Reversible Data Hiding for Encrypted Images Based on Statistical Learning

Abstract

In this paper, we propose a novel reversible data hiding (RDH) approach for encrypted images by using statistical learning. To hide the data, a new random permutation algorithm is proposed using a high-speed stream cipher to secure the data hiding process. A secret message is embedded into the permuted image blocks based on a checkerboard pattern, by modifying the least significant encrypted bits. To detect the hidden data, prior works utilize a single spatial correlation. In contrast, our approach is novel in that it uses a high-dimensional statistical feature vector upon which a new boosting algorithm for high reversibility is proposed. A complete encoding and decoding procedure of RDH for encrypted images is elaborated. The experimental results show that the proposed method can detect secret message bits and restore the original image simultaneously with $100\,\%$ reversibility with a higher capacity, significantly outperforming the state-of-the-art RDH methods on encrypted images.

Appendices

A Appendix

Spatial Smoothness Features

$$\begin{aligned} {v_{3}}=\sum \limits _{i=2}^{n-1}{\sum \limits _{j=2}^{n-1}{\left( {\left| {{c_{i,j}}-{c_{i-1,j}}}\right| +\left| {{c_{i,j}}-{c_{i+1,j}}}\right| +\left| {{c_{i,j}}-{c_{i,j-1}}}\right| +\left| {{c_{i,j}}-{c_{i,j+1}}}\right| }\right) }} \end{aligned}$$

(A.1)

$$\begin{aligned} {v_{4}}=\sum \limits _{i=2}^{n-1}{\sum \limits _{j=2}^{n-1}{\left( {\left| {{c_{i,j}}-{c_{i-1,j-1}}}\right| +\left| {{c_{i,j}}-{c_{i+1,j+1}}}\right| +\left| {{c_{i,j}}-{c_{i+1,j-1}}}\right| +\left| {{c_{i,j}}-{c_{i-1,j+1}}}\right| }\right) }} \end{aligned}$$

(A.2)

where $c_{i,j}$ is a pixel value in a $n\times n$ square image block. Natural images tend to have lower values of these features while unnatural images tends to have higher values.

Average pixel values:

$$\begin{aligned} {v_{5}}=\bar{c}=\frac{1}{{n^{2}}}\sum \limits _{i=1}^{n}{\sum \limits _{j=1}^{n}{\left| {c_{i,j}}\right| }} \end{aligned}$$

(A.3)

Absolute mean value difference between black and white marked pixels:

$$\begin{aligned} {v_{6}}=\left| {\sum {\sum \nolimits _{\left( {i+j}\right) \,\bmod \,2=0}{c_{i,j}}}-\sum {\sum \nolimits _{\left( {i+j}\right) \,\bmod \,2=1}{c_{i,j}}}}\right| \end{aligned}$$

(A.4)

Geometric mean value:

$$\begin{aligned} {v_{7}}={\left( {\prod \limits _{i=1}^{n}{\prod \limits _{j=1}^{n}{c_{i,j}}}}\right) ^{\frac{1}{{n^{2}}}}} \end{aligned}$$

(A.5)

Mean absolute deviation:

$$\begin{aligned} {v_{8}}=\frac{1}{{n^{2}}}\sum \limits _{i=1}^{n}{\sum \limits _{j=1}^{n}{\left| {{c_{i,j}}-{\bar{c}}}\right| }} \end{aligned}$$

(A.6)

Second to seventh order of the central sample moments:

$$\begin{aligned} {v_{7+k}}=M_{k}=\frac{1}{n^{2}}\sum \limits _{i=1}^{n}{\sum \limits _{j=1}^{n}\left| c_{i,j}-\bar{c}\right| ^{k}},\,k=2,\cdots ,7 \end{aligned}$$

(A.7)

where the second order statistic is also used in [10].

Discrete Cosine Transform. DCT coefficients constitutes $n\times n$ features:

$$\begin{aligned} {V_{u,w}}=\sum \limits _{i=1}^{n}{\sum \limits _{j=1}^{n}{{c_{i,j}}\cos \left[ {\frac{\pi }{n}\left( {i-\frac{1}{2}}\right) u}\right] }}\cos \left[ {\frac{\pi }{n}\left( {j-\frac{1}{2}}\right) w}\right] ,1\le u,w\le n. \end{aligned}$$

(A.8)

Other features include skewness $v_{15}$, kurtosis $v_{16}$, median value $v_{17}$, trimmed mean value excluding the outliers $v_{18}$, range of the values $v_{19}$ and interquartile range of the values $v_{20}$.

