## 5 Experimental Results
### Chi-Square Attack and RS Steganalysis  
In steganography, the main goal for a stego image is statistically undetectable [21] (p. 52). To test whether the embedded images are detectable, chi-square attack [23], RS steganalysis [24], and embedding efficiency [21] were conducted. Chi-square attack and RS steganalysis are discussed in this section, and embedding efficiency in Section 5.3. Each of chi-square attack and RS steganalysis has two experiments, one embedded \(15,000\times k\) bits (\(k=1,\ 2,\ 3\)) for [PERSON] et al. 's and the proposed methods. The other embedded \(15,000\times k\) bits (\(k=3\)) for [PERSON] et al. 's and the proposed methods. Chi-square attack employs [PERSON]'s chi-square test to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of an image, and it can detect whether a palette image is embedded by messages. The details of applying the chi-square attack to steganography methods are described as follows: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Block Size** & \multicolumn{2}{c}{**2 × 2**} & \multicolumn{2}{c}{**1 × 3**} \\ \hline
**Capacity** & \multicolumn{2}{c}{**12,288 × \(k\)**} & \multicolumn{2}{c}{**16,384 × \(k\)**} \\ \hline  & [16] & **Proposed Method** & [17] & **Proposed Method** \\ \hline Aerial & 33.07 & 38.35 & 29.22 & 36.91 \\ Airplane & 37.05 & 42.62 & 34.96 & 40.66 \\ Balloon & 35.82 & 42.13 & 33.78 & 40.52 \\ Earth & 38.94 & 45.60 & 39.14 & 43.99 \\ Girl & 33.53 & 38.89 & 30.55 & 37.33 \\ Lena & 34.34 & 40.30 & 32.52 & 38.61 \\ Parrots & 31.60 & 36.52 & 29.86 & 34.97 \\ Pepper & 32.72 & 37.58 & 30.72 & 36.05 \\ Mandrill & 30.81 & 35.91 & 27.00 & 34.25 \\ Couple & 35.48 & 41.76 & 32.75 & 40.16 \\ Sailboat & 33.92 & 39.52 & 31.18 & 37.99 \\ Milkdrop & 35.58 & 40.99 & 33.50 & 39.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: PSNRs of [PERSON] et al.’s [16], [PERSON] et al.’s [17], and the proposed methods with \(k=3\). * Let the palette indices of a palette image be divided into \(K\) categories, each category contains a pair of indices (\(2i\), \(2i+1\)), \(i=0\),..., \(K-1\). Let \(E_{i}\) and \(O_{i}\) be the theoretically expected and observed frequencies of pixels with index \(2i\) after embedding messages, respectively. Then \[E_{i}=\frac{\text{The number of pixels with color index }2i\text{ or }2i+1}{2},\] (17) \[O_{i}=\text{The number of pixels with color index }2i.\] (18)
* The \(\chi^{2}\) statistic is given as \[\chi^{2}_{K-1}=\sum\
olimits_{i=0}^{K-1}\frac{\left(O_{i}-E_{i}\right)^{2}}{E _{i}}\text{ with }K-1\text{ degrees of freedom}.\] (19)
* Let \(p_{v}\) represent the embedding probability, \(p_{v}\) can be calculated by the following equation: \[p_{v}=1-\frac{1}{2^{\frac{K-1}{2}}\Gamma\left(\frac{K-1}{2}\right)}\int_{0}^{ \chi^{2}_{K-1}}e^{-\frac{\chi}{2}}x^{\frac{K-1}{2}-1}\mathrm{d}x.\] (20)  
In chi-square attack, the results for three methods are \(p_{v}=0\); this means that the three methods can resist chi-square attack. There are two reasons: (1) These three methods are block-based; the embedding capacities are limited and lower than 1 bpp; this will make chi-square attack fail [21]; (2) chi-square attack is used to detect LSB-based methods, but these three methods are not LSB-based methods. Chi-square attack just uses sample counts and neglects spatial correlations among pixels in the stego image. [PERSON] et al. [24] introduced RS steganalysis for detection of LSB embedding that utilizes sensitive dual statistics derived from spatial correlations in image. In RS steganalysis, for a given image \(I\), through a given local mask, two flipping functions, and a discrimination function, all pixels of the image can be classified into three groups: Regular, singular, and unchanged. Given a non-negative mask \(m\), we can obtain the relative frequency of the regular group denoted as \(R_{+}\) and the relative frequency of the singular group denoted as \(S_{+}\).