## 3 Proposed Method
### Watershed-Based Feature Recognition
#### 3.1.2 Feature Recognition Using Watershed Technology  
The point on \(2 DCM\) is denoted by \(x_{i}\) that has a value of height function defined by root mean square curvature, and has a set of connected neighbouring points on \(2 DCM\). Let **X** be the set of all points on \(2 DCM\). For each \(x_{i}\subseteq\)**X**, 4-neighbour of \(x_{i}\) are employed for processing. We traverse each \(x_{i}\) on \(2 DCM\), the \(x_{i}\) moves around its 4-neighbour depending on the connectivity and curvature descent. The topology of the \(2 DCM\) is described by rectilinear grid. Thus, the surface segmentation results affected by the height function is \(2 DCM\). The procedure of watershed segmentation is shown as below:  
1. **Find the local maxima of 2 DCM, and allocate labels for them as initial markers**. Once the \(2 DCM\) is defined over T-spline surface, initial labels need to be specified for the whole surface. In order to figure out the problem of over-segmentation, a set of markers is positioned on \(2 DCM\)  
Figure 3: The two-dimensional curvature map (2 DCM) of the T-spline surface.  
with unique labels. The markers can be used if the main parts of the surface are known, which can obtain the desired segmentation efficiently. Each initial marker has a one-to-one relationship to a specific watershed sub-region, thus the number of initial markers is equal to the final number of watershed sub-regions. The initial marker can be selected automatically or manually, and is generally difficult to obtain automatically without any user interaction. Selecting markers is a crucial and complicated task for watershed technology. For the sake of simplification, local maximum points are considered as markers in the paper.
2. **Adopt descent algorithm for initial markers.** The watershed region is extracted from each initial marker by using a root-mean-square curvature descent algorithm on the _2 DCM_. Each point on the _2 DCM_ tested by the descent algorithm is labeled with the same specified label of the initial marker. The unlabeled points are searched along the 4-neighbour of each point on the _2 DCM_. As shown in Figure 4a, given a point indexed by (_i,j_) on the _2 DCM_, the position of 4-neighbour points can be obtained concisely: the up point is indexed by (_i,j_+1), down point indexed by (_i,j_-1), left point indexed by (_i_-1,_j_), right point indexed by (_i_+1,_j_). The descent algorithm is explained with the help of Figure 4b. For each initial marker (red point in Figure 4b), four areas are generated by 4-neighbour points and are handled using a similar method. Area A is taken as an example for more details. The green point is an arbitrary point \(p_{i,j}\) in area A, three candidate points are chosen for test of \(p_{i,j}\) that are marked in yellow, and the test is passed for \(p_{i,j}\) when equation 11 is true. \[k_{i,j}<=\frac{k_{i+1,j}+k_{i,j+1}+k_{i+1,j+1}}{3}\] (11) where \(k_{i,j}\) denotes the root-mean-square curvature of \(p_{i,j}\). When the test is passed, two neighbour points colored in blue are added for the next test until the point moves to another local region or the curvature is smaller than the background value specified by user. The yellow points and blue points in areas B, C, D can be obtained in a similar way. All watershed regions are found after all points on the _2 DCM_ are labeled.