## 2 Proposed Superpixel-Based Model
### Image Reconstruction  
Similar to the bilateral filtering, for each pixel at position \((i,j)\) that needs to be reconstructed, we use the following two measures as the weights to reconstruct its value: (1) the distance of the current pixel at \((i,j)\) to the center of its neighboring superpixel at \((k,l)\), and (2) the intensity difference between the center of the superpixel at \((i,j)\) and the center of its neighboring superpixel at \((k,l)\). Figure 2 shows a graphical illustration of the reconstruction, and the reconstructed image \(I_{low}(i,j)\) can be mathematically obtained as  
\[\omega(i,j,k,l)=\exp\left(-\frac{(i-k)^{2}+(j-l)^{2}}{2\sigma_{d}^{2}}-\frac{ ||I(i,j)-I(k,l)||^{2}}{2\sigma_{r}^{2}}\right), \tag{1}\]  
\[I_{low}(i,j)=\frac{\sum_{k,l}I(k,l)\omega(i,j,k,l)}{\sum_{k,l}\omega(i,j,k,l)}, \tag{2}\]  
Figure 1: Flowchart of the proposed method.  
where \(I(i,j)\) and \(I(k,l)\) are, respectively, the intensity values for the centers of the superpixels at \((i,j)\) and \((k,l)\). It is clear that \(\omega(i,j,k,l)\) represents the combined weight of the neighboring superpixel at \((k,l)\) with respect to the current pixel at \((i,j)\). As is typically done, we only pick the four neighboring superpixels in the up, down, left, and right directions, respectively. Furthermore, \(\sigma_{d}\) and \(\sigma_{r}\) are the scale parameters controlling the smoothing effect; higher values of which produce smoother reconstructed images. We empirically set \(\sigma_{d}=4\) and \(\sigma_{r}=0.3\) for all of the experiments in this work, as explained next.  
In this work, we set appropriate values of \(\sigma_{d}\) and \(\sigma_{r}\) in Equation (1) to prevent the unbalance of the distance- and intensity-related weightings for image reconstruction instead of a simple normalization. This is mainly due to a technical consideration that the setting of \(\sigma_{d}\) and \(\sigma_{r}\) should depend on the superpixel size in this study. For example, according to the analysis and discussion in Section 3.1, we set the superpixel size as about 7 pixels in this work. Therefore, for the image reconstruction with four neighboring superpixels, the maximal distance between the current point and the neighboring superpixel centers (i.e., the term \(\sqrt{(i-k)^{2}+(j-l)^{2}}\) in Equation (1)) is about 10 pixels (along the diagonal line). Therefore, we set \(\sigma_{d}=4\) to control the decrease rate of the distance-related weighting. On the other hand, the maximum intensity difference between two pixels (i.e., the term \(||I(i,j)-I(k,l)||\) in Equation (1)) is 1.0. Therefore, we set \(\sigma_{r}=0.3\) to control the decrease rate of the intensity-related weighting.  
In summary, the setting of \(\sigma_{d}=4\) and \(\sigma_{r}=0.3\) used in this work can well balance the distance- and intensity-related weightings (i.e., with similar decreasing rates) when reconstructing the image with the superpixel size as about 7 pixels. Please note that as mentioned above, the setting of \(\sigma_{d}\) and \(\sigma_{r}\) should be dependent on the superpixel size. Large \(\sigma_{d}\) and \(\sigma_{r}\) values are required for the image segmentation with large superpixel sizes, and vice versa.