# Chaotic Henon-Logistic Map Integration: A Powerful Approach for Safeguarding Digital Images
## 3 The Proposed Scheme
\[PSNR=10\log_{10}\frac{[MAX_{I}]^{2}}{MSE_{t}} \tag{6}\]

where \(MAX_{I}\) represents the maximum signal value in the original reference image, which is \"known to be good\". The UACI (Unified Average Changing Intensity) metric is essential for assessing the effectiveness of image encryption algorithms, specifically in altering pixel intensities. Equation (7) presents the formula for UACI, where \(M\times N\) denotes the dimensions of the images (\(I\), \(K\)) being compared. Additionally, the Number of Pixels Change Rate (NFCR) is another important measure, with its mathematical expression provided in Equation (8) [6]:

\[UACI=\frac{1}{M\times N}\left[\sum_{i,j}\left[\frac{|I(i,j)-K(i,j)|}{255} \right]\right]\times 100\% \tag{7}\]

\[NPCR=\frac{\sum_{i,j}D(i,j)}{M\times N}\times 100\% \tag{8}\]

where

\[D(i,j)= \begin{cases}0&\text{if }I(i,j)=K(i,j)\\ 1&\text{otherwise}\end{cases} \]

The Unified Average Changing Intensity (UACI) metric is a crucial instrument in image encryption. It evaluates the effectiveness of encryption algorithms and improves techniques for maximum security and attack resistance. UACI quantifies the average variation in pixel intensity between an original image and its encrypted counterpart, providing valuable insights into the uniform distribution of pixel values within the encrypted image. A more secure encryption result is produced when the encryption technology has effectively distributed pixel intensities, as indicated by a higher UACI score. A well-known statistic in image processing for assessing the effectiveness of image encryption methods is the Structural Similarity Index Measure (SSIM), which compares the structural similarity of the original and encrypted images. Unlike metrics such as the MSE or PSNR, which primarily focus on pixel-level discrepancies, the SSIM incorporatesfactors that align with human visual perception, including contrast, structural information, and luminance. An efficient technique for image encryption should provide encrypted images that are visually different from the original and devoid of any obvious patterns or connections. The SSIM is a quantifiable measure of encryption performance, assessing the structural similarity between the original and encrypted images. A low SSIM indicates that the encryption method has effectively obscured the structural details of the original image, thereby thwarting potential attackers from reconstructing the original image or extracting valuable information. As a result, the SSIM is an essential part of creating and assessing image encryption techniques, helping practitioners and academics optimize algorithms to improve security and protect privacy. Equation (9) expresses the SSIM calculation:

\[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}^{2}\mu_{ y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})} \tag{9}\]

In this context, \(\mu_{x}\) and \(\mu_{y}\) represent the pixel sample means of images \(x\) and \(y\), respectively. Similarly, \(\sigma_{x}^{2}\) and \(\sigma_{x}^{2}\) denote the variances of images \(x\) and \(y\), while \(\sigma_{xy}\) refers to the covariance between the two images. The terms \(c_{1}=(k_{1}L)^{2}\) and \(c_{2}=(k_{2}L)^{2}\) are constants introduced to prevent division by a very small denominator. Here, \(L\) represents the dynamic range of pixel values, calculated as \(2^{\text{bits per pixel}}-1\). By default, the constants are set to \(k_{1}=0.01\) and \(k_{2}=0.03\)..

Table 2 presents the two critical quality metrics, the Mean Square Error (MSE) and peak signal-to-noise ratio (PSNR), used to assess the performance of the encryption algorithm. A greater disparity between the original and encrypted images indicates improved encryption performance, denoted by a lower MSE value. This disparity suggests that the encrypted image is significantly different from the original, making decryption more difficult for unauthorized users. Conversely, a lower PSNR value denotes increased noise or distortion in the encrypted image compared to the original, unencrypted image.