M. Costa considers three cases in the context of digital communication with side-information [101]. In 1983, digital watermarking did not exist. These cases, depicted in Fig. 7.5, are translated to zero-bit watermarking terminology as follows:
1. Neither the embedder, nor the detector knows the host signal X. The embedder is not side-informed and the detector is blind. Switches (a) and (b) are open in Fig. 7.5. The embedder emits a signal W independent of X, then the channel adds first X and then Z. These sources of noise being Gaussian and independent, N = W + Z is distributed as N (0n, N In) with N = σX2 + σZ2. The performances depend on P and N .
2. The embedder knows X, but not the detector. The embedder is side-informed and the detector is blind. Switch (a) is closed but switch (b) is open in Fig. 7.5. Many watermarking applications follow this case.
3. The embedder and the detector knows X. Both switches are closed in Fig. 7.5. The detector is not blind and it removes X from the received signal R. That way, the embedder may not use X because W only suffers from one source of noise, N = Z. The performances depend on P and N = σ2Z.
From cases 1 to 3, we keep on taking into account more information (more switches are closed). Therefore, the performances of case 2 (under our best efforts) should lie in between the perfor-
7.6. Conclusion 87
mances of cases 1 and 3. In other words, cases 1 and 3 play the role of the lower and upper bounds.
In article [101], Costa considers communication with side-information at the emitter (i.e. a decoding problem). He measures performances by the channel capacity, and shows that:
C1 = 1 2log 1 + P σ2 X + σZ2 ≤ C2 ≤ C3= 1 2log 1 + P σ2 Z . (7.5)
Then he exhibits a scheme under case 2 whose achievable rate does not depend on σ2X and moreover matches C3. This proves that, thanks to the side-information at the emitter, C2= C3.
In our detection problem, we may measure the performances by the error exponent charac- teristic and the same rationale translates as
F1(Efp) ≤ F2(Efp) ≤ F3(Efp). (7.6)
The fundamental question in zero-bit watermarking is whether there exists a scheme such that F2(·) = F3(·). Indeed, we are more interested if such equality happens for high Efp. In the same
way, the characteristic F (·) can be replaced by more relevant measurements like EfpR or ¯σZ2 in
inequality (7.6) and the underlying fundamental question.
Chapter 8 investigates schemes under case 1. Their performances translate to case 3 replac- ing N = σ2X+ σZ2 by N = σZ2. Chapter 9 investigates zero-bit watermarking under case 2 in the noiseless scenario: σZ = 0, while Chapter 10 assumes σZ > 0.
7.6
Conclusion
As stated so far, this fundamental question is ill-posed w.r.t. to the specificities of watermarking listed in Sect. 7.4. We need to be careful about the working assumptions. This especially holds for the knowledge the embedder and the knowledge of the detector about parameters (P, σX2, σZ2). Two schemes can only be compared if they work under the same assumptions.
In the following chapters, these assumptions vary from one scheme to another, but they are always clearly stated. It is obvious that the schemes where the embedder is oblivious to σ2Z and the detector is oblivious to (P, σX2, σ2Z) are more practical in real-life applications.
Chapter 8
One unique source of noise
Before investigating zero-bit side-informed watermarking, this section elaborates on a simpler problem defined as:
H0 : R = N,
H1 : R = w + N,
with N ∼ N (0n, N In). This models two cases introduced in the previous chapter:
• Case 1: zero-bit watermarking without side-information at the embedding side and blind detection. Since the host and the noise sources are independent, N = σ2X + σ2Z. The host is not a side information but a source of noise and the watermark signal cannot depend on X. It is a constant vector of squared norm kwk2 = nP shared by the embedder and the detector.
• Case 3: zero-bit watermarking with a non-blind detection. The detector removes X and the embedder is not obliged to take it into account. In that case, N = σZ2.
8.1
Optimal Neyman-Pearson detector
Appendix 15 explains how to derive the error exponents with a probabilistic point of view using the moment generating function. It applies this method to Spread Spectrum in App. 15.1. It shows that by considering the Neyman-Pearson test (the score function is the likelihood ratio s(R) = p(R|H1)/p(R|H0)), and the Chernoff’s bound for both Pfn and Pfp which gets tighter as n
increases, the characteristic for this simple detection problem is given by:
Efn = r P 2N −pEfp + !2 , (8.1)
with |a|+:= a if a > 0, and 0 otherwise. This gives birth to left and right endpoints:
EfnL = EfpR= P
2N. (8.2)
When operating at Efp = E, the watermark is robust (in the sense that EfpR ≥ E) when
N ≤N/2P, which translates to Case 1: ¯σZ2 = P 2E − σ 2 X + , Case 3: ¯σZ2 = P 2E. (8.3)
u u
Figure 8.1: According to Shannon, the circular cone (right) is optimum in the sense that it maximizes Q(Ω) for a given solid angle Ω.
The bigger E, the less robust the watermark is. Roughly speaking, for a required Pfp, operating
at Efp = E implies that the dimension of the vectors is about n ≈ −log Pfp/E. The bigger E,
the shorter the vectors are. As a consequence, for instance under case 3, ¯σZ2 ≈ nP/2| log Pfp|. We
rediscover here the well-known rule of thumb of digital watermarking: the more spread (i.e. large n), the more robust the watermark is (large ¯σZ2).
Obliviousness to parameters (P, N ) prevents computing p(R|H0) and p(R|H1). But, by ap-
plying a suitable increasing function, the likelihood ratio indeed boils down to the simple sufficient statistic s(R) = R>u where u :=w/kwk. The detection region is thus a half-space delimited by
the hyper-plane R>u = τn. This simple detector achieves the characteristic function (8.1).
The main problem is that the threshold τn=
√
N Φ−1(1 − Pfp) to meet a prescribed proba-
bility of false positive depends on N usually unknown in practice. In the same way, for a targeted Efp = E, threshold τnmust scale as
√
2N En. The obliviousness to N prevents the use of the opti- mal Neyman-Pearson detector. Chapter 11 deals with other schemes sharing the same drawback.