## 2 Proposed Hybrid Framework for Underwater Image Enhancement
### Proposed Multi-Scale Detail Enhancement Unit (MSDE)
This decomposition process is performed within the framework of the following optimization problem:  
\[min_{Z,L,E}\left\|Z^{\star}\right\|+\left\|L^{\star}\right\|+\lambda\cdot\left\| E_{1}\right\| \tag{7}\]  
In this formulation, \(\left\|\cdot\right\|^{\star}\) represents the nuclear norm, which is the sum of a matrix's singular values and serves as a convex relaxation of the rank function. Its purpose is to encourage the primary components Z and L to be low-rank. The \(\left\|\cdot\right\|_{1}\) denotes the l1-norm, which is applied to the error term E to model it as a sparse matrix, effectively isolating noise and gross errors. The parameter \(\lambda>0\) is a balancing coefficient that controls the trade-off between the low-rank structure and the sparse error [24]. _Unsupervised Deep-Learning-Based Fusion (MU-Fusion)_: MU-Fusion represents one of the most critical stages of our proposed method. It is an unsupervised, deep-learning-based image fusion model that optimally combines the differently enhanced Y channels. In unsupervised fusion models, the training process continues by focusing only on the source image without transferring in-process outputs to subsequent epochs. However, in-process outputs may contain some clues for important information such as pixel intensity distribution, gradient information and structural similarity for image fusion. Owing to its memory-based architecture, MU-Fusion proposes a memory unit that allows the in-process outputs obtained in previous learning steps to affect the current output. In this way, not only the current inputs but also the in-process outputs contribute to learning; the network exhibits a more consistent and generalizable fusion performance [26]. The learning process of MUFusion is defined by a dual-component loss function that takes both content loss and memory loss into account:  
\[L_{total}=p_{1}\cdot L_{content}+p_{2}\cdot L_{memory} \tag{8}\]  
\[S_{A}=\frac{SSIM(O,I_{1})+SSIM(O,I_{2})}{2} \tag{9}\]  
\[I\in[0,\;1]^{H\times W} \tag{1}\]0  
\[I\in[0,\;1]^{H\times W} \tag{1}\]1  
\[I\in[0,\;1]^{H\times W} \tag{1}\]2  
\[I\in[0,\;1]^{H\times W} \tag{1}\]3  
\[I\in[0,\;1]^{H\times W} \tag{1}\]4  
In the above formula, Lcontent measures the degree to which the obtained fusion image preserves important content details in the source images; and Lmemory ensures the consistency and continuity of the learning process with the intermediate outputs produced in the previous epochs. The parameters \(p_{1}\) and \(p_{2}\) control the weighting between these two types of loss and can be determined adaptively according to the image type [26]. In the proposed method, MUFusion synthesizes both global contrast and local details optimally by performing information fusion between the structural detail component has been denoised with the Latent Low-Rank Representation (LatLRR) method and the Y channel to which adaptive gamma correction has been applied. The final Y channel obtained is recombined with the Cb and Cr color components obtained in the adaptive color correction step in order to preserve color accuracy; thus, the image created in the YCbCr space is converted to the RGB color space and the final enhanced underwater image is obtained. This last step ensures that the enhancement processes only affect the structural information and do not distort the color components, thus ensuring that natural, balanced and visually rich results are obtained. This approach, which does not require a reference image, increases the generalization ability of the method and provides consistent and high-quality image enhancement under different illumination, chromatic aberration or blur conditions.