Mathematical Models for 3D Reconstruction using Surface Triangulation

3D reconstruction is one of the most important research areas in computer version, which has a very wide application. Triangulation of simple polygon has been quite a matural mathematical approach in this field. Based on the bijection between triangulation and graph without loop, the problem is converted into the study of graph without loop. Firstly, enumerative equations of graph without loop for given number of vertex degree and edges are provided, including orientable, nonorientable and total of all surfaces. These differential equations are all Riccati type. No feasible and convenient way has been in sight for extracting an explicit solution to resolve the functional equations up to now. Next, the corresponding calculative functions of graphs with two parameters are extracted directly and the number can be derived by simple recursive formulae. The established mathematical model could provide a theoretical foundation for computerized algorithms, which can be used for 3D reconstruction.