# A Novel Method Using 3D Interest Points to Place Markers on a Large Object in Augmented Reality
## 2 Proposed Method for Placing Multiple Markers
### Scoring Step
#### 2.2.1 Centrality of Segmented Area  
There are various conventional methods for segmenting meshes, but most of the purposes are to classify them into meaningful parts as [16; 17; 18]. Among these segmentation methods, the main purpose of K-means clustering is to divide into k parts rather than to classify into meaningful parts [19; 20]. Therefore, this paper considers K-means clustering as a method for segmenting meshes. To compensate for the randomness of this method, K-means++ is used. This method ensures that the centers of the initial clusters are the farthest away from each other, resulting in a more even division. Clustering is performed using \(\mathbf{V}\), and the cluster of specific \(\mathbf{f_{i}}\) is determined to be the highest percentage cluster in the three \(\mathbf{v}^{\prime}\)s that constitutes it. Figure 5 shows the segmentation result using K-Means++ after preprocessing on a specific virtual large object (a car model). Figure 4: Determination of how to create faces based on ROI boundaries: (**a**) intersecting the ROI boundary through the face; (**b**) new faces created by vertex conditions. From Hypothesis 1, the specific \(i\) th \(\mathbf{f}\) in the \(j\) th area among the segmented areas (hereafter this \(\mathbf{f}\) is denoted as \(\mathbf{f}_{j,i}\)) is placed more evenly as it is closer to the \(j\) th CoA, making it an appropriate candidate. But the \(\mathbf{f}\)'s in the \(j\) th area (hereafter the set of these \(\mathbf{f}\)'s is denoted as \(\mathbf{F}_{j}\)) have three-dimensional geometric information, the mean position of \(\mathbf{F}_{j}\) may not represent the CoA. For example, if \(\mathbf{F}_{j}\) is shaped like a sphere convexly protruding from a circular plane, as shown in Figure 6, the CoA is expected to be the top of the sphere. However, the mean position of \(\mathbf{F}_{j}\) is located below the center of the sphere. As a result, the \(\mathbf{f}\) closest to this position is not the closest to the expected CoA. To solve this problem, the dimensionality of \(\mathbf{F}_{j}\) is reduced to \(\mathbb{R}^{2}\) using principal component analysis. Dimension reduction is performed by excluding the smallest eigenvector among eigenvectors for \(\boldsymbol{V}\). Based on this, the CoA is defined as the mean position of \(\mathbf{F}_{j}=\left\{\mathbf{f}_{j,i}\Big{|}\mathbf{f}_{j,i}\in\mathbb{R}^{ 2},\ 1\leq i\leq\mathrm{n}\big{(}\mathbf{F}_{j}\big{)}\right\}\), expressed as Equation (2). \[\mathrm{CoA}\big{(}\mathbf{F}_{j}\big{)}=\frac{1}{n}\sum_{i=1}^{n}\mathrm{ position}\Big{(}\mathbf{f}_{j,i}\Big{)} \tag{2}\]  
where \(n\) denotes \(\mathrm{n}\big{(}\mathbf{F}_{j}\big{)}\), \(\mathbf{f}_{j,i}\) denotes \(\mathbf{f}_{j,i}\) dimensionally reduced to \(\mathbb{R}^{2}\), and position is a function that obtains the position of \(\mathbf{f}_{j,i}\). The farther \(\mathbf{f}_{j,i}\) is from \(\mathrm{CoA}\big{(}\mathbf{F}_{j}\big{)}\), the less suitable it is as a candidate, and the closer it is, the more suitable it is as a candidate. We define the degree of this suitability as centrality of segmented area in this paper (hereafter CSA), as expressed by the normalized value in Equation (3). \[\mathrm{CSA}\big{(}\mathbf{f}_{j,i}\big{)}=1-\frac{\left\|\mathrm{position} \Big{(}\mathbf{f}_{j,i}\Big{)}-\mathrm{CoA}\big{(}\mathbf{F}_{j}\big{)} \right\|}{\max\Big{(}\forall\mathbf{f}^{{}^{\prime}}\in\mathbf{F}^{{}^{ \prime}}_{j},\ \left\|\mathrm{position}\Big{(}\mathbf{f}^{{}^{\prime}}\Big{)}-\mathrm{CoA} \big{(}\mathbf{F}_{j}\big{)}\right\|\Big{)}} \tag{3}\]  
where \(\mathbf{F}^{{}^{\prime}}_{j}\) denotes \(\mathbf{F}_{j}\) dimensionally reduced to \(\mathbb{R}^{2}\), and \(\mathbf{f}^{{}^{\prime}}\) denotes \(\mathbf{f}\) dimensionally reduced to \(\mathbb{R}^{2}\) by iterating through \(\mathbf{F}^{{}^{\prime}}_{j}\). Figure 7 shows the CSA computed by Equation (3) at a specific \(\mathbf{f}_{j,i}\). Figure 5: Example of segmenting a virtual object: (**a**) preprocessed source; (**b**) \(k=4\); (**c**) \(k=6\); (**d**) \(k=8\).