## 3 The Proposed Octree-Hiding Scheme
 \tag{1}\]  
where:  
\(N_{1}\) represents the length of the signal (in samples);  
\(N_{2}\) denotes the width of the signal (in samples);  
\(N_{3}\) is the depth of the signal (in samples);  
\(k_{1}\) ranges from 0 to \(N_{1}-1\);  
\(k_{2}\) ranges from 0 to \(N_{2}-1\);  
\(k_{3}\) ranges from 0 to \(N_{3}-1\);  
\(Y_{k_{1},k_{2},k_{3}}\) denotes the DCT coefficient at \((k_{1},k_{2},k_{3})\);  
Figure 1: **(a)** Applying the 2D quad-tree segmentation method on an image when setting a minimum block size of \(32\times 32\). **(b)** The 3D Octree segmentation method. Recent research has demonstrated that the magnitude of the DCT coefficients is less critical to the signal than the phase [26]. Consequently, the proposed method focuses on separating the magnitude from the phase, hiding information solely in the DCT magnitude of the host image, as illustrated in step 3 in Figure 2.  
Next, the quantization process begins to identify the insignificant DCT coefficients that will be used to embed the secret data. Figure 3 illustrates the distribution of significant DCT coefficients within a 3D-DCT matrix [27]. To model the shaded volume depicted in Figure 3, we first consider the \(x-y\) plane where \(t=0\). The shaded area in the \(x-y\) plane can be approximated as a hyperbola, such that we have the following:  
\[xy\leq C_{1} \tag{2}\]  
Similarly, the shaded areas in the \(x-t\) and \(y-t\) planes can be represented, respectively, by the following:  
\[xt\leq C_{2} \tag{3}\]  
\[yt\leq C_{3} \tag{4}\]  
where \(C_{1}\), \(C_{2}\), and \(C_{3}\) are constants. We combine the following equations: (2)-(4):  
\[(xyt)^{2}\leq C_{1}C_{2}C_{3} \tag{5}\]  
Thus, the important coefficients \(g(x,y,t)\) can be allocated by the following:  
\[g(x,y,t)=xyt\leq\sqrt{C_{1}C_{2}C_{3}} \tag{6}\]  
Since \(C_{1}\), \(C_{2}\), and \(C_{3}\) are constants, the significant coefficients \(g(x,y,t)\) can be founded by a single constant \(C^{\prime}\):  
\[g(x,y,t)=xyt\leq C^{\prime} \tag{7}\]  
The embedding process is carried out within the insignificant DCT coefficients of each cube. Given that these coefficients have relatively low values, the system, as depicted in step 6 in Figure 2, adjusts the amplitude of the secret signal by applying a scaling factor \(k\). This scaling aligns the secret signal's amplitude with the range of the insignificant DCT coefficients. This adjustment can be described by the following equation:  
\[\text{Rescaled Signal}=k\cdot\text{Secret Signal} \tag{8}\]  
Figure 2: General schematic representation of the proposed Octree steganography technique. where \(k\) is the scaling factor used to modify the amplitude of the secret signal so that it fits within the range of the insignificant DCT coefficients. Figure 2.0  
where \(D^{\prime}\) is the secret vector after rescaling and S denote the original secret data. Once the secret signal is rescaled using an appropriate scaling factor, the next step involves substituting the insignificant DCT coefficients with the rescaled secret signal (as shown in step 7 in Figure 2). Since the embedding is carried out only in the magnitude of the DCT coefficients, the original phase spectrum must be preserved and combined with the altered magnitude DCT to fully reconstruct the DCT structure, ensuring that the secret data is embedded correctly. The next step involves applying the inverse DCT to convert the cubes back into the spatial domain. It is important to note that these embedding procedures are performed on each cube segmented by the Octree technique. Once the secret data are embedded in all cubes and their spatial representations are obtained, the embedder assembles all the cubes back into their original positions within the video segment, resulting in the final carrier video that incorporates the hidden data.