ⎝ 0 8435 0 5372⎞ ⎟ = ⎛ ⎜ ⎝ sin( 36 06 ') ' sin36 06 '⎞ ⎟ ⎠
−
°
°
'⎠
0 5372 0 8435⎠
.
cos36 06
.
Also called a loading matrix in statistics, matrix G essentially rep-
resents the angle of rotation of the former spectral domain to the new
component domain (Fig. 6.22). In this particular case the angle of
rotation (q ) equals 36°06′. The element values in matrix G govern
how the pixel values in the raw band should be rearranged among