# A Learnable Viewpoint Evolution Method for Accurate Pose Estimation of Complex Assembled Product
## 2 Proposed Method
### Feasible Region Construction and Update
#### 2.2.4 Sparse Trajectory Approximation  
Let the projected \(\mathcal{V}\) be \(\mathcal{V}^{\perp}\). For each \(vp\), \(vp\in\mathcal{V}^{\perp}\), an unconstrained objective function can be constructed as  
\[f_{2}(\mathbf{T})=min\Bigg{\{}\sum_{w}^{W}\lVert\mathcal{H}\big{(}\mathbf{x}^{vp}- \pi(\mathbf{TP}^{w})\big{)}\rVert_{2}^{2}\Bigg{\}}, \tag{10}\]  
where \(\mathcal{H}(\cdot)\) is the kernel function to enhance the robustness of outliers, and [PERSON] is selected in this study. Taking \(\mathcal{V}^{\perp}\) as the initial state, (17) is solved in parallel based on the LM strategy, where the pose change can be recorded as \(tra^{h}=\left\{tra_{\mathbf{T}_{0}^{\perp}},tra_{\mathbf{T}_{1}^{\perp}}, \ldots,tra_{\mathbf{T}_{n}^{\perp}}\right\}\), \(tra_{\mathbf{T}_{0}^{\perp}}=\left\{\mathbf{T}_{v}^{\perp},\mathbf{T}_{v}^{1},\mathbf{T}_{v}^{2}\ldots,\mathbf{T}_{v}^{V}\right\}\), and \(V\) is the number of iteration steps of the \(v\)-th trajectory.  
Let the optimization direction of all trajectories in the \(z\)-th step be \(\{\Delta\mathbf{T}_{0}^{z},\Delta\mathbf{T}_{1}^{z},\ldots,\Delta\mathbf{T}_{ v}^{z}\}\), and define the trajectory affinity as  
\[\mathcal{A}_{m,n}^{z}=f_{d}(\mathbf{T}_{m}^{z},\mathbf{T}_{n}^{z})\cdot\big{(} \Delta\mathbf{\hat{T}}_{m}^{z}\Delta\mathbf{\hat{T}}_{n}^{z}\big{)}, \tag{11}\]  
Afterward, the horizontal interpolation is evaluated for \(\{\mathbf{T}_{1}^{z},\mathbf{T}_{2}^{z},\ldots,\mathbf{T}_{v}^{z}\}\). For all \(\mathcal{A}_{m,n}^{z}\), \(m,n\in tra^{h}\),  
\[tra^{v}=\Bigg{\{}li\bigg{(}\mathbf{T}_{m}^{z},\mathbf{T}_{n^{\prime}}^{z}\frac {1}{2}\Big{(}f_{d}(\mathbf{T}_{m}^{z+1},\mathbf{T}_{m}^{z})+f_{d}(\mathbf{T}_ {n}^{z+1},\mathbf{T}_{n}^{z})\Big{)}\bigg{)}\Bigg{|}\mathcal{A}_{m,n}^{z}> \frac{1}{v}\sum_{m}\sum_{n=m+1}\mathcal{A}_{m,n}^{z}\Bigg{\}}. \tag{12}\]  
Let \(\ell=\left(f_{d}(\mathbf{T}_{m}^{z+1},\mathbf{T}_{m}^{z})+f_{d}(\mathbf{T}_{ n}^{z+1},\mathbf{T}_{n}^{z})\right)/\left(2f_{d}(\mathbf{T}_{m}^{z},\mathbf{T}_{ n}^{z})\right)\), then \(li(\cdot)\) can be expressed as a spherical linear interpolation operation on \(\mathbf{q}\) and \(t\), respectively, in the following way.  
\[li_{\mathbf{q}}=\frac{sin[(1-\ell)\theta]\mathbf{q}_{m}+sin(\ell\theta)\mathbf{q}_{n}}{sin (\theta)},\;li_{\mathbf{t}}=\mathbf{t}_{m}+\ell\left(\mathbf{t}_{n}-\mathbf{t}_{m}\right). \tag{13}\]  
Through the above work, the discrete approximation \(tra_{i}=\left\{tra_{i}^{h},tra_{i}^{v}\right\}\) of \(\mathcal{F}_{i}\) can be obtained.  
During evolution, once the historical best \(vp^{q}|_{w}\) of any component is reset due to the generation of a child \(vp\), the pose feasible region update can be selectively started according to the change in the \(vp^{q}\). \(\mathcal{F}_{i+1}\) is updated from \(\mathcal{F}_{i}\), and the update work can be summarized into four steps, including the following: (a) simplifying \(\mathbf{x}^{w}\) based on structural stability to obtain \(\mathbf{x}^{vp},\mathbf{x}^{\prime}\); (b) determining \(\delta_{i+1}\) and \(\mathbf{\Gamma}_{i+1}^{j_{i+1},W}\) and selecting the support \(vp\) according to \(vp^{q}\) from \(\{\mathcal{P}_{i},\mathcal{C}_{i}\}\); (c) calculating the projection of the support \(vp\) to the boundary by Formulas (6)-(9); and (d) solving the optimization problem (10) and performing affinity-based interpolation with (11)-(13).