Spatial Image Steganography Incorporating Adjacent Dependencies

Abstract

Most of the existing model-based image steganographic schemes in spatial domain assume the independence among adjacent pixels, thus ignores the embedding interactions among neighbouring pixels. In this paper, we propose a new image steganographic scheme by taking advantages of the correlations between each pixel and its eight-neighbourhood and determine the embedding cost of the pixel through the minimization of the KL-divergence between the cover and stego objects. Experimental results demonstrate that our scheme show superior or comparable performance to two state-of-the-art algorithms: HILL and MiPOD in resisting steganalysis detectors.

Appendices

Appendix

A Decomposition of $p(0 \sim 8)$

Figure 1, illustrates 9 pixels of one block with index $\mathrm{{\{ 0}} \sim \mathrm{{8\} }}$, where every two pixels are adjacent if connected or independent if disconnected. Based on conditional independence, for two independent pixels, i, j, we can easily obtain:

$$\begin{aligned} p(i,j) = p(i|j) \cdot p(j) = p(i) \cdot p(j), \end{aligned}$$

(15)

Let $i \leftrightarrow j$ represents adjacent pixels i, j, we proceed with a group of pixels according to conditional independence:

$$\begin{aligned} \begin{aligned} p(i,{j_1}, \ldots {j_m})&= p(i|{j_1}, \ldots {j_m}) \cdot p({j_1}, \ldots {j_m})\\&= p(i|{j_k},i \leftrightarrow {j_k}) \cdot p({j_1}, \ldots {j_m}). \end{aligned} \end{aligned}$$

(16)

Then, it’s obvious to get (17) and subsequently we take the second term of former formula repeatedly and use (16) to simply them:

$$\begin{aligned} p(0 \sim 8) = p(5|0 \sim 4,6 \sim 8) \cdot p(0 \sim 4,6 \sim 8) = p(5|0,1,2) \cdot p(0 \sim 4,6 \sim 8), \end{aligned}$$

(17)

$$\begin{aligned} p(0 \sim 4,6 \sim 8) = p(6|0 \sim 4,7,8) \cdot p(0 \sim 4,7,8) = p(6|0,2,3) \cdot p(0 \sim 4,7,8), \end{aligned}$$

(18)

$$\begin{aligned} p(0 \sim 4,7,8) = p(8|0 \sim 4,7) \cdot p(0 \sim 4,7) = p(8|0,1,4) \cdot p(0 \sim 4,7), \end{aligned}$$

(19)

$$\begin{aligned} p(0 \sim 4,7) = p(7|0 \sim 4) \cdot p(0 \sim 4) = p(7|0,3,4) \cdot p(0 \sim 4), \end{aligned}$$

(20)

$$\begin{aligned} \begin{aligned} p(0 \sim 4)&= p(4|0,1,2,3) \cdot p(0,1,2,3) = p(4|0,1,3) \cdot p(0,1,2,3) \\&= \frac{{p(0,1,3,4)}}{{p(0,1,3)}} \cdot p(0,1,2,3), \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned} p(0,1,2,3) = p(1|0,2,3) \cdot p(0,2,3) = p(1|0,2) \cdot p(0,2,3) = \frac{{p(0,1,2)}}{{p(0,2)}} \cdot p(0,2,3), \end{aligned}$$

(22)

For the numerator and denominator in first term in (21):

$$\begin{aligned} p(0,1,3,4) = p(3|0,1,4) \cdot p(0,1,4) = p(3|0,4) \cdot p(0,1,4) = \frac{{p(0,3,4)}}{{p(0,4)}} \cdot p(0,1,4), \end{aligned}$$

(23)

$$\begin{aligned} p(0,1,3) = p(1|0,3) \cdot p(0,3) = p(1|0) \cdot p(0,3) = \frac{{p(0,1)}}{{p(0)}} \cdot p(0,3), \end{aligned}$$

(24)

Substitute (22), (23) and (24) into (21), (21) into (20), (20) into (19), (19) into (18) and (18) into (17) in turn, (17) can be expressed as:

$$\begin{aligned} p(0 \sim 8) = \frac{{p(1,0,2,5) \cdot p(3,0,2,6) \cdot p(3,0,4,7) \cdot p(1,0,4,8) \cdot p(0)}}{{p(0,1) \cdot p(0,2) \cdot p(0,3) \cdot p(0,4)}}. \end{aligned}$$

