## A Communications Approach to Image Steganography (2002)

Venue: | in Proceedings of SPIE: Electronic Imaging 2002, Security and Watermarking of Multimedia Contents IV |

Citations: | 16 - 0 self |

### BibTeX

@INPROCEEDINGS{Eggers02acommunications,

author = {Joachim J. Eggers and R. Bäuml and B. Girod},

title = {A Communications Approach to Image Steganography},

booktitle = {in Proceedings of SPIE: Electronic Imaging 2002, Security and Watermarking of Multimedia Contents IV},

year = {2002},

pages = {26--37}

}

### Years of Citing Articles

### OpenURL

### Abstract

Steganography is the art of communicating a message by embedding it into multimedia data. It is desired to maximize the amount of hidden information (embedding rate) while preserving security against detection by unauthorized parties. An appropriate information-theoretic model for steganography has been proposed by Cachin. A steganographic system is perfectly secure when the statistics of the cover data and the stego data are identical, which means that the relative entropy between the cover data and the stego data is zero. For image data, another constraint is that the stego data must look like a "typical image." A tractable objective measure for this property is the (weighted) mean squared error between the cover image and the stego image (embedding distortion). Two different schemes are investigated. The first one is derived from a blind watermarking scheme. The second scheme is designed specifially for steganography such that perfect security is achieved, which means that the relative entropy between cover data and stego data tends to zero. In this case, a noiseless communication channel is assumed. Both schemes store the stego image in the popular JPEG format. The performance of the schemes is compared with respect to security, embedding distortion and embedding rate.

### Citations

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Citation Context ...message dependent data mapping operates sample-wise in the simple case of IID cover data x. To enable binary embedding, two disjoint sets X0 and X1 have to be de£ned, where X0 ∪X1 = X and X0 ∩X1 = ∅. =-=(14)-=- These sets X0 and X1 can be interpreted as the representatives of two different quantizers, which emphasises the relationship of the new information embedding technique to QIM. The message u is encod... |

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Citation Context ...SCS. Each of the letters is embedded into the corresponding cover data elements xn. The embedding rule for the nth element is given by ( ) dn sn = xn + α (Q∆ {xn − an}−(xn − an)) , where an =∆ + kn , =-=(3)-=- 2 and Q∆ {·} denotes scalar uniform quantization with step size ∆. The key k is a pseudo-random sequence with kn ∈ (0, 1] which has to be derived from the secret key K. This embedding scheme depends ... |

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Citation Context ...llest input data elements in order to achieve a mapping with minimum MSE distortion. Next, all output data elements yn belonging to the second histogram bin 5have to be derived from the remaining hy =-=[2]-=- smallest input data elements. Proceeding this argumentation shows that all bins of the output histogram have to be £lled in increasing order by mapping the input data with values in increasing order.... |

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(Show Context)
Citation Context ...er than the MSE Dπ =(1/Lx) ∑Lx n=1 (xn − yπ(n)) 2 ,whereπ(n) denotes an arbitrary permutation of the element indices. Now, consider the £rst histogram bin i =1of the output data which must contain hy =-=[1]-=- data elements. Due to Theorem 4.1, all data elements yn belonging into this histogram bin have to be derived from the hy [1] smallest input data elements in order to achieve a mapping with minimum MS... |

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Citation Context ...b ˜b =0 · pr r| ˜ ] ( ) [ b =0 + Prob ˜b =1 · pr r| ˜ ] b =1 ⎧ ⎨ Prob( = ⎩ ˜ b=0) Prob(x∈X0) px [r] ;r∈X0 Prob( ˜ b=1) Prob(x∈X1) px [r] ;r∈X1 ( ) = px [r] iff Prob ˜b = i = Prob (x ∈Xi) ∀ i ∈{0, 1}. =-=(16)-=- We observe ( that ) the PMF of the cover data is not modi£ed by the proposed information embedding scheme if the probability Prob ˜b =1 of “1”-bits in the encoded message ˜b is equal to the probabili... |

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Citation Context ...x) = (9) 2 0 ; else. i=1 ˜hx (t) is shown for the given example by a dotted line in the upper right diagram in Fig. 3. We introduce Nx Nx ∑ ∑ hx (t) = hx [i] δ (t − i) and hy (t) = hy [j] δ (t − j) , =-=(10)-=- i=1 with δ (x) being a Dirac impulse, in order to obtain well de£ned integrals of the histograms hx [i] and hy [j]. The lower diagrams in Fig. 3 show the integrals ∫ t 0 hy (τ) dτ, ∫ t 0 hx (τ) dτ an... |

97 | Detecting LSB steganography in color and gray-scale images
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Citation Context ...le histograms. The quantizer Qt is de£ned as the function ⎧ ⎨ x Qt(t) = ⎩ (1) ; t ≤ t1 x (j) ; tj−1 <t≤ tj ∀j ∈{2, 3,... ,Nx − 1} x (Nx) (11) ; tNx−1 <t so that the operation yn = Qt(tn) =Qt(in − an) =-=(12)-=- produces the desired output data yn if the set T of decision thresholds full£lls the integral equations ∫ 1 ∫ t1 ∫ j hy (τ) dτ = ˜hx (τ) dτ and hy (τ) dτ = ˜hx (τ) dτ for j ∈{2, 3,... ,Nx − 1}. (13) ... |

