# A Two-Stage Gradient Ascent-Based Superpixel Framework for Adaptive Segmentation
## 3 Proposed Two-Stage Adaptive Image Segmentation Framework
### Adaptive Marker-Controlled Watershed Subdivision  
SLIC superpixel segmentation exposes some drawbacks due to its simple framework, not only resulting in redundant eigenvalue computation, but becoming a bottleneck of performance. For example, other than LSC superpixel [16], there is no global image property considered in local-based k-means clustering, which would lead to wrong gathering for some pixels during clustering, and eventually be mislabeled. What's worse, SLIC adopts a split-and-merge post-processing, and produces a large number of heterogeneous regions if isolated fragments aggregate without accurate guidelines [24]. Aimed at these situations, a feasible approach is to find the superpixels without uniformity and then subdivide them in a more precise way. For efficiently sieving the superpixels mentioned above, as well as keeping the external outliers of SLIC segmentation, a homogeneity criterion is put forward to define under-segmentation. Since almost all superpixels are merged and relabeled after post-processing, along with discarding their local information during clustering and aggregating, the criterion is then divided into two cases for different superpixels:  
Figure 1: Visual comparison of segmentation results. The first two rows display Simple Linear Iterative Clustering (SLIC) superpixels after 5 and 10 iterations respectively. The third row displays speeded-up Simple Linear Iterative Clustering (sSLIC) superpixels. In each image, the expected superpixel number in the upper left and lower right is 100 and 200 respectively. Alternating columns show each segmented image followed by zoom-in performance of the center. See Section 4 for qualitative and quantitative evaluations. * If a superpixel is still simply connected without merging neighboring isolated regions (in Figure 2c, they are filled with blue and green), namely \(\Omega_{k}^{m}=\Omega_{k}\), the inner difference in Lab color space can be calculated by: \[d\Big{(}C_{i}^{c}\Big{)}=\|C_{k}^{c}-C_{i}^{c}\|,\] (3) \[C_{i}^{\min}=\arg\min_{C_{i}\in\Omega_{k}}\Big{(}d\Big{(}C_{i}^{c}\Big{)}\Big{)},\] (4) \[C_{i}^{\max}=\arg\max_{C_{i}\in\Omega_{k}}\Big{(}d\Big{(}C_{i}^{c}\Big{)}\Big{)},\] (5) where \(C_{i}^{\min}\) and \(C_{i}^{\max}\) are a pair of pixels with the minimum and maximum distances from cluster center \(\Omega_{k}\) respectively. The superpixel is considered heterogeneous if: \[\|C_{i}^{\min}-C_{i}^{\max}\|>\varepsilon.\] (6)
* If a superpixel \(\Omega_{k}^{m}\) merges neighboring isolated regions (cyan and yellow parts in Figure 2c), the mean value of Lab color space is obtained as a region vector \(C_{ki}^{c}=[l_{k},a_{k},b_{k}]\) to the neighboring region \(\Omega_{ki}\). The sum of inner difference between \(C_{k}^{c}\) and each \(C_{ki}^{c}\) is computed for determining the heterogeneity of \(\Omega_{k}^{m}\) if \[\sum d\Big{(}C_{ki}^{c}\Big{)}>2\varepsilon.\] (7)  
In the next step, the adaptive marker-controlled watershed approach is adopted to subdivide the under-segmentation superpixels (green and yellow regions in Figure 2e). In the first case, a set of markers for topographic surface flooding is defined as \(C_{i}^{\min}\Big{|}C_{i}^{\max}\) by (4)-(6). Another situation is \(C_{ki}|\Omega_{ki}\), where \(C_{k}\) is the cluster center of \(\Omega_{k}\) and \(\Omega_{ki}\) represent(s) the entire partition of isolated region(s) with relative large color difference by (7) and (8):  
\[d\Big{(}C_{ki}^{c}\Big{)}>\frac{\varepsilon}{2}. \tag{8}\]  
notice that it is reasonable for the proposed sSLIC to skip subdividing the regions that converge in advance, which nearly all manifest homogeneity and do not need evaluating (white parts in Figure 2e). It is also implied that sSLIC preserves strong homogeneity from SLIC since only a few superpixels are  
Figure 2: The schematic diagram of proposed two-stage image segmentation framework.