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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

- **) , **

M 2

t

SUM Ei =

E E —

—> min.

(2)

i=l

N

2

SUM (Bj -

**) — **

—> min.

(3)

j=l

N

where ** **

= SUM Bk / N.

(4)

k=l

(Bj - **) can be expressed by the following **

equation.

(Bj - **) **

= (-1/N, — 1/N 1-1/N. -1/N,

—1/N) B (5)

Let us consider the following Q,

t

Q = (B1 - **, B2 - ****,...) **

= 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 - **) = (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.

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