## 4 Proposed Method
### Next Best View Pose Optimization  
In robotic active reconstruction, quickly identifying the most uncertain view for NBV pose selection is crucial. The existing methods rely on fixed candidate views, requiring time-consuming rendering from all predefined viewpoints. We propose a gradient-based NBV pose optimization method that reduces the rendering time by selecting NBV poses with fewer renderings. By using gradient information to pinpoint where uncertainty increases most rapidly, our approach achieves an efficient NBV with minimal renderings. To generate an object-centric differentiable pose, we define the transformation matrix \(\mathcal{T}_{\mathbf{\Theta}}\) using spherical coordinates \((\phi,\theta,r)\) where \(\mathbf{\Theta}:=(\phi,\theta)\) indicates the learnable parameters. First, we define the translation vector \(\mathbf{t}\) by converting the spherical coordinates to Cartesian coordinates:  
\[\mathbf{t}_{\mathbf{\Theta}}=[x,y,z]^{\intercal}=[r\sin\theta\cos\phi,r\sin \theta\sin\phi,r\cos\theta]^{\intercal}. \tag{10}\]  
Next, we define the rotation axes to ensure an object-centric view:  
Figure 3: Comparison of uncertainty in the original NeRF and uncertain alpha voxel grid for the unseen view. We divide the views into (**a**) a seen view, which is similar to the train view, and an unseen view, which represents an out-of-distribution view, to investigate the overfitting issue of entropy-based uncertainty. As illustrated in (**b**), the original NeRF model fails to accurately capture the uncertainty in the unseen view due to the overfitting issue when trained using only one side of an object (**bottom right**). However, by applying \(V^{(\text{unc,\alpha})}\) to a voxel-based model, it can effectively capture the uncertainty in the unseen view (**top right**). * The \(z\)-axis of the camera frame denoted as \(\vec{w}\) is oriented toward the object origin: \[\vec{w}=-\frac{(x,y,z)}{\sqrt{x^{2}+y^{2}+z^{2}}}.\] (11)
* The \(x\)-axis of the camera frame denoted as \(\vec{u}\) is aligned parallel to the \(xy\)-plane of the global object frame: \[\vec{u}=\frac{(-y,x,0)}{\sqrt{x^{2}+y^{2}}}.\] (12)
* The \(y\)-axis of the camera frame denoted as \(\vec{v}\) is derived as the cross product of the rotation \(z\)-axis and the rotation \(x\)-axis: \[\vec{v}=\vec{w}\times\vec{u}.\] (13)  
The rotation matrix \(\mathbf{R}\) is formed by these orthogonal vectors, \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\):  
\[\mathbf{R_{\Theta}}=\Big{(}\vec{u}\quad\vec{v}\quad\vec{w}\Big{)}^{\mathsf{T}}. \tag{14}\]  
Finally, the \(4\times 4\) transformation matrix \(\mathcal{T_{\Theta}}\) is defined as follows:  
\[\mathcal{T_{\Theta}}= \begin{pmatrix}\mathbf{R_{\Theta}}&\mathbf{t_{\Theta}}\\ 0&1\end{pmatrix} . \tag{15}\]  
By this construction, the transformation matrix \(\mathcal{T_{\Theta}}\) is differentiable since all the pose construction steps are tractable. To ensure diverse exploration during the NBV pose optimization process and avoid overfitting to the existing training set poses, we introduce a simple regularizer loss \(\mathcal{L}_{\text{reg}}\) as follows:  
\[\mathcal{L}_{\text{reg}}=\frac{1}{N_{t}}\sum_{i=1}^{N_{t}}\text{err}(\mathcal{ T_{\Theta}},\mathcal{T_{i}}), \tag{16}\]  
where \(N_{t}\) is the number of poses in the existing training set, and \(\text{err}(\cdot,\cdot)\) refers to the absolute value of the pose error, including both the euclidean distance between the translations and the rotational error between the orientations of the two poses. The final NBV loss \(\mathcal{L}_{\text{NBV}}\) is calculated as the negative sum of the uncertainty map as in Equation (9) observed from \(\mathcal{T_{\Theta}}\) and \(\mathcal{L}_{\text{reg}}\):  
\[\mathcal{L}_{\text{NBV}}=-\lambda_{1}\mathbf{U}(\mathcal{T_{\Theta}})-\lambda _{2}\mathcal{L}_{\text{reg}}, \tag{17}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) are scaling parameters.