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Replacing sums with integrals and pmf’s with pdf’s (where applicable), we can obtain a proof for the maximal achievable rate region for the case where • I = Yˆ = Z = R. d. zero-mean Gaussian with variance PI . The distribution of I is denoted pG I for simpicity. • d(x, y) = dA (x, y) = (x − y)2 . d. 13). , by max γ∈Γ(RQ ,D,DA ) min RQ − 1 1 PW (γ) 1 − log(γ), log 1 + 2 2 DA γ . d. 3. The quantities γ, PW (γ), PYˆ are defined similarly here. , Z = 0. Observe that If PYˆ = E(Yˆ 2 ) ≤ DA then pZ|Yˆ (pYˆ |I )(z|ˆ n 2 ˆ ˆ2 pZ|Yˆ ∈ MA (pG I , (pYˆ |I ) , DA ) since E[(Y − Z) ] = E(Y ) ≤ DA .

2: The watermarking/quantization system with Gaussian attacks combined with scaling. 5) of achievable rates in the continuous case is then established The rate region Rgauss D by the following theorem. 3: The rate region Rgauss of achievable rate pairs (RQ , RW ). 6) ∆ and [·]+ = max{·, 0}. 3. 3. Its upper boundary is composed of: D • The segment AB on the straight line RW = RQ − 1 2 log PI D + . , the projection of BC on the RQ -axis. • The half-line C∞ which is parallel to the RQ -axis and has vertex C.

32) depends only on the pmf’s pZj ,Yˆj ,Ij (z, yˆ, i) = pZ|Yˆ (z|ˆ y ) pYˆj |Ij (ˆ y |i) pI (i), for each time instant j = 1, . . , n (pZ|Yˆ and pI do not depend on j due to memorylessness). Furthermore, RQ − I(Ij ; Yˆj ) and I(Zj ; Yˆj |Ij ) are both concave with respect to pYˆj |Ij . 35) j=1 was used. 36) ≤ max          pYˆ |I : ∃ pYˆ |I ... t. 36). 37) → 0, we conclude the proof of the converse. Direct Theorem We now show that Rdsc D is achievable. 16)). We use a random coding argument, where the watermark index W is assumed uniformly distributed in {1, .

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Digital Watermarking, Fingerprinting and Compression An Information-Theoretic Perspective by Karakos

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