Full text: XVIIIth Congress (Part B7)

> sampling 
ation from 
assification 
The article 
thode der 
eitgestellt. 
en werden 
rektur des 
in crop is 
of the ML 
mation on 
the study 
ssification 
he results 
ncy table: 
.e. winter 
e number 
se fallible 
eloped by 
in (1972) 
; the pro- 
(1) 
  
where p is the true proportion and 0 and $ are probabi- 
lities of misclassification and q is 1 — p. This equation 
applies to the binomial case with only one land use 
category. 
In case of k mutually exclusive land use categories, 
(multinomial case) the result for the category j is 
k 
mi =) pili (2) 
i=1 
where 0;; is the probability that a unit, which belongs 
to the category ? is classified to the category j. The 
bias in the estimates 7; — p; is corrected ba applying 
the following equation 
P—AxII (3) 
where P is & x 1 column vector of the estimates p;, À is 
kxk Matrix with the terms nj; — aij/a.; and II is the 
kx 1 column vector with the results of the maximum 
likelihood classification. To correct the misclassified 
results of the ML classifier given by the vector II the 
error matrix À must be known. 
Let us consider the results of the ML classificati- 
on with multitemporal ERS-SAR GTC data for five 
major crops. In the study area following contingency 
table is available from which A can be derived: 
fallible ML classifier 
Crop 1 2 3 4 5 6 3 
  
  
T 1. 2751. 794. 907. 419. 120 2019... 7010 
r à 186 ..511...166. .211...41.;-515...1630 
u 3 681 352 824 263 107 868 3095 
e 4 211 915 420 2368 460 2256 6630 
5 55 184 94 368 111 464 1276 
GIS 6 305 412 242 341 65 589 1954 
> 4189 3168 2653 3970 904 6111 21595 
Where: 1 - winter wheat, 2 - winter barley, 3 - summer 
barley, 4 - sugar beets, 5 - maize and 6 - others 
The disadvantage of the above procedure is obvious. 
The true classifications have to be provided for the 
entire area. But this data is not available. This is the 
am of the image classification. 
Some information can, however, be gained using 
sampling techniques. Suppose a small sample can be 
provided with true classifications. The size of that 
sample is denoted n. The entire area consisting of N 
pixels is classified using only the ML- classifier. Thus 
the n members of the sample are classified twice. 
The results give following contingency table: 
fallible maximum 
  
  
likelihood 
classifier 
Crop s3da 2 k 
1 311 9122 ‘+ ay aj. 
True 2 ag; 822 Ak a2. 
classifier 3  a31 aa23 -:- ask as. 
(GIS) 
k' “ayy Vago ev. Cag | ar. 
an An par n 
  
X1 Xo X3 Xk N-n 
  
where a;; is the number of pixels in the sample whose 
true category is ¢ and whose fallible category is j. X = 
(X1, X2, ---, Xx) is a vector of frequencies, where X; 
is the number of pixels in the image with N units 
whose fallible category is 7. 
After Tenebein (1972), estimates of p and the 
misclassification probabilities 0 can be derived as fol- 
lows 
k 
pi =) aij(X; + a;)/(a;N) (4) 
j=1 
ó; = (X; + a j)aij/ (a.5.N pi) (5) 
The results of the maximum likelihood classificati- 
on of the N - members of the image (X; + a ;) are 
corrected by multiplying with the ratios a;;/a.; and 
summing over j. In the case of k land use categories, 
the estimates are derived as given in equation 3. 
In addition a coefficient of reliability K is defined. It 
measures the strength of the relationship between the 
true measurements and maximum likelihood classifier 
for each category. 
  
  
: k 
K: = pi(Y 68/73 — 1)/gi (6) 
j=1 
The variance of p; is 
: Didk 7 Piqk 
iz 1 —E; K; 
vig) =a ky + BK) 
If K = 0, the true classification is the only way to 
obtain reliable estimates for the fraction of considered 
crop area. If K = 1, the ML classifier is not fallible, 
thus the true classifications are not required. In samp- 
ling optical data K is within the range of 0.3 - 0.8 ( 
Smiatek, 1993). 
It is necessary, however, to accept the sampling er- 
ror as the error probabilities are estimated from a sam- 
ple. But, the required sample size can be estimated 
according to the desired accuracy criteria. This is the 
great advantage of the sampling procedure proposed 
here. 
625 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996 
 
	        
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