the syncronization accuracy, the signal transfer, A/D con- 
version, the quality of the sensors, the temperature, the 
humidity and so on[5]. Since each part of the error is quite 
small and independent, we can suppose it to have a nor- 
mal distribution according to the central limit theorem of 
mathematical statistics, that is 
e(i, j) + N(n,0?) (24) 
Usually wu should be zero. If u is not equal to zero, we can 
take a simple translational transformation to make u equal 
to zero. So the error &(?, j) can be supposed : 
e(i,j) ~ N(0,0*) (i,j) €R (25) 
The equation (25) is also called the random model of the 
image. 
3.2 The Function Model 
An image region Ry is defined here as a set of the con- 
nected pixels p;;, which true grayvalues §(z, 7) satisfy a cer- 
tain function: 
fi((4,7,3(, 7) =0 (26) 
If we really know the light intensity function of an object, we 
should use this function as the segmentation function. We 
must linearize the function at first if it has a unlinearized 
form. Otherwise let us use the first consistent principle for 
the segmentation, namely suppose there is a homogeneity 
inside a region. That means the grayvalues in the same 
region are homogeneous. So we can suppose the grayvalues 
in a region satisfy a planar equation: 
9(2,3) 2 axi + bj t e (27) 
or more simply a horizontal plane: 
3(53) = xx (28) 
Refer to the equation (4), we can set the linear estimation 
equation as: 
v(i,j) = axi bj ek —9(53) | (G3) € R& (29) 
for the whole region we have 
V=AX-L with weight P = I (unit matrix) (30) 
where X = (a4, by, cy )T. 
According the principle of the least square method, we can 
obtain the optimal estimates of X 
X — (ATA)! AT], (31) 
: |ZL-XTATL 
3.3 Criterion of Segmentation 
and the variance 
  
We know that n pixels (pi, p2,:::, pa) belong to the same 
region. From the equation (31) and (32) we can get the 
region parameters X and the variance &. Now the question 
is how to determine whether the pixel p,4,1 belongs to the 
region. If the pixel p,41 belongs to the region, it satisfies 
the equation (29). Otherwise there is a model error, or we 
   
can say the grayvalue gn+1 has a different expectation than 
the grayvalues (g1, 92,‘ ** , Jn)- We can represent it as 
In+1 + Un+1 = Okin41 + ÖkIn+ı fc; +5; (33) 
So we can determine the pixel p,41 by a hypothesis test: 
Ho: E(s./Ho) = 0 (34) 
H, : E(s,/H;) = En (35) 
In the equation (18), let H = (0,0,---,0,1)7, P = I, we 
have the statistic variable 
Ti i ia. ^^ x?(1,6?) (36) 
020, rong i 
where v,+1 is the correction to g,,1 under the primary hy- 
pothesis and can be obtained from 
V-AX-L (37) 
with V = (v1, 921 Un, Uni? L= (91,92, ri Um Ja) 
Qv, 41,41 18 the element of the matrix Qyy in the n + 1 line 
and n + 1 column, which is computed from 
Qvv 2 I — A(AT A)! AT (38) 
where A is same as in the equation (37). 
The equation (36) can be simplified as the standardized 
normal distribution under the primary hypothesis: 
T 
ns Pf 
O A don 419541 
In applications we can use 6 computed from the first n 
pixels as the o in the equation (39), we can also use the 
statistic variable T? from the equation (21): 
^ N(0,1) (39) 
5 [vn 41 | 
i10 = 17 & Tt i(n —t— 1,0 40 
ie dn ) (40 
where 
  
Qun+19n+1 
2 
se rv mt (41) 
The others are the same as in the equation (39). 
3.4 Separability of Regions 
  
In order to determine which hypothesis is correct, a risk 
level x must be given. This risk level o is the probability of 
incorrect rejection of the primary hypothesis. For a given 
a, we can find a correspondent critical value K, from the 
table of the probability density function. If the value of 
statistic variable (39) or (40) is greater than this K,, the 
primary hypothesis is rejected and the alternative hypothe- 
sis is accepted. The probability of correct acceptance of the 
alternative hypothesis is called the power of test B, There 
is another risk in hypothesis test, namely the probability of 
incorrect acceptance of the primary hypothesis. It is equal 
to 1 — B. The power of test B is not only dependent on o 
(the smaller o the smaller 8), but also on the magnitude of 
the model error sy. 
Now we have to answer the question, how great a model 
error should be, in order that it can be found by the hy- 
pothesis test under the given risk error a and power of test