Capacity and codes for embedding information in gray-scale signals

Gray-scale signals can be represented as sequences of integer-valued symbols. If such a symbol has alphabet {0,1,...,2^B-1} it can be represented by B binary digits. To embed information in these sequences, we are allowed to distort the symbols. The distortion measure that we consider here is squared error, however, errors larger than m are not allowed. The embedded message must be recoverable with error probability zero. In this setup, there is a so-called "rate-distortion function" that tells us what the largest embedding rate is, given a certain distortion level and parameter m. First, we determine this rate-distortion function for m=1 and for m→∞. Next we compare the performance of "low-bits modulation" to the rate-distortion function for m→∞. Then embedding codes are proposed based on i) ternary Hamming codes and on the ii) ternary Golay code. We show that all these codes are optimal in the sense that they achieve the smallest possible distortion at a given rate for fixed block length for any m.