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(Show Context)
Citation Context ...imative) implementation is outlined. Let x denote a discrete random variable with the £nite alphabet X = {x (1) ,x (2) ,x (3) ,... ,x (Nx) }, where Nx = |X | < ∞ and x (1) <x (2) <x (3) <...<x (Nx) . =-=(6)-=- The PMF px [x] of x can be estimated from an observation x of length Lx by normalizing its histogram ˆpx [x] by the length Lx of the observed data sequence x. Considering the histogram ˆpx [x] is pre... |

50 |
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(Show Context)
Citation Context ... (12) produces the desired output data yn if the set T of decision thresholds full£lls the integral equations ∫ 1 ∫ t1 ∫ j hy (τ) dτ = ˜hx (τ) dτ and hy (τ) dτ = ˜hx (τ) dτ for j ∈{2, 3,... ,Nx − 1}. =-=(13)-=- 0 0 j−1 tj−1 The meaning of these integral equations is illustrated in Fig. 3 by the dashed lines in the lower diagrams. Note that the highest threshold t10 is redundant. We summarize that input data... |

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35 | G.W.Wornell, “Preprocessed and postprocessed quantization index modulation methods for digital watermarking - Chen |

31 | Detecting Steganographic Messages in Digital Images
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(Show Context)
Citation Context ...m variable a with uniform support over the range [0, 1). The PDF of the random variable t is proportional to the function Nx ∑ ( ˜hx (t) = hx [i] rect t + 1 ) { 1 ; −1/2 <x≤ 1/2 − i , with rect (x) = =-=(9)-=- 2 0 ; else. i=1 ˜hx (t) is shown for the given example by a dotted line in the upper right diagram in Fig. 3. We introduce Nx Nx ∑ ∑ hx (t) = hx [i] δ (t − i) and hy (t) = hy [j] δ (t − j) , (10) i=1... |

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(Show Context)
Citation Context ...ise variance of an AWGN attack. Decoding of the message û from the received data r is based on the extracted received data y. The extraction rule for the nth element is yn = Q∆ {rn − kn∆}−(rn − kn∆), =-=(4)-=- where |yn| ≤∆/2. yn should be close to zero if dn =0was sent, and close to ±∆/2 for dn =1. Although most of the work on SCS considers AWGN attacks, it has been shown as well that SCS is also robust a... |

27 | A robust optimization solution to the data hiding problem using distributed source coding principles - Chou, Pradhan, et al. - 2000 |

22 | Robustness of a Blind Image Watermarking Scheme
- Eggers, Su, et al.
- 2000
(Show Context)
Citation Context ...roduces as little distortion as possible. Here, a mean squared error (MSE) distortion measure DMap = 1 Lx ∑ Lx n=1 d(xn,yn) = 1 Nx Nx ∑ ∑ Lx i=1 j=1 ( γijd x (i) ,x (j)) , with d(xn,yn) =(xn− yn) 2 , =-=(7)-=- is assumed. An important consequence of this distortion measure is that the mapping x → y with minimum MSE must preserve the relation between different data elements. That is, for two input data elem... |

22 | Wornell, “Achievable performance of digital watermarking systems
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- 1999
(Show Context)
Citation Context ...ates the amount of data that has been mapped from values of the set X1 and X2, respectively. Formally, the conditional PMFs are given by [ pr r| ˜ { ] px [r] b = i = Prob(x∈Xi) ; r ∈Xi for i ∈{0, 1}. =-=(15)-=- 0 ;r/∈Xi The unconditional PMF pr [r] of the steganographic data r is given by ( ) [ pr [r] = Prob ˜b =0 · pr r| ˜ ] ( ) [ b =0 + Prob ˜b =1 · pr r| ˜ ] b =1 ⎧ ⎨ Prob( = ⎩ ˜ b=0) Prob(x∈X0) px [r] ;r... |

21 |
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(Show Context)
Citation Context ...e SCS watermark signal w has a uniform distribution of width α∆ and is statistically independent from the cover data x. Thus, the PDF of the watermarked signal s is given by ps (s) =px (s) ∗ pw (s) , =-=(5)-=- where “∗” denotes linear convolution. In general, a non-zero relative entropy between x and s has to be accepted, meaning perfect security against statistical steganalysis cannot be achieved. However... |

17 | Performance of a practical blind watermarking scheme
- Eggers, Su, et al.
- 2001
(Show Context)
Citation Context ...d output histogram hy [j]. Let in denote the symbol index in {1, 2,... ,Nx} of a given input data element xn. This index is randomly mapped on a continuous valued random variable t with tn = in − an, =-=(8)-=- where an is drawn from a continuous valued random variable a with uniform support over the range [0, 1). The PDF of the random variable t is proportional to the function Nx ∑ ( ˜hx (t) = hx [i] rect ... |

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5 | Wornell, "Achievable performance of digital watermarking systems - Chen, W - 1999 |

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3 |
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- 2001
(Show Context)
Citation Context ...unt of information that can be embedded per cover data element (steganographic capacity) is given by the binary entropy function H(P 1 )=P 1 log 2(P 1 )+(1− P 1 )log 2(1 − P 1 ) [bits/cover element]. =-=(17)-=- 8The steganographic capacity is 1 bit/(cover element) for P 1 =0.5and decreases for P 1 =0.5. The capacity is zero for P 1 =0 and P 1 =1. So far, the histogram preserving embedding method has been ... |

3 | Iterative decoding of digital watermarks - Kesal, Mihçak, et al. - 2000 |

1 | Optimal histogram modi£cation with MSE metric - Mes¸e, Vaidyanathan - 2001 |

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