Feature Extraction from Hyperspectral Data Using ICA
In this case, the norm involved in (8.4) becomes:
Ilu - Wxl12 = IIWxo - Wxl12 = IIW (xo - x) 112
= (W(xo _x))T (W(xo -x))
= (xo - x) W W (xo - x) .
T( T )
The conditional pdf can then be rewritten as:
1
I
T( T
( I ) - l'
- 2a2 (xo-x) W w)(xo-x)
P u x - 1m (
)m e
.
aL .. O J2rro2
We multiply and divide the entire expression by J det (WWT):
For x = Xo, we have:
205
(8.6)
(8.7)
(8.9)
The limit corresponds to a dirac delta function in x around Xo. Putting this
result into (8.3), for u in the image of Wx, we obtain:
f
1
p(xo)
p(u) =
c5(x-xo)p(x)dx=
.
Jdet(WW T )
yldet(WW T )
(8.10)
Note that in the case when W is a square matrix, i.e., when m = n, we get
back the relationship in (8.2) since det (W) = det (WT).