Digital Watermarking using Phase Dispersion Algorithm
It’s an algorithm for robust, invisible watermarking.
Use the characteristics of spread-spectrum to hide and extract information.
Embedding an iconic image
The embedding process can be defined as the following equation:
I ' ( x, y )   ( M ( x, y ) * C ( x, y ))  I ( x, y )
(1)
where I(x,y) is the original image.
I’(x,y) is the watermarked image.
M(x,y) is the watermark.
C(x,y) is the Carrier function.
 is an arbitrary constant which is chosen to make the embedded message invisible.
M(x,y) is a small iconic image and has typical size of 128x128 or 64x64 or 32x32. In
order to make it hard to detect, a lower energy image is better candidate than a higher
energy image.
C(x,y) It’s a carrier function with random phase.
Extracting an iconic image
M ' ( x, y )  I ' ( x, y )  C ( x, y )   ( M ( x, y ) * C ( x, y ))  C ( x, y )  I ( x, y )  C ( x, y ) (2)
where M’(x,y) is the extracted watermark.
Ideally, the cross correlation of the carrier function with itself should be a delta function,
while cross correlation of the carrier function with the original image should be zero.
C ( x, y )  C ( x, y )   ( x, y )
I ( x, y )  C ( x , y )  0
M ' ( x, y )   ( M ( x, y ) *  ( x, y ))   ( M ( x, y ) ,
(3)
(4)
(5)
where  ( x, y ) is the Dirac Delta function.
Carrier function design consideration
Carrier function is the most critical here to make embedding and extraction possible. It
needs to have the characteristics so equations (3), (4) can be true and very close. This can
be accomplished by setting the Fourier magnitude of the carrier image to a constant at all
frequencies.
(6)
C( , )  constant,
The Carrier function is generated when the user specify a piece of ‘key’ information. This
information is needed for reconstructing the Carrier function in the extraction process.
Without the key, the watermark can not be extracted and seen.
The design of the carrier is a tradeoff between the visual transparency of the embedded
message, the optimal extracted signal quality and robustness to image processing.
Contrast Sensitivity Function (CSF):
Reciprocal of the CSF can be used to determine the carrier magnitude needed at a given
frequency to bring the embedded signal just below the threshold of detectability by an
average observer.
Make it robust by embedding each watermark into multiple-tile image
Embedding processing illustration
Robustness and testing
Two factors are used here to describe the level of change. CF (Correlation Factor) for the
image similarity and MSE (Mean-square Error) for image difference.
CF 
M ' ( x, y )  M ( x, y )
M ( x, y )  M ( x, y )
1
N
M
( I ' (i, j )  I (i, j )) 2

i 1 i 1
NM
The following tests were done for evaluate the performance of this algorithm.
MSE ( I , I ' ) 
In the test, 2048x2048 ‘Lena’ is used for the test image. The 128x128 ‘Team 1’ image is
used as the watermark.
2048x2048 256 grey scale ‘Lena’ image
128x128 watermark
image
Affected by low-pass filter
When a lowpass filter is applied on a watermarked image, the extracted watermark is
blurred just as the filter is applied on the watermark. From the following equation, the
extracted logo is equivalent to original log convolve with a low pass filter.
Affected by JPEG lossy compression
The following are the test summary for how JPEG compression affects the detectability
of the extracted image:
No compression
Compression ratio of
7
Compression ratio of
13
Compression ratio
of 20
Affected by noise
The following are the test summary for how noise affects the detectability of the
extracted image:
10% zero mean noise
20% zero mean noise
Extracted watermark
Extracted watermark
50% zero mean noise
100% zero mean noise
Affected by cropping
The following are the test summary for finding out how cropping affects the detectability
of the extracted image:
50% cropping
90% cropping
Affected by multiple watermarks
The following are the test summary for how multiple watermarks affect the detectability
of the extracted image:
Watermark 1 with a key
Watermark 2 with a
different key
Two watermarks with the
same key
From the test, the algorithm is found to be able to handle multiple watermarks. With
different key, the watermarks can be extracted separately. With the same key, a
composite watermark is extracted.
Affected by halftone printing
The following are the test summary for how halftone printing affects the detectability of
the extracted image:
Image after printed and rescanned
Extracted watermark
After the image is printed and scanned, because of halftoning, the correlation is