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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.