# Repair of Voids in Multi-Labeled Triangular Mesh
## 2 Related Works  
Generally speaking, the mesh repairing started around the 1990s, when mesh technology was just taking off. For example, the paper [5] published by [PERSON] and [PERSON] is one of the important foundations of volume reconstruction. The work of [PERSON] [6] and [PERSON] [7] in the 1990s is a typical example in the early stage of mesh repairing.  
The existing studies on model repair mainly focus on the single-domain model repairing methods, which are roughly divided into two categories: the voxel-based repairing methods and the surface-based repairing methods [8].  
The voxel-based repairing methods [9; 10; 11; 12] first convert the mesh surfaces into voxels, then repair the meshes in the voxel space, and then use polygon extraction technology to convert the voxels into meshes. [PERSON] et al. [13] first constructed a signed distance function, then applied a diffusion process to extend the function through the volume to repair holes. [PERSON] [14] used an octree grid to reconstruct a closed surface by contouring. [PERSON] et al. [15] converted polygonal models to a volumetric representation and repaired the model through the conversion between the model and the volume domain. [PERSON] [16] built an adaptive octree for the input model and then reconstructed the model through voxel topology simplification. In general, the voxel-based repairing methods have a good performance in efficiency, but some original detailed features may be lost between model conversions, and the repaired model may have more redundancy triangular patches compared with the original model.  
The surface-based repairing methods [17; 18; 19; 20; 21] generally first identify the location of the holes, and then directly perform mesh repairing on the local areas. These kinds of methods are roughly divided into three categories. The first are the repairing algorithms based on interpolation. The interpolated meshes can be generated using simple polynomial functions [22], triangular B-splines [23] or radial basis functions [24]. This kind of algorithms is suitable for repairing simple holes, but not for holes with complicated situations with grooves and islands. The second are the algorithms based on triangulation. [PERSON] [25] and [PERSON] [26] first triangulated the hole area, and then continuously adjusted the newly generated meshes through mesh refinement and smoothing to make the new meshes similar to the surrounding mesh density. [PERSON] et al. [27], [PERSON] et al. [28] and [PERSON] [29] first segmented the complex holes into multiple sub-holes along the characteristic curve, and then repaired the complex holes by repairing these relatively smooth small holes. This kind of algorithms can quickly achieve mesh repair, but it is often difficult to obtain the detailed features of the model. The third are the template-based repairing methods [30; 31; 32]. The idea of this type of algorithms is to search for a patch similar to the feature of the hole in the template library or the model itself and realize the repair by copying and pasting. This type of algorithms consumes relatively high time, and it is often difficult to repair the meshes in complex situations.  
The above studies are all single domain mesh repairing algorithms. Scholars in the field of multi-domain mesh repairing also put forward some ideas. [PERSON] et al. [33] presented a scene reconstruction method. The method used RGB-D data to reconstruct a 3D scene mesh model, then segmented different types of surfaces and objects in the scene, and then deployed an adaptive hole-filling algorithm to recover the original meshes. [PERSON] et al. [34] proposed a method to repair non-manifold boundaries of segmented simplicial meshes, and developed algorithms about relabeling, point insertion and simulated annealing to reconstruct models.