# A Novel Adaptive Image Data Hiding and Encryption Scheme Using Constructive Image Abstraction
## 3 Experimental Results and Analysis
### Analysis of Security
#### 3.2.2 Analysis of Histogram  
An image's histogram provides valuable statistical information, making it a prime target for statistical attacks by malicious eavesdroppers. A well-designed encryption algorithm should hide statistical information and prevent them from being revealed. Figure 6 displays histograms of the original, intermediate abstracted, abstracted stego, and ciphered images. Our encryption algorithm makes the histogram of abstracted stego uniform and completely different from that of the plaintext in all the channels, making it impossible to reveal any meaningful information. The results of the histogram analysis indicate that the proposed method is effective in preventing statistical attacks.  
The uniformity of a histogram can be quantitatively evaluated using the Variance of Histogram (VOH), which can be computed using Equation (9). The equation uses  
Figure 5: The visual perception of the HDR test image “507” (**a–f**) and LDR test image “NCHU Main Gate” (**g–j**). (**a**) Directly displaying the original HDR image without any tone-mapping (TM) operation. (**b**) The tone-mapped HDR image. (**c**) The abstracted stego image concealed with around 8.487 million secret bits. (**d**) The edges of the abstracted stego image. (**e**) The ciphered image. (**f**) The deciphered TM image. (**g–j**) The corresponding results of LDR test image “NCHU Main Gate,” where the abstracted stego image (**h**) has concealed more than 5.66 million secret bits.  
Figure 6: Histogram analysis of the encryption scheme on images, “507,” (top row) and the “NCHU Main Gate,” (bottom row). (**a**) The histograms of the original images. (**b**) The histograms of the intermediate abstracted images. (**c**) The histograms of the abstracted stego images. (**d**) The histograms of the encrypted stego images.  
\(Z=\{z_{0},z_{1},\ldots,z_{255}\}\) to represent the number of counts in each bin, while \(\mu\) is the mean count of all bins. A smaller VOH indicates a more uniform histogram.  
\[VOH(Z)=\sum_{i=0}^{255}E\Big{[}(z_{i}-\mu)^{2}\Big{]} \tag{9}\]  
We use the Chi-square (\(\chi^{2}\)) test, in addition to VOH, to determine any statistically significant difference between the expected and observed frequencies of the histogram. If considering the present scenario, the degree of freedom is 255. Assuming a significance level (\(\alpha\)) of 0.05, the threshold would be 293.25. If the calculated \(\chi^{2}\) value from the histogram is less than the given threshold, it would lead us to accept the null hypothesis, which indicates that the histogram of the stego abstracted image is statistically indistinguishable from a uniform distribution.  
In Table 4, we have presented the values of VOH and \(\chi^{2}\) for two sample stego abstracted images, namely \"507\" and \"NCHU Main Gate\". The table presents these values in both plain and ciphered status, offering a comparison of VOH and \(\chi^{2}\) values before and after encryption. In particular, VOH values drop to less than \(4\times 10^{-3}\%\) of the plaintext, and \(\chi^{2}\) values are less than the threshold value (293.25) in all the channels. Moreover, we have performed \(\chi^{2}\) test for all 10 test images to gain more insights, and the results are plotted in Figure 7. From the graph, it is evident that the \(\chi^{2}\) values of all 10 test images in every channel are below the dashed-line threshold.  
Based on the findings displayed in Table 4 and Figure 7, we can conclude that our algorithm significantly reduces VOH and \(\chi^{2}\) values during the encryption process. In addition, the output images generated by our encryption algorithm passed the statistical hypothesis test. Our scheme ensures that the histogram of the encrypted image in all channels has a uniform distribution, thus providing a high level of security for the ciphered images.