Test Statistic I
To determine the quality of the watermarking process, you can test the outcome. One metric is the difference between the population means of A and B rather than their values per se. The standardized test statistic is given by the equation in Figure 2(a).
Figure 2: Test statistic I.
Both sample means can be assumed normally distributed under the Central Limit Theorem, because of the large amount (M>>30) of Fourier coefficients. Therefore, '' is normally distributed and you can make the estimations '','' and . The independence of the random variables and , from the same set AUB of Fourier coefficients, leads to the approximations 0 and 0.
Therefore, you can formulate the mutually exclusive propositions:
- H0: (z)=N(0,1)
- H1: m(z)=N(m,1), m(k/ 1 +k2)x1/
with N(,s2) the normal distribution, mean , and the variation coefficent defined in Figure 2(d). m can be derived with the approximation in Figure 2(b). With these equations, the threshold can be calculated as T=z1-PI. According to the symmetry of the normal distributions (z) and m(z), k can be calculated from mT=z1PII and the approximation for m, see above. This is a reasonable assumption because and are both estimators calculated from the mixed sets A and B; see Figure 2(c).
Therefore, you can define the probabilities of correct detection (1-PII), rejection (1-PI), measure the variation coefficient in Figure 2(d) of the mean of set AUB, and calculate the necessary embedding factor k from the equation in Figure 2(c). But a factor k, ensuring the probabilities of correct detection or rejection according to Figure 2(c), may result in audible distortions. To satisfy both requirements, you can define the probabilities of correct detection (1PII), rejection (1PII), use the effective embedding factor k according to the psychoacoustic model, and calculate the number of Fourier coefficients from Figure 2(c).
Test Statistic II
According to the equation in Figure 2(a), you can extend the test statistic by normalizing the random variable z from equation >Figure 2(a) to the mean ('+')/2 of the whole set AUB; see Figure 3. Therefore, you expect z to be a random variable with mean 0 in the unmarked and an approximated mean of m2k in the marked case; again, see Figure 3.
Figure 3: Test statistic II.
One disadvantage of this kind of metric is the complicated PDF is not suitable for the calculation of the threshold and embedding parameter. I measured m for different k and verified the linear relation of equation in Figure 3.
Properties of the Algorithm
The general approach in quality evaluation is to compare the original signal with the watermarked signal for different k during subjective listener tests. To ensure the quality, the power of the watermarked noise is shifted just below the masking threshold. This results in an average embedding factor that is equivalent to k=0.15 derived without using the psychoacoustic model. Shifting the watermark noise to higher power levels decreases the quality and increases the average embedding factor. The embedding factor k=0.10 was used for robustness tests.