## 3 Methodology
### Image Reconstruction  
Given two consecutive frames \(I_{r}\) and \(I_{s}\) from a monocular video, the depth network \(\Theta\) predicts the depth map \(D_{r}\) for the reference frame, while the pose network \(\Phi\) estimates the relative camera motion (R,t) between the two frames, where R is the rotation matrix and t is the translation vector. Using this information, the source frame \(I_{s}\) is synthesized by warping the reference frame \(I_{r}\) according to the predicted depth and camera motion:  
\[I_{r\gets s}=W_{arp}(I_{s}D_{r},R,t) \tag{11}\]The warping process projects the pixels from the source frame \(I_{s}\) onto the reference frame \(I_{r}\), generating a synthesized view of \(I_{s}\) from the perspective of \(I_{r}\). In our framework, the photometric loss \(L_{photo}\) combines SSIM and pixel-wise intensity differences to penalize deviations between the reconstructed and reference images. The photometric consistency loss is then computed as the pixel-wise difference between the reference frame \(I_{r}\) and the synthesized source frame \(I_{r\gets s}\):  
\[L_{photo}=\sum_{p}(\alpha\cdot|I_{r}(p)-I_{r\gets s}(p)|+(1-\alpha) \cdot SSIM(I_{r}(p),I_{r\gets s}(p))) \tag{12}\]  
where is \(L_{photo}\) the photometric loss, combining SSIM and the absolute intensity difference to measure the reconstruction quality, where \(p\) denotes the pixel index, and SSIM is the structural similarity index that accounts for differences in local image structure and luminance. The parameter \(\alpha\) controls the trade-off between pixel-wise intensity differences and structural similarity, typically set to \(\alpha\) = 0.85. This loss encourages the predicted depth map to align with the image geometry by penalizing discrepancies between the real and synthesized frames. Monocular depth estimation can sometimes produce noisy depth maps, particularly in textureless or homogeneous regions of the image where depth information is ambiguous. To mitigate this issue, we introduce an edge-aware depth smoothness loss that encourages smooth depth variations in regions where the image has low gradients, while preserving depth discontinuities at object boundaries. In our method, the smoothness loss \(L_{smooth}\) enforces spatial consistency in depth predictions while preserving edges, leveraging the image gradients \(\
abla I_{r}(p)\) as weights. The depth smoothness loss \(L_{smooth}\) is defined as follows:  
\[L_{smooth}=\sum_{p}|\
abla D_{r}(p)|e^{-\
abla I_{r}(p)} \tag{13}\]  
where \(L_{smooth}\) is clearly defined as the edge-aware depth smoothness loss and matches the context in which it is introduced, where \(\
abla D_{r}(p)\) is the spatial gradient of the predicted depth map, and \(\
abla I_{r}(p)\) s the spatial gradient of the reference image. The exponential term \(e^{-\
abla I_{r}(p)}\) ensures that the smoothness loss is weighted more heavily in regions with low image gradients and less in high-gradient areas, such as object boundaries. This loss encourages the depth predictions to be smooth in areas where there is little visual variation, while allowing for sharp depth changes at the edges, such as the boundaries of objects. One of the core challenges in monocular depth estimation is scale ambiguity, particularly in dynamic scenes where objects are in motion. The depth scale for moving objects often differs from that of the static background due to the independent motion of these objects. To address this issue, the DSA module ensures that the depth scales of dynamic objects are consistent with the static background. The scale-alignment loss \(L_{scale}\) is designed to minimize the difference in depth scales between dynamic and static regions.