Raw reversibility of information hiding on the basis of VQ systems

Abstract

Up to now, the VQ-based steganographic methods are all lossy approaches since they cannot recover the cover images. Prior studies on reversibility in VQ-based steganography only show the ability to recover the quantized cover images. This paper presents a new two-phase steganographic protocol of genuine VQ-based reversibility, referring to the ability of using a VQ codebook to recover a target cover image and to retrieve all hidden data from the stego-images. The protocol consists of two phases. The Phase 1 generates a stego-image of excellent visual quality; the Phase 2 provides high visual quality and high embedding capacity of the stego-image. Both the stego-images used in the two phases can pass Chi-square test. To evaluate the performance of our method, we have conducted several experiments and compared Phase 1 and Phase 2 of the proposed protocol with two prior studies in reversible steganography based on VQ. The results show that Phase 2 achieves significant improvement on both the embedding capacity and the visual quality of the stego-images. On the other hand, Phase 1 achieves the best visual quality and the lowest embedding capacity of the stego-images. Together with the two phases of the proposed protocol, we can achieve genuine recovery of the cover image of Phase 1. This property cannot be achieved by any combination of prior studies since they all provide low embedding capacity of the stego-images and high difference between the cover image and the stego-image.

Appendices

Appendices

In this section, we describe two VQ-based (1, 1) reversibility methods, Chang et al.’s method [26] and Huang et al.’s method [27].

Appendix 1: Chang et al.’s method [26]

In 2006, Chang et al. [26] proposed a SMVQ reversible data hiding scheme which embeds a secret bit into each SMVQ-coded block $H_{i}$. The output format is a regular image, not index table or codestream. The key point of Chang et al.’s reversible method is used two codewords to extract the secret data correctly and also reverse the original SMVQ index. If the to-be-embedded secret bit is 0, the codeword $k_{a}$ found by the SMVQ approach is used to replace $H_{i}$. Otherwise, the to-be-embedded secret bit is 1 and $H_{i}$ is replaced with $\left\lfloor {\frac{2\times k_a +1\times k_b }{3}} \right\rfloor$, where codeword $k_{b}$ is the closest codeword of $k_{a}$ in the sub-codebook. The detailed embedding algorithm is shown as follows:

• Input: Cover image H, secret data B, codebook C.

• Output: Stego-image $S'.$

• Step 1: Partition H into n non-overlapping $m \times m$ blocks: $H=H_{0}, H_{1}, {\ldots }, H_{n-1}$.

• Step 2: Blocks in the first row or the first column are encoded by VQ and do not participate in the embedding phase.

• Step 3: From left to right and from top to down, process each remained block $H_{i }$ by executing Step 4 $\sim$ Step 6.

• Step 4: The upper block and left block of $H_{i}$ are used to generate sub-codebook $K = (k_{0}, k_{1}, {\ldots }, k_{N-1})$ from C, where $k_{j}$ is the jth codeword and $j = 0, 1, {\ldots }, N-1$. Find out the closest codeword $k_{a}$ of $H_{i }$ from sub-codebook K.

• Step 5: If to-be-embedded secret bit b is 0, $S_{i}'$ is set to $k_{a}$.

• Step 6: If to-be-embedded secret bit b is 1, $S_{i}'$ is set to $\left\lfloor {\frac{2\times k_a +1\times k_b }{3}} \right\rfloor$, where codeword $k_{b}$ is the closest to $k_{a}$ in sub-codebook K.

Figure 6 shows the flowchart of processing a non-seed block X in Chang et al.’s method, where the gray blocks of the cover image are seed blocks and are encoded by VQ.

Fig. 6

Flowchart of Chang et al.’s method

Appendix 2: Huang et al.’s method [27]

In 2012, Huang et al. proposed a new SMVQ reversible data hiding scheme with images as outputs is proposed by improving Chang et al.’s scheme [26]. Compared with Chang et al.’s scheme, Huang et al. adjusted the direction from $\overline{k_a k_b }$ to $\overline{k_a H_i }$. The improvement of Huang et al.’s method is to raise the image quality in PSNR and also keep the embedding capacity. Figure 7 shows the concept with Huang et al.’s method. Figure 7a shows the concept of the proposed method, the codeword $k_{a}$ is the closest codeword to the block $H_{i}$, and the codeword $k_{b}$ is the closest codeword to the codeword $k_{a}$. If codeword $k_{a}$ is as a center, two spheres are created. The radius of the external sphere is equal to the distance between $k_{a}$ and $k_{b}$. The radius of the internal sphere is half of the radius of the external sphere. Codeword $k_{a}$ is the closest to all of the points in the internal sphere. Figure 7b and c shows the methods to generate the stego-images in Chang et al.’s method and Huang et al.’s method. The stego-image of Huang et al.’s method will closer to cover image H than Chang et al.’s method.

Fig. 7

The relationships between three key elements, $H_{i}$, $k_{a}$, and $k_{b}$, in different methods [26, 27]. a Three key elements, b Chang et al.’s method [26], c Huang et al.’s method [27]