Conclusion
This work proposes a novel parameterization method, based on discrete surface Ricci flow and optimal mass transportation theories. The algorithm is capable of finding area-preserving parameterizations for genus zero surfaces with multiple boundaries. The achieved mapping is diffeomorphic and invariant under conformal transformations. Conventional Kantorovich’s approach has O(n2) unknown variables, whereas the proposed method has only O(n) variables, where n is the number of vertices on the mesh. Furthermore, the algorithm gives users full control of the area of each part on the texture domain. Both Ricci flow and Optimal Mass Transport Map are based on convex optimizations. Our experimental results demonstrate the efficiency and efficacy of the proposed method. In practice, conformal mapping and area-preserving mapping have different advantages and disadvantages. Depending on the real applications, users can choose one of them or a combination of them. In future, we will explore the way to generalize the current framework to high genus surfaces, and find more applications in various engineering and medical fields.
