# Repair of Voids in Multi-Labeled Triangular Mesh
## 1 Introduction  
Polygonal representations of three-dimensional geometry modeling are widely used in computer graphics and practical applications [1]. However, due to limitations of the modeling methods, the models with multiple meshes often cannot be directly utilized by applications that require manifold meshes (topologically correct, e.g., free of singular edges and intersections) without voids as input. In this paper, based on several polygonal meshes without labels, we consider studying the representation method of the multi-labeled mesh and the model repairing method according to specific rules. Despite the extensive research on mesh repairing, few works have addressed the topological correctness of the meshing repairing involving several input meshes. To describe clearly, the mesh repairing involving single material is called as single-domain model repairing, the mesh repairing involving multiple materials is called as multi-domain model repairing. Compared with the single-domain model repairing, the multi-domain model repairing not only needs to consider the mesh quality of each sub-model, but also needs to consider the adjacent rules among each sub-model according to the practical applications. In this paper, we mainly focus on the multi-domain model repair method used to deal with the voids between models caused by insufficient data sampling or modeling algorithm defects. For example, when each sub-model representing different materials is modeled separately, there may be voids overlaps and other unexpected defects between sub-models in the modeling results. For a geological body with specific geological interfaces, the voids between geological interfaces, which may be regarded as 'new' geological interfaces,are undesired defects in geological modeling. In implicit geological modeling [2; 3], it is easy to determine a point is inside or outside the implicit model by computing the sign of the function value. However, for a void combined by several implicit functions, it is difficult to determine whether an exterior point is in the void. We consider repairing the voids between several combined surface meshes, which is useful for the methods of meshing multiple domains. Different from the single-domain model repairing method, which focuses on the quality of the mesh, the multi-domain model repairing method is mainly concerned with the topology adjacent relationship and the corresponding adjacent rules among each sub-domain model. Considering that each sub-mesh model can be repaired by the traditional single-domain model repairing method, we assume that the input meshes of the multi-domain model repairing method satisfy the characteristics of closed manifold, and the triangular topologies of the boundaries of the meshes are completely consistent. Based on the assumption, we can convert the multiple input meshes into the multi-labeled mesh which is convenient for searching the topology adjacent relationship among sub-meshes. It is worth noting that the mesh topology should be adjusted through preprocessing steps when the topology of overlapped triangular facet between multiple input meshes is inconsistent. In this paper, we seek a robust method for repairing the voids existing in multiple triangular meshes, so that the meshes are topologically consistent (without undesired material, namely the interior voids). Similar to the basic idea of the model repairing method based on multi-labeled voxels, the model repairing method based on multi-labeled mesh ensures the topological consistency between polygonal meshes by the strategy of greedy searching. Different from the multi-domain model repairing method based on multi-labeled voxels [4], the novel method repairs the polygonal mesh directly. The former repairs the model by searching the voids in the multi-domain model represented as labels, and the latter repairs the model by searching the voids in the multi-domain model represented as polygon mesh.