In this example, the same football image is watermarked with the character ‘T’. Unlike the other cases, the watermarking is far less obvious, as the RGB values of the difference image are just scaled versions of the ASCII value of ‘T’, whose ASCII value is 84. Obviously, we know what the scale factor that was applied to the discrete cosine transform of ‘T’ (it is 0.5, so the r-g-b values of each pixel in the difference image is 42-42-42), but if we didn’t know this, more advanced watermark detection techniques such as those in the next section would be needed.
An example of watermarking a picture with something other that a picture is shown below:
With this scale factor, the difference image is mostly black, even though we can tell that the watermarked image is clearly different than the original one. Of course, this is still not the most robust way of watermarking, as lowering the scale factor also affects the amount that the image is actually changed. That being said, the difference image still doesn’t really show the watermark, so this example will require other methods in order to determine what the watermark is.
This watermarked image has the same problem as the first example, as the difference image tells us what the watermark is. However, this is using a scale factor of 0.25 on the watermark’s frequency domain coefficients (the first example also uses this scale factor), and using a slightly lesser scale factor of 15% results in the following image:
The example above illustrates a basic watermarking of one picture into another using frequency domain methods (in this case, we used the discrete cosine transform). Obviously, the difference image, which is the original image subtracted from the watermarked image, is a darkened version of the watermark, which means that it was very easy for us to determine what the watermark is without even using more advanced ways of detection. However, this is the case for a lot of visible watermarks.
Initial Findings for Inserting Watermarks
Using the images shown below, we started to look at watermarking techniques in the space-domain and frequency domain. Listed below in the next sections are the various methods for watermarking, and the methods for trying to find the watermark after the fact and remove the watermark.
Below is another similar example using a birthday cake with the same football image.
This method of transforming both the original image and the watermark to the frequency domain and the adding the transforms into one image and then taking the inverse transform of that back into the spatial domain is the same end result of just taking the hidden image and adding it (with a scaling factor) to the original image. This can be explained by Parseval's identity, where the sum of the discrete signal (in this case, the image) can be fully represented in the frequency domain. Adding the images in one domain is the same as adding the images in a different domain.