approach by the subspace method is explained,
along with an experimental results derived from
Compact Airborne Spectral Imager (CAST) data to
compare this new method with conventional
approaches.
2. UNMIXING BY SUBSPACE METHOD
2.1 Statistical unmixing methods
Conventional statistical unmixing methods
assume that the mixel spectral vector is a
weighted mean of the class spectral vector which
constitutes the mixel. Within each mixel, there are
several mixed classes with area fractions which
correspond to the weights of the model. These
weights are estimated by the unmixing method.
In a remotely sensed image with p channels Æ
land cover classes exist in the image and area
fraction of classw(1) is fi . A linear mixel model
assumes that the observed p dimensional vector r
is expressed as
K
r=Mf+n=> fm +n (D)
i=l
Where M is a p x p matrix which has class
spectral vectors mi as column vectors, fis a vector
which has £i as components, and the n stands for
noise vector.
Statistical unmixing methods include, unmixing
by least squares, factor analysis and singular
value decomposition (Malinowski, 1991; Settle and
Drake 1993) By comparison, unmixing by
subspace method does not assume a linear
statistical model.
2.2 The Principle of subspace method
The basic idea behind the subspace method is
that the class spectral vector lies mainly in a small
class specific subspace instead of within the entire
dimension of the spectral space. If the class
subspace is determined from the training sample
of each class, class membership values can be
calculated by the projection of the mixel
observation spectral vector from the corresponding
subspaces from which the training samples were
drawn(Watanabe 1969; Kohonen 1977).
There are 3 ways of calculateing the subspace in
the subspace method. These are algebraic,
statistical and learning subspace method (Oja,
1984). In this paper, a statisitical subspace method
called | CLAFIC(CLAss-Featuring Information
Compression) algorithm is used. This method is
known to be fast and effective in the case where
782
the volume of training data is moderate.
2.3 Subspace determination by enhanced
CLAFIC method
The CLAFIC algorithm determines the class
subspace in order to maximize the projection of the
class vector on the corresponding class subspace.
However, by maximizing the projections for all
classes at the same time, the separation between
the similar classes decreases.
In order to avoid this drawback, we have
employed the Enhanced CLAFIC algorithm which
maximizes the projection on the class subspace to
which the training vector belongs and also
minimizes the projection on the other subspaces at
the same time. In the following, the Enhanced
CLAFIC algorithm is described.
In the enhanced CLAFIC method, the class
subspace L® which corresponds to a land cover
classesw(i) (I-1..,K) , is determined so as to
maximize the expected projection of vector x which
belongs to the classw(1) . It also minimizes the
expected projection of vector x which belongs to
the other classes is 99 ( #1). The problem here
is to determine the subspace L9 to satisfy these
conditions at the same time as formulating the
next minimization problem.
K
V EC P? dx e aq? ) — E(x' P? xax e?) (2)
j*i
=
where P% is the projection matrix to the L@
The first term of equation (2) is the expected
projection of sample vectors which do not belong to
the class w(i), and the second term is the expected
projection of vectors which belong to the class w(2).
By minimizing term (2), we can determine the
subspace L/? which minimizes the first term and
maximizes the second term of (2).
Projection matrix Æ is expressed using
orthogonal normal bases |ui ,...,upe 9 | of
subspace Lf as
(i)
PO — Y uu (3)
k=]
By substituting equation (3) into (2) and
rewriting (2) using base vector ux? (k=1,...,p9),
K (0 0)
S$ EG? yIx eq?) S$ Ecru? Ixew”) (4)
2 k=1 k=1
Calculating the expectation first, (4) becomes,
K (1) 0}
3 $0000 t Y uo goo (5)
Zi k=1 k=1
f
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
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