B Appendix

Inspired by [15, 16], a new two-class Adaboost algorithm is developed here based on additive logistic regression and empirical loss:

$$\begin{aligned} \mathop {\min }\limits _{\mathbf{F}\left( \mathbf{x}\right) }\;\sum \limits _{i=1}^{N}{\exp \left( -\frac{1}{2}{} \mathbf{y}_{i}^{T}{} \mathbf{F}\left( \mathbf{X}_{i}\right) \right) } \end{aligned}$$

(B.1)

with the symmetric constraint $\small F_{1}\left( \mathbf{X}_{i}\right) +F_{2}\left( \mathbf{X}_{i}\right) =0$, where $\small \mathbf{F}\left( \mathbf{X}\right)$ is a continuous-valued Adaboost classifier. We consider $\small \mathbf{F}\left( \mathbf{X}\right)$ that has the following forward stage additive modelling form:

$$\begin{aligned} \mathbf{F}\left( \mathbf{X}\right) =\sum \limits _{l=1}^{L}{\beta ^{\left( l\right) }g^{\left( l\right) }\left( \mathbf{X}\right) } \end{aligned}$$

(B.2)

where $\small \beta ^{\left( l\right) }\in \mathfrak {R}$ are weighting coefficients and $\small g^{\left( l\right) }\left( \mathbf{X}\right)$ are basis functions satisfying the symmetric constraint $\small g_{1}\left( \mathbf{X}\right) +g_{2}\left( \mathbf{X}\right) =0$. The additive model is $\small \mathbf{F}^{\left( l\right) }\left( \mathbf{X}\right) =\mathbf{F}^{\left( l-1\right) }\left( \mathbf{X}\right) +\beta ^{\left( l\right) }{} \mathbf{g}^{\left( l\right) }\left( \mathbf{X}\right)$ thus the optimization can be represented as

$$\begin{aligned} \left( \beta ^{\left( l\right) },\mathbf{g}^{\left( l\right) }\right)= & {} \arg \;\mathop {\min }\limits _{\beta ,\mathbf{g}}\sum \limits _{i=1}^{N}{\exp \left( -\frac{1}{2}{} \mathbf{y}_{i}^{T}\left( \mathbf{F}^{\left( l-1\right) }\left( \mathbf{X}\right) +\beta ^{\left( l\right) }{} \mathbf{g}^{\left( l\right) }\left( \mathbf{X}\right) \right) \right) }\nonumber \\= & {} \arg \;\mathop {\min }\limits _{\beta ,\mathbf{g}}\sum \limits _{i=1}^{N}{w_{i}\exp \left( -\frac{1}{2}{} \mathbf{y}_{i}^{T}\beta ^{\left( l\right) }{} \mathbf{g}^{\left( l\right) }\left( \mathbf{X}\right) \right) } \end{aligned}$$

(B.3)

where $\small w_{i}=\exp \left( -\frac{1}{2}{} \mathbf{y}_{i}^{T}{} \mathbf{F}^{\left( l-1\right) }\left( \mathbf{X}\right) \right)$ is the current sample weight. In this work, basis function $\mathbf{g}\left( \mathbf{X}\right)$ is defined as:

$$\begin{aligned} g_{k}\left( \mathbf{X}\right) =\frac{1}{A}\cdot \frac{2\exp \left( -d_{k}\left( \mathbf{X}\right) \right) }{1+\exp \left( -d_{k}\left( \mathbf{X}\right) \right) }-\frac{1}{2} \end{aligned}$$

(B.4)

where A is a normalization factor and $\small d_{k}\left( \mathbf{X}\right)$ is defined as the l1-norm distance between the feature vector $\small \mathbf{X}$ and the k-th class of the detection vectors as follow

$$\begin{aligned} d_{k}\left( \mathbf{X}\right) =\frac{1}{\left| \mathbf{Y}_{k}\right| }\sum \limits _{\mathbf{V}\in \mathbf{Y}_{k}}\left\| \mathbf{X}-\mathbf{V}\right\| ,\,k=0,1 \end{aligned}$$

(B.5)

where $\small \mathbf{V}$ is a feature vector, $\small \mathbf{Y}_{k}$ is the set of the feature vectors in the k-th class, and $\left| \cdot \right|$ denotes the cardinality of a set. A small $\small d_{k}\left( \mathbf{X}\right)$ means $\small \mathbf {X}$ is close to the k-th class, and $\small g_{k}\left( \mathbf{X}\right)$ will be large.