Image Processing Reference
In-Depth Information
where
O
=[
f
1
,
,
f
N
]
,
···
O
=[
f
1
,
···
,
f
K
]
,
B
N
=[
ψ
1
,
···
,
ψ
N
]
(15.21)
are the juxtaposed vector coordinates corresponding to the data, a subset of the basis,
and the new data, respectively. The resulting data vectors have
N
coordinates for
f
k
,
down from the original
M
for
f
k
, and yet achieve an approximation of the original
data with the minimal error of Eq. (15.9). Since
B
M
has full rank with orthogonal
columns, using the full set in Eq. (15.20) would rotate the original coordinates. Con-
sequently, Eq. (15.20) is a truncated version of the rotated coordinates, where the
corresponding basis vectors are numbered according to their ability to represent the
observed data.
Before the rotation, it is often wise to translate the data to the mass center, es-
pecially in equally quantized feature spaces (e.g., images are usually quantized uni-
formly to yield 256 gray values), to avoid numerical problems stemming from the
limited dynamic range. Data translation can also be preferable from a pattern dis-
crimination viewpoint too, because the basis rotated at the mass center to yield the
minimal error of representation is not the same as a basis attached elsewhere and
rotated there. The rotation at the mass center will rank the directions according to the
maximal distance of the data to the mass center and the directional variation of the
data (which we discuss at the end of this section). If two classes are to be discrimi-
nated in a certain direction in the new basis, it is better that there is a great variation
in that direction than no variation, since the latter suggests that the classes leave the
same footprint, and hence cannot be separated in that particular direction.
Translating the observed vectors to the centroid:
K
1
K
f
k
←
(
f
k
−
f
c
)
,
where
f
c
=
f
k
,
(15.22)
k
=1
and searching for a new basis minimizing the approximation error
K
1
K
f
k
2
f
k
−
(15.23)
k
=1
among the bases attached to the mass center can be conveniently achieved by pre-
processing. The scatter matrix obtained after shifting the data to the mass center is
also known as the
covariance matrix
. Alternatively, the covariance matrix can be
computed as
K
1
K
f
c
)
T
C
=
(
f
k
−
f
c
)(
f
k
−
(15.24)
k
=1
where
f
k
is the unshifted observation data. The eigenvectors of
C
are frequently
called the
principal components
, whereas the rotated coordinates
O
, relative to the
mass center, are called the Karhunen-Loeve (KL) transform.
We could use the same arguments as above to find analogous results for the di-
mension reduction of the complex vector space,
C
M
, allowing us to formulate the
following lemma, which summarizes the conclusions so far.
