# A Learnable Viewpoint Evolution Method for Accurate Pose Estimation of Complex Assembled Product
## 2 Proposed Method
### Feasible Region Construction and Update  
#### 2.2.1 Problem Planning  
Projection matrix (or pose) \(\mathbf{T}\) contains seven variables if the representation approach of quaternion and translation is employed. The crossover and mutation in the traditional genetic algorithm are too exploratory to ensure the optimization of \(\mathbf{T}\). The commonly used numerical optimization is limited by the observed \(\{\mathbf{x},\mathbf{p},\mathbf{P}\}\) and prone to fall into local optimum. Thus, it is expected to provide a range constraint in the solution space for parent \(\mathcal{P}_{i}\) to children \(\mathcal{C}_{i}\) based on observed \(\{\mathbf{x},\mathbf{p},\mathbf{P}\}\), which can be continuously updated and contracted to the ideal solution with the adjustment of \(\mathbf{x},\mathbf{P}\) during the evolution. The range in this study is defined as pose feasible region \(\mathcal{F}\), and for \(\mathcal{F}_{i},\mathcal{F}_{i}=\Big{\{}\mathbf{T}\Big{|}F_{i}^{i,l}(\mathbf{T })\leq 0,\mathbf{T}\in\text{SE}(3)\Big{\}}\), where  
\[F_{i}^{\delta_{i},W}(\mathbf{T})=\sum_{w}^{W}\big{\|}\mathbf{x}^{vp^{q}}-\pi( \mathbf{T}\mathbf{P}^{w})\big{\|}_{2}^{2}-\delta_{i}^{2}. \tag{4}\]  
\(\mathbf{x}^{20^{\prime}}\) represents the pixel coordinate corresponding to the best \(vp^{q1}|_{w}\) achieved by the \(w\)-th visible component in the previous \(i\) round. \(\delta_{i}=E^{\delta}(\overline{vp}^{qi})\), \(\overline{vp}^{qi}=(1/P_{i})\Sigma_{k}^{P_{i}}(vp^{q}_{k}),vp_{k}\in\mathcal{ P}_{i+1}\), \(\mathcal{P}_{i+1}\) is the initial parent selected from \(\{\mathcal{C}_{i},\mathcal{P}^{\prime}_{i}\}\) in the \(i\)-th round, and \(\mathcal{P}^{\prime}_{i}\) refers to \(\mathcal{P}_{i}\) after migration to \(\mathcal{F}_{i}\). \(E^{\delta}\) is the evolution function of the tolerance factor for constructing \(\mathcal{F}\). To provide a feasible constraint to the pose search, the expression of \(\mathcal{F}\) should be solved. This expression problem can be formulated as  
\[min_{\mathbf{T}}\Bigg{\{}\frac{W}{\sum_{w}^{W}}\Big{\|}\mathbf{x}^{w^{\prime}}-\pi (\mathbf{T}\mathbf{P}^{w})\Big{\|}_{2}^{2}+\left(-\frac{1}{i}\right)\log\!\left( -F_{i}^{\delta_{i},W}(\mathbf{T})\right)\Bigg{\}}, \tag{5}\]  
where \(-\log\!\left(-F_{i}^{\delta_{i},W}(\mathbf{T})\right)\) is the logarithmic barrier function and \(i\) denotes the approximation factor to the indicative function. This is an optimization technique that uses the interior point method to remove inequality constraints. In practice, however, the optimization process of (5) is too complicated. Given different initial values, repeated iterations and optimizations of \(i\) are computationally expensive. To this end, the assumption of structural stability is again applied as a basis for simplification. Specifically, only the matching relation corresponding to the best advantage in \(\left\{\mathbf{p}^{w},\widetilde{\mathbf{x}}^{w\prime}\right\}\) is retained to sparse \(\mathbf{x}^{w\prime}\) to reduce the scale of problem (5). To alleviate the tedious optimization work, a combined strategy based on boundary projection and sparse trajectory approximation is provided below.