(25)

B 2D and 4D Steganographic Fisher Information matrix

We mimic Eq. (3), where one-dimensional steganographic Fisher Information proposed, to construct 2D and 4D steganographic Fisher Information matrix in this proof. First, elements in 2D steganographic Fisher Information matrix are approximated as:

$$\begin{aligned} \begin{aligned} I_{ij}^{(12)}(0)&= I_{ij}^{(21)}(0) = {\sum _{m,n}}\frac{1}{{{p_{m,n}}}}(\frac{{\partial {q_{m,n}}({\beta _i},{\beta _j})}}{{\partial {\beta _i}}}{|_{{\beta _i},{\beta _j} = 0}}) \cdot (\frac{{\partial {q_{k,l}}({\beta _i},{\beta _j})}}{{\partial {\beta _j}}}{|_{{\beta _i},{\beta _j} = 0}}), \\&I_{ij}^{(11)}(0) = {\sum _{m,n}}\frac{1}{{{p_{m,n}}}}{(\frac{{\partial {q_{m,n}}({\beta _i},{\beta _j})}}{{\partial {\beta _i}}}{|_{{\beta _i},{\beta _j} = 0}})^2}, \\&I_{ij}^{(22)}(0) = {\sum _{m,n}}\frac{1}{{{p_{m,n}}}}{(\frac{{\partial {q_{m,n}}({\beta _i},{\beta _j})}}{{\partial {\beta _j}}}{|_{{\beta _i},{\beta _j} = 0}})^2}. \end{aligned} \end{aligned}$$

(26)

${p_{m,n}}$, the abbreviation of $p_{m,n}^{i,j}$, is the probability $P\{ {i} = m,{j} = n\}$, which can viewed as a point $f({x_i = m},{x_j = n})$ in the two-dimensional Gaussian distribution. Although ${x_i},{x_j}$ are both discrete integer variables, we suppose $f({x_i},{x_j})$ is continuous for differentiate:

$$\begin{aligned} f({x_i},{x_j}) = \frac{1}{{2\pi {\sigma _i}{\sigma _j}\sqrt{1 - \rho _{ij}^2} }}\exp ( - \frac{1}{{2(1 - \rho _{ij}^2)}}(\frac{{x_i^2}}{{\sigma _i^2}} - \frac{{2{\rho _{ij}}{x_i}{x_j}}}{{{\sigma _i}{\sigma _j}}} + \frac{{x_j^2}}{{\sigma _j^2}})). \end{aligned}$$

(27)

Given ${p_{m,n}} = f(m,n),{p_{m \pm 1,n}} = f(m \pm 1,n),{p_{m,n \pm 1}} = f(m,n \pm 1)$, considering the Taylor approximation at points $(m \pm 1,n),(m,n \pm 1)$, we have:

$$\begin{aligned} f(m \pm 1,n) \approx f(m,n) \pm \frac{{\partial f({x_i},{x_j})}}{{\partial {x_i}}}{|_{{x_i} = m,{x_j} = n}} + \frac{1}{2} \cdot \frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_i}^2}}{|_{{x_i} = m,{x_j} = n}}, \end{aligned}$$

(28)

$$\begin{aligned} f(m,n \pm 1) \approx f(m,n) \pm \frac{{\partial f({x_i},{x_j})}}{{\partial {x_j}}}{|_{{x_i} = m,{x_j} = n}} + \frac{1}{2} \cdot \frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_j}^2}}{|_{{x_i} = m,{x_j} = n}}. \end{aligned}$$

(29)

$$\begin{aligned} - 2{p_{m,n}} + {p_{m - 1,n}} + {p_{m + 1,n}} \approx \frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_i}^2}}{|_{{x_i} = m,{x_j} = n}}, \end{aligned}$$

(30)

$$\begin{aligned} - 2{p_{m,n}} + {p_{m,n - 1}} + {p_{m,n + 1}} \approx \frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_j}^2}}{|_{{x_i} = m,{x_j} = n}}. \end{aligned}$$

(31)

After substituting (5), (27), (30) and (31) into (26), and taking variables m, n, s, t continuous, we can finally derive (32)–(34). We give an example formula for element ij in 4$\,\times \,$4 Fisher Information matrix as (35), which can be easily generalized to other elements, their calculation results do not list because of the length limit.

