Fig. 1. Theoretical upper bound on bit error probability in the absence of attacks. The bound has been obtained by fixing the detection threshold in such a way to minimize P_b. Results refer to the watermarking of a 512×512 image (n=16 384).

Fig. 2. Dependence of the upper bound on P_b upon M. To better appreciate such a dependence, only a limited range of γ values is considered. Results refer to the watermarking of a 512×512 image (n=16 384).

Fig. 3. Bound on bit error probability vs. number of marked coefficients.

Fig. 4. Dependence of P_bound upon M and n. Results refer to an image watermarked with γ=0.1.

Fig. 5. Comparison between P_bound and true bit error probability (n=16 384, image size =512×512).

Fig. 6. Bit error probability in the presence of JPEG coding with decreasing quality factor. The watermark was inserted by letting γ=0.1,0.08,0.05 and n=16 384.

Fig. 7. Watermark detector response in the presence of JPEG coding with decreasing quality factor. The five highest responses over all possible watermark shifts are given along with the detection threshold (n=16 384, γ=0.1).

Table 1. False alarm probabilities in ensuring a false detection probability in equal to 10⁻⁶. The corresponding missed detection probability in are reported as well. The results have been obtained by letting γ=0.1, and n=16 384, and by averaging P_m over a set of 300 random watermarks

Table 2. Watermark payload for different values of M (n=16 384)

Table 3. True bit error rate and P_bound for different values of n. To speed up the experimental analysis, we set γ=0.03, so to obtain a higher bit error probability