# Pose Tracking and Object Reconstruction Based on Occlusion Relationships in Complex Environments
## 2 Methods
### Neural Object Field  
To extend the method to track arbitrary objects, a novel neural object field (NOF) [12] is introduced to enhance the optimization process. Unlike prior instance-level or category-level tracking methods, this approach eliminates the need for specific CAD models or geometric priors associated with object categories. The neural object field represents a neural signed distance field (SDF) centered on the object, which learns the object's multi-view consistent 3D shape and appearance while optimizing the poses of memory frames. Since the model is trained independently for each video, it is capable of learning effectively for each new object without the need for pre-training.  
Similar to BundleSDF [12], we apply multi-resolution hash encoding [20] to the input \(x\) before passing it to the network. The surface normal at a point in the object field can be derived by taking the first-order derivative of the signed distance field, expressed as \(n(x)=\frac{\partial\Omega(x)}{\partial x}\). This is implemented using automatic differentiation in PyTorch.  
Given the object pose \(\xi\) of a memory frame, an image is rendered by emitting rays through each pixel. The 3D points along each ray are sampled at different positions, defined as \(x_{i}(r)=o(r)+t_{i}d(r)\), where \(o(r)\) is the ray origin (i.e., the camera focal point) and \(d(r)\) is the ray direction, both of which depend on the object pose \(\xi\). The parameter \(t_{i}>0\) controls the position along the ray. The color \(c\) of a ray \(r\) is computed by integrating over the near-surface regions.  
\[c(r) =\int_{z(r)-\lambda}^{z(r)+0.5\lambda}\omega(x_{i})f(x_{i},n(x_{i }),d(x_{i}))\,dx \tag{1}\] \[\omega(x_{i}) =\frac{1}{(1+e^{-\alpha\Omega(x_{i})})(1+e^{\alpha\Omega(x_{i})})} \tag{2}\]  
where \(\omega(x_{i})\) is the bell-shaped probability density function [21] that depends on the distance from the point to the implicit object surface.  
To achieve efficient rendering, we adopt the ray sampling strategy proposed in [12]. Initially, we uniformly sample \(N\) points within the occupancy voxel bounds and \(z(r)+0.5\lambda\)  
Figure 1: Overview of our system. First, we compute the image mask of the target object, then feed the mask segmentation result into the feature-matching network to obtain a coarse pose estimation using the Umeyama [19] algorithm with the feature-matching result of the previous frame. Second, we use a dual-layer mask-based strategy to select frames from the key-frame pool that have a strong co-visibility relationship with the current frame and perform joint optimization with the current frame to obtain the final pose estimation result.  
To further improve the quality of reconstruction, additional samples are distributed around the surface. Rather than employing importance sampling based on SDF predictions, which would require multiple forward passes through the network [22; 23], we sample \(N^{\prime}\) points from a normal distribution centered on the depth reading \(\mathcal{N}(z(r),\lambda^{2})\). This approach results in a total of \(N+N^{\prime}\) samples, avoiding the computational overhead of querying multi-resolution hash encodings or the network itself.