A Novel Feature Selection Model for JPEG Image Steganalysis

Abstract

Image steganalysis is a very important research topic in the field of information security. The existing feature based image steganalysis methods have achieved the appealing performance. The performance of them greatly depends on the quality of the hand-crafted steganalysis feature vectors, such as Cartesian Calibration PEV (CC-PEV), DCT Residuals (DCTR), and so on. However, these feature vectors may contain some redundant elements that will reduce the discrimination power and increase the computation cost. In this paper, a novel feature selection model is proposed for JPEG image steganalysis. Specifically, the proposed model imposes an $l_{2,1}$-structural constraint on the projection matrix for feature selection. Further, to make the model insensitive to noises and outliers, a capped $l_2$-norm based loss function is adopted. Moreover, a graph-based manifold regularization term which exploits the intrinsic local geometric structure of the data is added into the objective function to select the effective feature elements. Finally, an alternately iterative optimization algorithm with proven convergence is given to solve the proposed model. The extensive experiments on three state-of-the-art JPEG steganographic algorithms with 0.1 and 0.2 embedding rates and two JPEG quality factors show that the proposed model can effectively remove some irrelevant and redundant elements meanwhile retaining high detection accuracy.

Appendix

Proof of Theorem 1. According to the step $\mathbf {1}$ of Algorithm 1 in the t-th iteration and Eq. (11), we have

$$\begin{aligned} W_{(t+1)}&=\hbox {arg}\min _W \sum _{i=1}^n u_i\Vert W^T \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T X) L (W^T X)^T]+\lambda _2\Vert W\Vert _{2,1}\nonumber \\&=\hbox {arg}\min _W \sum _{i=1}^n u_i\Vert W^T \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T X) L (W^T X)^T]\nonumber \\&\quad +\,\lambda _2tr(W^T D_{(t)} W) \end{aligned}$$

(17)

which indicates that

$$\begin{aligned}&\sum _{i=1}^n u_i\Vert W^T_{(t+1)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t+1)} X)L (W^T_{(t+1)} X)^T ]\nonumber \\&\quad +\,\lambda _2tr(W^T_{(t+1)} D_{(t)} W_{(t+1)})\nonumber \\&\le \sum _{i=1}^n u_i\Vert W^T_{(t)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t)} X) L (W^T_{(t)} X)^T]\nonumber \\&\quad +\,\lambda _2tr(W^T_{(t)} D_{(t)} W_{(t)}) \end{aligned}$$

(18)

According to Eq. (12), the following equations hold:

$$\begin{aligned} tr(W^T_{(t+1)} D_{(t)} W_{(t+1)})&=\sum _{i=1}^d\frac{\Vert \varvec{w}^i_{(t+1)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2}\end{aligned}$$

(19)

$$\begin{aligned} tr(W^T_{(t)} D_{(t)} W_{(t)})&=\sum _{i=1}^d\frac{\Vert \varvec{w}^i_{(t)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2} \end{aligned}$$

(20)

Thus, Eq. (18) can be transformed into the following form:

$$\begin{aligned}&\sum _{i=1}^n u_i\Vert W^T_{(t+1)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t+1)} X) L (W^T_{(t+1)} X)^T]+\lambda _2 \sum _{i=1}^d\frac{\Vert \varvec{w}^i_{(t+1)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2}\nonumber \\&\le \sum _{i=1}^n u_i\Vert W^T_{(t)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t)} X) L (W^T_{(t)} X)^T]+\lambda _2\sum _{i=1}^d\frac{\Vert \varvec{w}^i_{(t)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2} \end{aligned}$$

(21)

Further, the above inequality can be rewritten as

$$\begin{aligned}&\sum _{i=1}^n u_i\Vert W^T_{(t+1)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t+1)} X) L (W^T_{(t+1)} X)^T]\nonumber \\&\quad +\,\lambda _2 \sum _{i=1}^d\left( \frac{\Vert \varvec{w}^i_{(t+1)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2}+\Vert \varvec{w}^i_{(t+1)}\Vert _2-\Vert \varvec{w}^i_{(t+1)}\Vert _2\right) \nonumber \\&\le \sum _{i=1}^n u_i\Vert W^T_{(t)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t)} X) L (W^T_{(t)} X)^T]\nonumber \\&\quad +\,\lambda _2\sum _{i=1}^d\left( \frac{\Vert \varvec{w}^i_{(t)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2}+\Vert \varvec{w}^i_{(t)}\Vert _2-\Vert \varvec{w}^i_{(t)}\Vert _2\right) \end{aligned}$$

(22)

By replacing a and b in Lemma 1 with $\Vert \varvec{w}^i_{(t+1)}\Vert _2^2$ and $\Vert \varvec{w}^i_{(t)}\Vert _2^2$, we get

$$\begin{aligned}&\Vert \varvec{w}^i_{(t+1)}\Vert _2 - \frac{\Vert \varvec{w}^i_{(t+1)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2}\le \Vert \varvec{w}^i_{(t)}\Vert _2 - \frac{\Vert \varvec{w}^i_{(t)}\Vert _2^2}{2\Vert \varvec{w}^i_{(t)}\Vert _2} \end{aligned}$$

(23)

By adding Eqs. (22) and (23) on both sides (note that Eq. (23) is repeated for $1\le i \le d$), we have

$$\begin{aligned}&\sum _{i=1}^n u_i\Vert W^T_{(t+1)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t+1)} X) L (W^T_{(t+1)} X)^T]+\lambda _2 \sum _{i=1}^d\Vert \varvec{w}^i_{(t+1)}\Vert _2\nonumber \\&\le \sum _{i=1}^n u_i\Vert W^T_{(t)} \varvec{x}_i-\varvec{y}_i\Vert _2 + \lambda _1tr[(W^T_{(t)} X) L (W^T_{(t)} X)^T]+\lambda _2\sum _{i=1}^d\Vert \varvec{w}^i_{(t)}\Vert _2 \end{aligned}$$

(24)

That is to say,

$$\begin{aligned} Obj(t+1)\le Obj(t) \end{aligned}$$

(25)

Since the values of objective function (7) decrease monotonously and are greater than zeros, so Algorithm 1 is convergent. Theorem 1 is proven.