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