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generalization. Suppose we have the function model of pa- 
rameter estimation as the following: 
Anxt Xıx1 = E(L)nxa (1) 
where L is the observations, E(L) is the expectation of L, 
X is the unknown parameters, n should be greater than t, 
A is the coefficients matrix and has full rank, and 
n Ty Qu 412 cC Qu 
2 T2 az 22 az 
pe Xe AS 
In Zt Qni Qn2 ‘** Ant 
(2) 
In the equation (1) A and L are already known, but X and 
E(L) are unknown. 
The differences between observations L and expectations 
E(L) are called true errors €, the negative € is called cor- 
rections V: 
e = L — E(L), V2-e (3) 
From the equation (1) we can have 
V=AX-1 (4) 
Suppose the observations L have a normal distribution, the 
variance of L is X, the correspondent weight is P. Then the 
probability density function of L is 
fh, ll, 10) = 
arts — 512 — (DEL — ES) 
and 
yp (6) 
where c is called unit weight variance. 
The least square method estimates the unknown parameters 
X under the conditions: 
VT PV — minimum (7) 
The estimates of X then are 
X 2 (ATPA) ! ATPL (8) 
The estimates of o and the variance of X are 
  
: VTPV 
os 5 —1 (9) 
Xxx 2o?(ATPA)'! (10) 
The estimates X are the best linear unbiased estimates and 
the Xxx is minimal if the observations have a normal dis- 
tribution. 
2.2 Hypothesis Test 
By means of hypothesis test we can determine if there are 
any model errors in the function model (1). Suppose the 
primary hypothesis is 
Ho: E(L/Ho) - AX (11) 
the alternative hypothesis is 
Ho E(LJH,) S AX HS (12) 
  
In the equation (11) and (12) A4; and H4 are the known 
coefficients matrixes, X;x; are the unknown parameters, 
Sıpx1 are the unknown model error parammeters, Lnx1 are 
the observations, and 
L ^» N(E(L),X) (13) 
We can rewrite the equation (12) as 
L+V=AX+ HS withweigh Pzeo!E!..(14) 
Suppose n > t + k, and [A H] has full rank, then we can 
obtain the estimates S from the equation (14) with the least 
square method[4]: 
S = P;lHT PQvv PL (15) 
where 
Quy £73 = AAT PAYA” (16) 
Pss = HT PQyy PH (17) 
If o? is known, then the following statistic variable T; has 
the uncentralized 
x?-distribution with k degrees of freedom and uncentralized 
parameter 6: 
T, = —(L7BL) ~ x2 (k,8%) (18) 
where 
B = PQvv PHP; HT PQvvP (19) 
1 
6? = —(ST PssS) (20) 
o 
If the unit weight variance c is unknown, we can use the 
following statistic variable T»: 
LTBL 
WI 
If the primary hypothesis is right, the uncentralized param- 
eter § is zero, so we can use the statistic variable Tj or T2 
under a certain risk level a to determine if there are any 
model errors. 
F(k,n —t— k, 6°) (21) 
3. IMAGE SEGMENTATION 
3.1 Image Error Analysis 
  
An digital image R is usually described by a two dimen- 
tional array G. Each element g(i,j) of the array G repre- 
sents the grayvalue of the image at the position —line and 
j—column. An image pixel can be therefore defined by its 
position and its grayvalue: 
The grayvalue g(i, j) is composed of two parts, namely the 
true grayvalue ÿ(i, j) and the true error e(i, j), which both 
are unknown: 
The error e(i, j) of the pixel p(i, j) is caused by the instru- 
ment and the enviroments of the image acquisition. There 
are a variety of influence aspects to the error e(i,j), eg. 
if a CCD camera is used to acquire a digital image, then 
the error sources can be among others the lens distortion,