## 2 Proposed Hybrid Framework for Underwater Image Enhancement
### Proposed Multi-Scale Detail Enhancement Unit (MSDE)  
The weak structural details, low contrast and blurred edges in underwater images make the detail enhancement phase a critical step in the image enhancement process. In this study, the proposed detail enhancement unit consists of two consecutive sub-steps: multi-scale detail enhancement and Latent Low-Rank Representation (LatLRR)-based structural decomposition. Using this module, the structural integrity is preserved, details are inherently emphasized, and artifacts are effectively attenuated. Figure 1: Flow chart of the proposed hybrid framework. **Multi-Scale detail enhancement:** In order to enhance the structural details, the Y channel is smoothed with Gaussian filters with different standard deviation values; then, for each scale, detail layers are created by taking the difference between the original image and its smoothed version. These detail layers contain edge and structure information of the scene at different scales. Small sigma values reveal thinner edges, while large sigma values expose wider structures. The obtained multi-scale layers are combined with certain weights to form a single detail component. This detail component is added to the input Y channel with a certain gain factor in the last stage to obtain the strengthened Y channel. Hence, this ensures that both thin edges and global details become more distinct at the same time. The mathematical modeling of the stages of the method is as follows:  
The input image, represented as a normalized grayscale (Y-channel) component, is defined as follows:  
\[I\in[0,\;1]^{H\times W} \tag{1}\]  
To extract structural information at multiple spatial resolutions, the image is convolved with Gaussian kernels of varying standard deviations:  
\[G_{\sigma_{i}}=I\star\mathcal{G}(\sigma_{i}),\qquad i=1,2,\ldots,N \tag{2}\]  
Here, \(\mathcal{G}(\sigma_{i})\) denotes a 2D Gaussian filter with standard deviation \((\sigma_{i}),\;\text{and}\;(*)\) represents the convolution operation. For each scale, a detail layer is computed by subtracting the smoothed image from the original input:  
\[D_{\sigma_{i}}=I-G_{\sigma_{i}} \tag{3}\]  
These multi-scale detail maps are aggregated using scale-specific weights to form a unified detail component:  
\[D=\sum_{i=1}^{N}w_{i}\cdot D_{\sigma_{i}} \tag{4}\]  
Finally, the enhanced image is obtained by adding the scaled detail component to the original image:  
\[I_{enh}=I+\alpha\cdot D \tag{5}\]  
where \((\alpha)\) is a gain factor controlling the strength of the detail enhancement. _Latent Low-Rank Representation (LatLRR):_ Although detail enhancement processes highlight structural information in the image, amplification of spurious effects such as noise and excessive emphasis may also occur. Therefore, in the proposed unit, the Latent Low-Rank Representation (LatLRR)-based structural decomposition method was preferred in order to reduce these negativities that may arise from using the detail contribution in its raw form and to extract only meaningful structural components and transfer them to the fusion stage. LatLRR enables us to represent the basic information components in the image in a more dense and separate way by decomposing the input image into three separate subspaces, namely, principal features, salient features and sparse noise [24]. In the method, only the principal features' subspace is included in the fusion process; thus, spurious patterns or noises that may occur in the detail enhancement process are largely removed from the image. This selective decomposition approach is a critical step that supports the detail enrichment and deep-learning-based fusion process to produce more consistent results. The LatLRR method decomposes an image matrix into three components, modeled as follows:  
\[I=I_{Z}+I_{L}+E \tag{6}\]Here, I represents the input image, \(I_{Z}\) represents the principal features' subspace, \(I_{L}\) represents salient features, and E represents sparse noise.