Watermarks

Consider the following image

Now, consider this second image

While they may look the same, they are not. The first version is an original photo downloaded from SLU's website. The second version is a modified version in which we have hidden additional information using a technique known as digital watermarking, explained below.

This particular photo has 800 x 600 pixels, with each pixel having "true color". Namely, each pixel has its own "rgb" value, with 8 bits to describe the intensity of red, 8 bits to describe the green, and 8 bits to describe the blue. This means that each color is described using a magnitude from 0 to 255. In MATLAB, the array representing this image is such that A(:, :, 1) contains the red channel intensities, A(:, :, 2) green, and A(:, :, 3) blue.

Our watermarking scheme is based on the fact that a viewer is very unlikely to notice minor discrepencies in those magnitudes. In particular, it is nearly impossible to visually detect the difference, for example, between a red magnitude of 187 versus 188. So we can reset the lowest-order bit for each of the three colors, for each of the pixels in the image. This allows us to embed additional binary data.

For today's exercise, you are to retrieve the following three types of information from the watermarked version of the above photo.

1. red channel
   We have used the lowest order bits of the red channel to embed a bitmap image. For each (row,column) of the original photo, if that low order bit is a zero it represents white in the hidden image, and if the low order bit is one it represents black in the hidden image.


2. green channel
   In the green channel, we manipulate the low order bits to encode a text representation. In particular, the 600x800 pixel image gives us 480000 low-order bits in the green channel. So we have used those bits to write a message in ASCII.

   Specifically, we consider the one-dimensional sequence of bits that results if taking the low-order green bit in "column major" order (that is, the first column from top to bottom, then the second colum from top to bottom). Our reason for using that convention is that it is exactly what you get is you "reshape" the green subarray into a one-dimensional array. That is, if you have the green "subarray" as an array G, syntax G(r,c) can be used to index the array by row and column, but G(k) is the k^th entry of the array when viewed one-dimesionally.

   The text can be reassembled by taking each group of 8 consecutive bits and recomputing the binary integer value that they represent (each being from 0 to 255). Once you have those integers, you can have MATLAB view those as characters using the char( ) function.



3. blue channel
   In the blue channel, we again use the column-major order of pixels to hide a binary sequence, this time encoding an audio clip (in mono). The audio clip uses 16 bits per sample and a sample rate of 8000 samples per second. While similar in style to our ASCII encoding of characters, there is an additional challenge for retrieving the audio.

   When dealing with text, the 8-bit integers were interpretted naturally as "unsigned" integers, namely having non-negative value from 0 to 255. In the case of audio, the 16-bits represent the amplitude of a sample as a signed integer, allowing for the possibility of negative numbers. The standard representation for signed integers uses a convention known as twos-complement.

   In short, if you first compute the unsigned interpretation of a 16 bit number, you can convert it to the signed interpretation as follows:

   • If the unsigned integer is a value val that is strictly less than 2¹⁵, the signed interpretation is also val.
   • If the unsigned integer is a value val that is greater than or equal to 2¹⁵, then it should be interpretted as the negative number (val - 2¹⁶).

Other digital watermarking techniques

Although our algorithm for watermarking worked for hiding information within the photo, this would not suffice for the purpose of copyright protection. The problem is that the owner of the photograph could trivially randomize those lower order bits, destroying our information without significantly degrading the image quality. The goal of watermarks in industry is to embed information in a file in a way that remains permanent, even when non-destrutive modifications are made. For more information, see additonal resources such as the Wikipedia entry.