Remote Sens. 2017, 9, 1094 4 of 20
Figure 1. The flowchart of the H
2
F based method.
2.1.1. RGF
Although the raw pixel spectral vectors could directly be used for training and classification,
they do not perform well. Moreover, since we need sub-feature sets from spectral features, we must
extend the pixels spectra to a group of features. Motivated by the effectiveness of RGF and its
improvement in HSI classification [
37
], in this paper, we use RGF to obtain the sub-feature set using
spectral information.
Let
Q
p
denote filtering result for the
p
th band of an hyperspectral image, we conduct guided
filtering [54] by
Q
p
i
= a
p
k
G
i
+ b
p
k
, ∀i ∈ ω
k
, (1)
where
G
is a guidance image,
i
is one of a pixel in
G
,
ω
k
is a window around pixel
i
,
k
is one of a pixel
in
ω
k
, and
a
p
k
and
b
p
k
are coefficients to be estimated. Usually,
G
is the first principal component of HSI
data. Please note that
G
only works as the guidance image, and it will not reduce the dimensionality
of the filtered results. Then, minimize the following energy function:
E(a
p
k
, b
p
k
) =
∑
i∈ω
k
((a
p
k
G
i
+ b
p
k
−I
p
i
)
2
+ ea
p
k
2
), (2)
where
I
is the input HSI data, and
e
is a hyper-parameter. Equation
(2)
can be solved directly by linear
ridge regression [55]:
a
p
k
=
1
|w|
∑
i∈ω
k
I
p
i
G
i
−µ
k
I
p
k
σ
k
2
+ e
,
b
p
k
= I
p
k
− a
p
k
µ
p
k
,
(3)
where
µ
k
and
σ
k
denote the mean value and standard variance of
G
in
ω
k
,
I
p
k
is the mean value of
I
in
ω
k
, and |ω| is the number of pixels in ω
k
.
Equation
(1)
is the optimization problem in guidance filtering, and
a
and
b
are the values need
to be optimized. Equation
(2)
is the optimization object function, and Equation
(3)
is the solution.
Rolling operation refers to replace
I
by
Q
and conduct Equations
(1)
and
(2)
repeatedly. In each rolling,
we can obtain a new HSI data. Therefore, using RGF we are able to generate a series of features based