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3. Context classification featuring proportion
estimation
3.1 Procedure
The proposed contextual classification method
is an element of the proposed ICAI system. First,
homogeneous areas are extracted and classified by
using homogeneity measure proposed in ECHO
classification method(Ref. 4). After that spatial
features such as line-likeness, area-likeness, etc.
and properties such as directionality, length,
width, etc. of heterogeneous areas are calculated
followed by referencing of contextual information
together with relational information. Instead of it,
it is also possible to extract the pixels which have
macroscopic spatial properties by using the
aforementioned spatial features. If the confidence
interval at the designated confidence level in terms
of classification accuracy is not enough, then a
mixing ratio of the pixel of interest for the
classes of the surrounding pixels is to be a
classification result as one of classification for
Fuzzy pixels. Namely proportion would be enough for
classified result of MIXELs.
3.2 Proportion estimation based on Inversion Problem
Solving * 1
Based on Inversion Problem Solving by
Towmey(Ref.5). a proportion estimation method is
derived.
Let observed vector be I with the
dimensionality of M, mixing ratio vector be B with
the number of classes of N and the matrix
representing the spectral response of each class be
A.
1 = A B + E (1)
where E denotes error vector. Since 1 is a given
vector, if A is assumed then B is determined under
the constrains that minimizing 2
of square error:E , and variance
2
of the solution:(B
- <B>) ,
M 2
t
SUM Ei =
E E —
—> min.
(2)
i=l
N
2
SUM (Bj -
<B>) —
—> min.
(3)
j=l
N
where <B>
= SUM Bk / N.
(4)
k=l
(Bj - <B>) can be expressed by the following
equation.
(Bj - <B>)
= (-1/N, — 1/N 1-1/N. -1/N,
—1/N) B (5)
Let us consider the following Q,
t
Q = (B1 - <B>, B2 - <B>,...)
= C B (6)
where
C=I 1-1/N -1/N -1/N ....-1/N I (7)
I -1/N 1-1/N -1/N .... -1/N I
I -1/N -1/N -1/N ... 1-1/N I
Cij = { 1-1/N (i.oq. j) (8)
-1/N (¡.no. j).
Then minimizing the following equation.
N 2 t t t
SUM (Bj - <B>) = (C B) (C B) = B C C B
j=l > min. (9)
Solution B is obtained based on eq. (1) under the
constrains of eq. (2) and (9). It is not always that
solution is existing so that the following r is
introduced.
t t t
R = EE + rBCCB > min. (10)
Let us consider the following differentiation of R
about B
t t t
0R/0B = @ {(1 —AB) (I-AB) + rB C C B(
/ 0B (11)
where
t t t t t
CCB + BCC = CCB+(CB)C
t t t
= C C B + (C C B) (12)
t t
- A (I - A B) - (1 - A B) A
t t t
= - A (I - A B) - (A (I - A B)} (13)
then the following equation is reduced,
t t
0R/0B = rC CB - A (I-AB)
t t t
+ (rC CB - A (I-AB)} = 0 (14)
From above equation,
t t
rC CB - A (I-AB) = 0 (15)
results in
t t -1 t
B = (A A + rC C) A I. (16)
Proportion vector B is estimated with given
observation vector I, previously designated spectral
response matrix A and determined r for convergence.
4. Experiment
4.1 Data used
Landsat-5 TM data of Ushizu-machi, Saga, Japan
observed in May 1986 was used. Topographic map and
image of TM band-4 are shown in Fig. 2. The study
area includes mostly paddy field with tiny pond,
partially residential area and creek, road and
railway networks.
4.2 Procedure
4.2.1 Designation of classes and their anchor points
In order to show just an example, three
classes, paddy field, road and water body. Their
anchor points were determined as the corners of the
triangle corresponding to the three classes, as all
the sampled data for the designated classes are to
be included, in 3-D feature space with TM band-1, 3
and 4 as is illustrated in Fig. 3.