$$\begin{aligned} \begin{aligned} I_{ij}^{(11)}(0)&= \sum \limits _{m,n} {\frac{1}{{{p_{m,n}}}}{{( - 2{p_{m,n}} + {p_{m - 1,n}} + {p_{m + 1,n}})}^2}} \\&\approx \int \limits _m {\int \limits _n {\frac{1}{{f\mathrm{{(}}{x_i},{x_j}\mathrm{{)}}}}{{\mathrm{{(}}\frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_i}^2}}{|_{{x_i} = m,{x_j} = n}}\mathrm{{)}}}^\mathrm{{2}}}} } dmdn = \frac{2}{{\sigma _i^2{{(1 - \rho _{ij}^2)}^2}}}. \end{aligned} \end{aligned}$$

(32)

$$\begin{aligned} \begin{aligned} I_{ij}^{(22)}(0)&= \sum \limits _{m,n} {\frac{1}{{{p_{m,n}}}}{{( - 2{p_{m,n}} + {p_{m,n - 1}} + {p_{m,n + 1}})}^2}} \\&\approx \int \limits _m {\int \limits _n {\frac{1}{{f\mathrm{{(}}{x_i},{x_j}\mathrm{{)}}}}{{\mathrm{{(}}\frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_j}^2}}{|_{{x_i} = m,{x_j} = n}}\mathrm{{)}}}^\mathrm{{2}}}} } dmdn = \frac{2}{{\sigma _j^2{{(1 - \rho _{ij}^2)}^2}}}. \end{aligned} \end{aligned}$$

(33)

$$\begin{aligned} \begin{aligned} I_{ij}^{(12)}(0)&= I_{ij}^{(21)}(0) \\&= \sum \limits _{m,n} {\frac{1}{{{p_{m,n}}}}( - 2{p_{m,n}} + {p_{m - 1,n}} + {p_{m + 1,n}}) \cdot ( - 2{p_{m,n}} + {p_{m,n - 1}} + {p_{m,n + 1}})} \\&\approx \int \limits _m {\int \limits _n {\frac{1}{{f\mathrm{{(}}{x_i},{x_j}\mathrm{{)}}}}} } \mathrm{{(}}\frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_i}^2}}{|_{{x_i} = m,{x_j} = n}}\mathrm{{)}} \cdot \mathrm{{(}}\frac{{{\partial ^2}f({x_i},{x_j})}}{{\partial {x_j}^2}}{|_{{x_i} = m,{x_j} = n}}\mathrm{{)}}dmdn \\&= \frac{{2\rho _{ij}^2}}{{{\sigma _i}{\sigma _j} \cdot {{(1 - \rho _{ij}^2)}^2}}}. \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned} \begin{aligned} I_{ijkl}^{ij}(0)&= \sum \limits _{m,n,s,t} \frac{1}{{{p_{m,n,s,t}}}} \cdot (\frac{{\partial {q_{m,n,s,t}}}}{{\partial {\beta _i}}}{|_{{\beta _i},{\beta _j},{\beta _k},{\beta _l} = 0}}) \cdot (\frac{{\partial {q_{m,n,s,t}}}}{{\partial {\beta _j}}}{|_{{\beta _i},{\beta _j},{\beta _k},{\beta _l} = 0}})\\&= \sum \limits _{m,n,s,t} \frac{1}{{{p_{m,n,s,t}}}} \cdot ( - 2{p_{m,n,s,t}} + {p_{m - 1,n,s,t}} + {p_{m + 1,n,s,t}}) \\&\cdot ( - 2{p_{m,n,s,t}} + {p_{m,n - 1,s,t}} + {p_{m,n + 1,s,t}}) \\&\approx \int \limits _m {\int \limits _n {\int \limits _s {\int \limits _t {\frac{1}{{f({x_i},{x_j},{x_k},{x_l})}}(\frac{{{\partial ^2}f({x_i},{x_j},{x_k},{x_l})}}{{\partial {x_i}^2}}{|_{{x_i} = m,{x_j} = n,{x_k} = s,{x_l} = t}})} } } } \\&{{\cdot ( \frac{{{\partial ^2}f({x_i},{x_j},{x_k},{x_l})}}{{\partial {x_j}^2}}{|_{{x_i} = m,{x_j} = n,{x_k} = s,{x_l} = t}})dsdtdmdn}}. \end{aligned} \end{aligned}$$

(35)