CS 432 Project 3: Digital Watermarking *

In this project you are going to experiment with a technique for hiding information in an image known as digital watermarking. Digital watermarks can encode many different kinds of information in an image yet remain invisible. The most obvious application is to use watermarks to encode information which can prove ownership, e.g., copyrights. However, digital watermarks can also be used to encode copy or read permissions, time stamps and quality control information. They can be used to encode latitude, longitude, altitude and sensor characteristics in images collected by satellite. Finally, they can be used to encode content tags useful for addressing image database queries or indications of whether or not an image is suitable for viewing by children.

While a caption can become detached from an image, a watermark cannot be. To prevent a watermark from being removed (either intentionally or unintentionally) it should be robust to those operations commonly performed on images. These include 1) cropping and undersampling; 2) encoding and decoding operations used in image compression (e.g., JPEG); 3) geometric and point operations (e.g., rotation and histogram equalization) and 4) re-watermarking.

Technical Background

We are going to experiment with a simple technique which uses a symmetrical watermark to modulate the amplitudes of an image's frequency spectrum. Recall that the Fourier transform of a real-valued function possesses Hermite symmetry. That is, f(x,y)=f*(-x,-y), where the "*" indicates the complex conjugate (see figure below). The watermark must possess the same symmetry---the use of a non-symmetrical transfer function would create imaginary components in the watermarked image. The watermark is constructed by summing reflected copies of a natural image of one quarter size and normalizing the grey levels of the resulting image to a tiny sub-range centered on unity (i.e., one plus/minus some epsilon). Because the transfer function is real-valued, it changes only the amplitudes (and not the phases) of the Fourier transform coefficients. Consequently, it allows a significant amount of energy to be inserted into the image without creating visible image artifacts.

As an example, we can take an image of Microsoft chairman Bill Gates (after the notorious Belgium pie throwing incident) of size 127 x 127 and use it to construct a watermark for a 256 x 256 image of Leonardo da Vinci's Mona Lisa.**

Step 1: Step 2: Step 3:

Note: Grey pixels in the above figure represent 1's.

Note that this watermark possesses four-fold symmetry and that this is more symmetry than is (strictly speaking) necessary. To see an example of a watermark possessing the minimum symmetry necessary, click here. The watermarks are constructed so that they are symmetric about the zero-th frequency, or DC term, of the discrete Fourier transform. In the above, we follow the common convention of offseting the DC term by (N/2, N/2). Note that the zero-th row and column of the watermarks also equal one. Because of the (N/2, N/2) offset, this row and column represent harmonic signals with x or y component equal to N/2 (equiv. -N/2), the so called folding frequency. The following images show the Mona Lisa and its Fourier transform:

The following images show the Fourier transform after it has been multiplied by the Bill Gates watermark. Because the watermark image is very nearly one everywhere, there is no visible change. The watermarked Mona Lisa image is then computed by inverse Fourier Transform. The watermark is invisible.

To extract the watermark, we divide the Fourier transform of the watermarked image by the Fourier transform of the original image. This division produces the watermark which was previously inserted. The watermark extracted from the Mona Lisa image is shown below:

Project Outline

In this project, you will use Cantata to build three encapsulated workspaces called Make Watermark, Insert Watermark, and Extract Watermark:***

Make Watermark---Inputs an image (of unspecified size and type) to be used to generate the watermark together with a value for epsilon and outputs a watermark image of size 256 x 256 and type DOUBLE. You are free to use either of the two generation schemes outlined above (or a scheme of your own invention) to create the necessary symmetry in your watermark.
Insert Watermark---Inputs an image of size 256 x 256 (of unspecified type) to be watermarked together with the watermark generated by Make Watermark and outputs a watermarked image of type DOUBLE.
Extract Watermark---Inputs a watermarked image generated by Insert Watermark together with an unwatermarked image of size 256 x 256 (of unspecified type) and outputs the extracted watermark.

A Cantata workspace showing the three glyphs as they would be typically used is shown below.

After implementing the three encapsulated workspaces described above, you will experiment with the watermarking system to determine the best value for the epsilon parameter. To do this, you need to bound the parameter from above and below. To bound it from above, you need to find the largest value of epsilon (i.e., epsilon_max) which does not introduce "unnacceptable" visible artifacts into the image. [Note: Clearly, "unnacceptable" is a somewhat subjective notion. All that is required is that you use common sense and be prepared to defend your choice.] A binary search is probably the best systematic procedure for doing this. To bound the parameter from below, you need to find the smallest value of epsilson (i.e., epsilon_min) which will allow the watermark to survive (in detectable form) the various image processing operations which you will subject it to. These include:

Cropping -- You will demonstrate that the watermark is still detectable in a subregion of size 128 x 128. [Note: To see how this is done, click here. ]
Undersampling -- You will demonstrate that the watermark is still detectable after you reduce the size of the image by a factor of two.
Rotation -- You will demonstrate that the watermark is still detectable after you have rotated the image by 45 degrees.
JPEG Encoding -- You will demonstrate that the watermark is still detectable after it has been encoded into JPEG with image quality set to 75%.

At this point, you may find that epsilon_min > epsilon_max. This indicates that there is no acceptable value for epsilon which will work for your choice of image and watermark. If you believe that this is the case, just say so. There is no "right" answer. However, be prepared to defend your conclusion(s). It may be that a little bit more experimentation (effort) will allow you to recover the watermark in a recognizable (albeit highly degraded) form. This is all that is necessary. It doesn't have to be pretty.

Hints

You are likely to find that Flip, Pad, Extract and Inset are useful for creating your watermark.
When possible, the key to recovering the watermark from a transformed image is to invert the transformation.
You can use xv to encode/decode your image into/from JPEG.
Real Part and Imaginary Part allow you to extract the real or imaginary parts of a complex image.
Rather than display a complex image with Edit Image it is best to display only the real part.
Reducing the range of grey values by clipping (e.g., using Mega Clip) is very useful for displaying recovered watermarks.

What You Should Hand In

You should hand in a concise (less than 8 page) research report describing your project. This report should include a description of the process you used to bound epsilon from above and a discussion of the criterion you used to judge image quality. It should also include short descriptions of the methods you used to recover the watermark after cropping, undersampling, rotation and JPEG encoding. The following should be included either in the body of the report (where appropriate) or in an appendix:

Screen dumps showing the component glyphs and links of each of the three encapsulated workspaces, Make Watermark Insert Watermark and Extract Watermark.
Hardcopy of an unwatermarked size 256 x 256 test image of your choice.
Hardcopy of the image which you used to create your watermark.
Hardcopy of your watermark.
Hardcopy of the test image watermarked at epsilon = (epsilon_min + epsilon_max)/2.
Histogram of the imaginary part of the watermarked test image together with the maximum and minimum imaginary value.
Hardcopy of the 128 x 128 subregion and recovered watermark.
Hardcopy of the undersampled image and recovered watermark.
Hardcopy of the rotated image and recovered watermark.
Hardcopy of the JPEG encoded image and recovered watermark.

When You Should Hand It In

The above should be handed in at the beginning of class, Fri. April 14.

Of Possible Interest

* This webpage is located at http://cs.unm.edu/~williams/cs432/project3s99.html
** These examples are in honor of Gates' multi-million dollar purchase of the Codex Leicester, da Vinci's treatise on fluids in motion.
*** The glyphs for encapsulated workspaces can be used in Cantata just like those of kroutines and xvroutines. The first chapter of the Cantata Visual Programming Manual includes detailed instructions for creating encapsulated workspaces.