Wavelet Transform Digital Watermarking decomposes a source cover image into a number of 2-D critically sampled subbands. There are four subbands created at the end of each stage of the wavelet transformation, defined in the vertical and horizontal directions: Low-Low, High-Low, Low-High and High-High. Subsequent stages apply a similar transform to the Low-Low band of the previous stage. The original signal can be completely reconstructed by performing the Inverse Wavelet Transform.
Watermarking insertion happens in any chosen subset of the cover image wavelet subbands except in the Low-Low subband of the last stage. Assuming that a digital watermark is a small binary image, the process is based on slight changes to high energy wavelet coefficient values in chosen subbands by the amount proportional to the watermark:
C’n,i,j = Cn,i,j +λwi,j|Cn,i,j|,
where:
- w – is a digital watermark,
- Cn,i,j – wavelet coefficient of the n-th subband,
- i and j are coordinates of the coefficients within the subband.
Here the energy of the subband wavelet coefficient is modeled as its absolute value. Instead of |Cn,i,j| the value Cn,i,j2, may be utilized.
In one implementation w may take values 0 and 1, in another ±1.
There is also a method where, instead of direct changes to the subband coefficients, weighted Gaussian noise energy proportional to w and Cn, is added to Cn:
C’n = Cn +λ∙w|Cn|∙N
The reason to use Gaussian Noise is it improves transparency and is quite robust to collusive attacks.
The parameter λ controls the level of the watermark. It has to be chosen based on a good compromise between transparency and robustness.
Human eyes are not sensitive to the small changes in edges and textures of the cover image but are very sensitive to the small changes in the flat or smoothareas of an image. Therefore making change proportional to the coefficient energy is a way to avoid proportionally big changes in the areas of small coefficients that in many cases correspond to the flat or smooth areas.
After watermarking insertion into the subbands, the inverse wavelet transform is applied to form the watermarked image.
The procedure for watermark extraction from a watermarked image utilizes the original cover image and watermark. First the wavelet transform is performed. Then, in subbands where watermarking is embedded, the energy of the coefficients is normalized to the energy of the original image in the same subband. The difference between subbands of watermarked and original images contains the watermark. Lastly, there may need to be post processing, depending on the particular method of insertion.
The advantage of wavelet digital watermarking over other transforms such as DCT is a robustness to common image distortions such as: additive noise, rescaling/stretching, half toning and wavelet compressions based on embedded zero-trees.