# A Learnable Viewpoint Evolution Method for Accurate Pose Estimation of Complex Assembled Product
## 2 Proposed Method
### Viewpoint Performance Assessment  
For each viewpoint \(vp\), the performance \(vp^{q}\) reflects the average similarity of virtual-real ROI of all visible components. The geometric feature is adopted to overcome texture sparsity. Inspired by [41], as shown in Figure 3, a self-adaptive evaluation line-based feature matching and evaluation algorithm that balances structural stability and multiple considerations is designed to enhance robustness against virtual-real differences, as well as uneven correspondence. For each visible component \(w\), where \(w\) belongs to the set of all visible components \(W\) under \(vp\), the virtual-real feature is denoted as \(\{\mathbf{p}^{w},\mathbf{x}^{w}\}\) and \(\mathbf{p}^{w}\) is projected from 3D coordinate \(\mathbf{p}^{w}\). \(\mathbf{p}^{w}_{l}=\{\mathbf{p}^{w}_{l,1},\ldots\mathbf{p}^{w}_{l,m}\}\) represents the set of points on the \(l\)-th projected line segment of \(\mathbf{p}^{w}\), and the same applies to \(\mathbf{x}^{w}_{r}\). Figure 3: _Cont._  
Figure 2: Method framework. For any \(\mathbf{p}_{l,m}^{w}\) in \(\mathbf{p}_{l}^{w}\), a rectangular search region with changeable size along the normal direction is expanded, where the size depends on \(\sigma=E^{s}(vp^{q}),vp^{q}\) is the performance of the parent view of \(vp\), and \(E^{s}\) denotes the evolution function of the search factor \(\sigma\). In the rectangular region, any point belonging to \(\mathbf{x}_{r}^{w}\) can be selected as a candidate matching point for \(\mathbf{p}_{l,m}^{w}\) as long as the angle between \(\mathbf{p}_{l}^{w}\) and \(\mathbf{x}_{r}^{w}\) satisfies the preset range. After searching, a redundant candidate point set \(\mathbf{\overset{\infty}{x}}^{w}\) of \(\mathbf{p}^{w}\) is established. Theoretically, there is always an ideal transformation \(\mathbf{T}_{\text{2D}}\) that can minimize the error of \(\left\{\mathbf{T}_{\text{2D}}\mathbf{p}^{w},\mathbf{x}^{w}\right\}\) with the constraints provided by root \(vp\). During the evolution stage, the translational part in \(\mathbf{T}_{\text{2D}}\) is the main factor that narrows the gap between \(\left\{\mathbf{p}^{w},\mathbf{x}^{w}\right\}\). Based on the above assumptions, the quadrant is divided uniformly to roughly determine the potential direction of translation, and a pair of matching points is defined as \(\left\{\mathbf{p}_{l,m}^{w},\mathbf{x}_{r,n}^{w}\right\}\). According to  
\[arccos\left(\left[\mathbf{x}_{r,n}^{w}-\mathbf{p}_{l,m}^{w}\right]\left[ \begin{matrix} 1\\ 0\end{matrix} \right]/\left\|\mathbf{x}_{r,n}^{w}-\mathbf{p}_{l,m}^{w}\right\|\right\), \tag{1}\]  
the candidate set of \(\mathbf{p}^{w}\) in each slope range can be obtained. For each slope range, the number ratio \(ra\) and the distance variance \(var^{res}\) between the candidate set and \(\mathbf{p}^{w}\) can be counted. The direction advantage is defined as \(norm(ra)+(1-norm(var^{res}))\), where \(norm\) represents normalization operation (all slope ranges), and candidate matching points are reassigned to \(\mathbf{p}^{w}\) in order of advantage. During the reassignment process, if there is a one-to-many situation, the candidate point belonging to the slope range that reduces the variance of the current matching distance is preferentially selected. Afterward, we can obtain \(\left\{\mathbf{p}^{w},\mathbf{\overset{\infty}{x}}^{w}\right\}\), \(cp^{ra}\), and \(cp^{res}\), where \(\mathbf{\overset{\infty}{x}}^{w}\) represents an incomplete set, that is, part of \(\mathbf{x}\) finds no corresponding \(\mathbf{p}\), \(cp^{ra}\) is the number ratio, and \(cp^{res}\) denotes the distance average of all matched pairs. Evaluation criteria that rely solely on \(cp^{res}\) are easily affected by background noise, incomplete geometric features, and virtual-real differences. Therefore, a ridge function mapping strategy is further embedded with \(cp^{ra}\) as an additional evaluation factor.