The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
The result of detection: 
k=n—\ 
c = Y*W = {y(k)*w(k)} = ^y(k)*w(k) 
k=0 
c=£A* w(k) * w(k) + X g{k) * w(k) 
k=0 k=0 
When n —» +oo , 
k=n-1 
c=X*n+ X g(k)*w(k) 
k=0 
C ~ N(/l * n, n(jU 2 + (7 2 )) (1) 
Case 2: Non-contain watermark 
To-be-detected information: 
V = {y(k)} = G = (g(*)} 
The result of detection: 
C =X y(k)*w(k)= X g(k)*w(k) 
k=0 k=0 
c = X ^ w * W ( k ) ~ ^(°’ + cj2 )) (2) 
k=0 
Because the formula (1) and (2) have same variance, 
discrimination rule can be set as follow (Wang Xuemin, 1999): 
x 10" 
-1000 -500 0 500 1000 1500 2000 
n 
Figure 1. The probability distributing of autocorrelation 
detection 
A*n 
=500, so discrimination rule as follow 
2 
icontain watermark, if c>500 
[non-contain watermark, if c<500 
Then missing detection probability 
e, =e, 
= <D 
f 
V 
A*\fn 
2 *ja 2 +ju\ 
<D(-1.4142) = 0.0787 
contain watermark, 
< 
non-contain watermark, 
if c> 
if c< 
A*n 
A*n 
Then flow detection probability is 
e \ e 2 
A,*yfn 
2*^<j 2 +/u 2 J 
Because X,n is variable, it is not convenient to compare 
^ * n with detect result c to judge whether the to-be-detected 
2 
information contain watermark or not, we can normalize c to 
solve this problem. 
(3) 
Let Z = , the discrimination rule can be set as follow: 
X*n 
contain watermark, if z>0.5 
When a = 10, /y = 5,n = 1000,2 = 1.0 , the probability 
distributing of (1) and (2) is as figure 1 show. 
non-contain watermark, if z<0.5 
3. THE RELATIONSHIP BETWEEN WATERMARK 
LENGTH „ EMBEDDING STRENGTH AND 
ATTACK STRENGTH 
From the formula (3) we know that: on the same condition of 
attack, watermark length(n) > embedding strength( X ) and 
2 2 
attack strength(decided by (7 + fl jdecide the rate of false 
alarms. The rate of false alarms is proportional to the noise