Agriculture Reference
In-Depth Information
using statistical methods such as principal component analysis (PCA, Jolliffe
2002
).
The main idea behind PCA is to reduce the dimensionality of a data set that
consists of a large number of correlated variables, while preserving as much as
possible of the overall variability. This is achieved by defining a new set of vari-
ables, the principal components (PCs). They are uncorrelated, and ordered by the
fraction of the total information that they contain. Because digital RS images are
numeric, this technique can reduce their dimensionality. In multi-band RS images,
the bands are the original variables. PCA can also be used as a change detection
technique for RS (Jensen
2004
; Muchoney and Haack
1994
; Munyati
2004
).
Now, consider PCA in the particular context of image analysis. A rectangular
Nrow
by
Ncolumn
image can be expressed as an
N
¼
Nrow
Ncolumn
dimensional
vector
t
¼
ð
x
1
x
2
x
N
Þ
;
ð
:
Þ
x
4
13
where the columns of the pixels in the image are placed one after the other to form a
one-dimensional image. Note that the values in the vector are intensity brightness
values. In this particular case, each image represents one variable. Suppose that we
have
M
images (i.e.,
M
bands representing
M
variables), and define the data matrix
to be
2
3
x
11
x
12
x
1
M
...
4
5
:
x
21
x
22
x
2
M
...
X
¼
ð
4
:
14
Þ
...
...
... ...
x
N
1
x
N
2
x
NM
...
The matrix X represents the starting point for PCA analysis. Applying image
compression with PCA, our aim is to reduce the
M
initial images into
p
new images
(
p
M
) by maximizing the information represented by the total variance of the
initial images. So, we can define the new first artificial images as
<
X
M
y
k
1
¼
a
11
x
k
1
þ
a
21
x
k
2
þ
...
þ
a
M
1
x
kM
¼
a
i
1
x
ki
,
k
¼
1, 2,
,
N
;
ð
4
:
15
Þ
...
i¼
1
or in matrix notation as
y
1
¼
Xa
1
;
ð
4
:
16
Þ
t
are the coefficients. The PC with the maximum
Var(
y
1
) is chosen as the first component (i.e., it accounts for the majority of the
observed variations). Likewise, the
p
-th component is defined y
p
¼
where a
1
¼
ð
a
11
a
21
a
M
1
Þ
...
Xa
p
, where the
t
a
1
p
a
2
p
a
Mp
vector a
p
¼
ð
...
Þ
is chosen so that Var(
y
p
) is the maximum,
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