Improved RobustWatermarking Based on Rational Dither Modulation - Advances in Image and Graphics Technologies - page 308

where [ . ] is the round function, and mod ( . ) denotes the modulo function. It is easy to
see that the watermarked ratio r y is an even or odd multiple of ʔ .
To get the watermarked ratio r y , we modify x 1 and x 2 to y 1 and y 2 respectively.
Suppose x 2 is larger than x 1 , then the following equation must be satisfied:
r y = y 1

y 2 = x 1 + d 1

(9)

x 2 + d 2

where d 1 and d 2 are the modification strength of x 1 and x 2 , respectively.
In watermarking algorithms, robustness and transparency are always two conflicting
factors. It is generally accepted that high transparency will decrease robustness and
high robustness will limit transparency on the other hand. So there must be a tradeoff
between the two factors. In our scheme, at a given quantization step size, we want that
the embedding distortion which results from the sample modification will be as small
as possible. To this aim, we define several modification rules as follows:
- Decrease x 2 and increase x 1 ,if r y is larger than r x ;
- Increase x 2 and decrease x 1 ,if r y is smaller than r x ;
- The amount of modification of x 2 should be larger than the modification of x 1 .
Because it is widely accepted that larger coefficients allow greater modification
strength.
To satisfy the above modification rules, we let d 1 and d 2 meet the following equation:

x 1
x 2 d 2

d 1 = − (10)

Combined with (9) and (10), d 1 and d 2 can be calculated as:

x 1 − x 1 x 2 r y

x 1 + x 2 r y

, 2 = x 1 x 2 −

x 2 r y
x 1 + x 2 r y

d 1 = − (11)

Afterwards watermarked samples y i are obtained. At the decoding end, the water-
marked signal y may be attacked and changed to z . The watermark bit m k is decoded

by the minimal distance decoder:

m k = arg min

m k ||

r z − Q m k ( r z ) || ,m k ∈{ 0 , 1 } . (12)

Now, let us see why our method can increase the quantization step size. As previously
said, only one sample can be modified in the basic RDM algorithm, but two samples
can be modified in our method. Without loss of generality, we suppose that x 1 , x 2 , d 1 ,
d 2 meet the following conditions:

x 1 = x 2 ,d 1 = − d 2 (13)

As l 1 norm is used, we can suppose that the function of past watermarked samples
g ( y k− 1

k−L ) is approximately equal to x 2 . It is clear that the following inequality is satis-
fied:

x 1 + d 1
x 2 + d 2 −

x 1
x 2 |

x 1 + d 1
x 2

x 1
x 2 |≈|

x 1 + d 1
g ( y k− 1

x 1
g ( y k− 1

| > | −

k−L ) − k−L ) |

(14)

Thus the quantization step size in our method is larger than that of the basic RDM
watermarking method.