## 3 The Proposed Octree-Hiding Scheme  
The Pixogram segmentation technique outlined in Section 2 clearly utilizes temporal redundancy. However, it overlooks another source of redundancy: spatial redundancy. Given that a video segment is essentially a sequence of static images (frames), there is a potential for significant redundant data within each frame. To address this, the Octree segmentation method is proposed to leverage both spatial and temporal redundancies. The Octree segmentation technique is a 3D extension of the quad-tree image segmentation method. While the quad-tree method focuses on the spatial homogeneity of pixel intensities to divide an image into non-overlapping, homogeneous blocks, the Octree method extends this concept to three dimensions. It segments a 3D volume into non-overlapping, homogeneous blocks based on spatial correlations within the pixel intensities. In [23; 24], audio steganography techniques are proposed that share a similar rationale to the technique discussed in this paper. These methods segment the cover signal into correlated segments and exploit the strong compaction property of the discrete cosine transform (DCT) within these segments. However, the segmentation in these audio-based techniques was performed in the 1D domain, whereas the proposed technique in this paper employs a 3D segmentation approach tailored for video steganography. This distinction underscores the progression of segmentation strategies from audio to video, illustrating the versatility of using DCT compaction properties in different domains for robust steganographic embedding. The quad-tree process begins by dividing the image into four equal squares. Each square is then evaluated to determine if it meets the homogeneity criteria. If the block does not meet the criteria, it is further subdivided into four smaller squares. This process of division is repeated iteratively until all blocks satisfy the homogeneity requirement. A straightforward homogeneity criterion involves checking the difference between the highest and lowest gray-level values within a block. If this difference is below a predefined threshold \(T_{quad}\), the block is considered homogeneous. The quad-tree algorithm can be configured with both a minimum and maximum block size. This ensures that the algorithm does not create blocks smaller than the minimum specified size (e.g., \(8\times 8\)) or larger than the maximum specified size (e.g., \(128\times 128\)). Even if larger blocks meet the homogeneity threshold, they will be split to adhere to the maximum block size constraint. By the end of the quad-tree process, the image will be segmented into variable-sized blocks that reflect the pixel correlations. Figure 1a illustrates the application of the quad-tree segmentation technique. The Octree method functions in a similar manner. As shown in Figure 1b, the Octree segmentation technique extends the quad-tree approach to 3D signals, such as video segments. Instead of dividing a 2D image into correlated blocks, the Octree segments a 3D video segment into correlated cubes. This segmentation process follows the same principles as the quad-tree technique but operates in three dimensions. A homogeneous cube thus captures both spatial and temporal redundancies. As depicted in Figure 2, the process begins with selecting a homogeneous block obtained through the Octree segmentation technique. Following this, the 3D discrete cosine transform (3D-DCT) is applied to the homogeneous block. The 3D-DCT is computed using the formula given in [25]:  
\[ \begin{split} Y_{k_{1},k_{2},k_{3}}=\sum_{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\sum_{n_{3}=0}^{N_{3}-1}y_{(n_{1},n_{2},n_{3})}\cos\biggl{[} \frac{\pi}{N_{1}}\left(n_{1}+\frac{1}{2}\right)k_{1}\biggr{]}\\ \cos\biggl{[}\frac{\pi}{N_{2}}\left(n_{2}+\frac{1}{2}\right)k_{2 }\biggr{]}\cos\biggl{[}\frac{\pi}{N_{3}}\left(n_{3}+\frac{1}{2}\right)k_{3} \biggr{]}\end{split}