## 3 Background  
Three-dimensional representation with NeRF [11] generally utilizes multilayer perceptrons (MLPs) to model complex 3D scene geometry. Specifically, the MLPs take 3D coordinates \(\mathbf{x}=(x,y,z)\) and 2D viewing directions \(\mathbf{d}=(\theta,\phi)\) to predict the RGB colors \(\mathbf{c}\) and volumetric density \(\sigma\) as  
\[\mathrm{MLP}(\mathbf{x},\mathbf{d})=(\mathbf{c},\sigma):\mathbb{R}^{3}\times \mathbb{R}^{2}\mapsto\mathbb{R}^{4}. \tag{1}\]  
Then, once the NeRF MLP is trained, \(\mathbf{c}\) and \(\sigma\) can be used to synthesize images from novel viewpoints through the volume-rendering method [45]. In the field of robotics, the volumetric density \(\sigma\) can further be used to extract the geometric information of objects, which can then be utilized for manipulation tasks. In volume rendering, the color of each pixel is rendered as the expected color computed along with the ray from the camera's origin to the pixel location. Since the NeRF dataset consists of pairs of ground truth RGB images \(\mathcal{I}\) and camera poses \(\mathcal{T}\in\text{SE}(3)\), denoted as \(\{\mathcal{I}_{i},\mathcal{T}_{i}\}_{i=1}\), the origin and viewing direction of the camera can be obtained from the corresponding camera pose \(\mathcal{T}\). Given the camera's origin \(\mathbf{o}\) and viewing direction \(\mathbf{d}\) based on the pixel location, the ray is defined as a half-line starting from the origin, i.e., \(\mathbf{r}(t)=\mathbf{o}+t\mathbf{d}\) for \(t\geq 0\). The expected color \(C(\mathbf{r})\) along with the ray \(\mathbf{r}\) is computed as  
\[C(\mathbf{r}):=\int_{0}^{\infty}T(t)\sigma(\mathbf{r}(t))\mathbf{c}(\mathbf{r} (t),\mathbf{d})dt, \tag{2}\]  
\[T(t):=\exp\biggl{(}-\int_{0}^{t}\sigma(\mathbf{r}(s))ds\biggr{)} \tag{3}\]  
where \(T(t)\) is the accumulated transmittance along the ray \(\mathbf{r}\). While Equation (2) provides the theoretically grounded manner to calculate the expected pixel color, the integration in Equation (2) is often intractable when \(\mathbf{c}\) and \(\sigma\) are modeled as the MLP. Hence, numerical estimation [46] of the expected color is widely employed. The integration in Equation (2) is approximated using a set of \(N\) discrete points \(\{p_{i}\}_{i=1}^{N}\) sampled along the ray \(\mathbf{r}\), i.e., \(p_{i}:=\mathbf{o}+t_{i}\mathbf{d}\) for some \(t_{i}\in(0,\infty)\). Then, the estimated color can be computed as follows,  
\[\hat{C}(\mathbf{r})=\sum_{i=1}^{N}w_{i}\mathbf{c}_{i},\ w_{i}=T_{i}\alpha_{i}, \ T_{i}=\prod_{j=1}^{i-1}(1-\alpha_{j}), \tag{4}\]  
where \(\mathbf{c}_{i}\) and \(\sigma_{i}\) are a color and density at \(p_{i}\), i.e., \((\mathbf{c}_{i},\sigma_{i})=f_{\theta}(p_{i},\mathbf{d})\); \(w_{i}\) is a ray weight at \(p_{i}\); \(T_{i}\) represents the accumulated transmittance; \(\alpha_{i}=1-\exp(-\sigma_{i}\delta_{i})\) represents the ray termination probability at \(p_{i}\); and \(\delta_{i}=t_{i+1}-t_{i}\) is the distance between adjacent samples. \(\hat{C}(\mathbf{r})\) is computed as the weighted sum of color samples \(\mathbf{c}_{i}\) with ray weight \(w_{i}\). Despite NeRF being highly effective in 3D scene representation, its dependency on volume density fields for novel view synthesis poses significant challenges for extracting high-quality surfaces. To mitigate this limitation, an existing approach combines neural implicit representation with signed distance functions (SDFs), leading to the development of neural SDF representation. In this framework, the SDF value \(f(x)\) at a given 3D point \(x\) is modeled by the neural network. In [30], the employment of SDF values introduces a distinct formulation of the opacity term, \(\alpha_{i}\), different from the approach utilized in NeRF. Specifically, in [30], \(\alpha_{i}\) is defined as  
\[\alpha_{i}=\max\biggl{(}\frac{\Phi_{s}(f(p_{i}))-\Phi_{s}(f(p_{i+1}))}{\Phi_{ s}(f(p_{i}))},0\biggr{)}, \tag{5}\]  
where \(f(p_{i})\) represents the SDF value at 3D point \(p_{i}\), and \(\Phi_{s}(x)=(1+e^{-sx})^{-1}\) denotes the sigmoid function.