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Quadrilateral Mesh Generation III : Optimizing Singularity Configuration Based on AbelJacobi Theorynull (Ed.)This work proposes a rigorous and practical algorithm for quadmesh generation based the AbelJacobi theory of algebraic \textcolor{red}{curves}. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quadmesh, in terms of the decktransformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a Tmesh; second, the edge lengths of the Tmesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quadmesh. In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the AbelJacobi map for a given divisor; optimize the divisor to satisfy the AbelJacobi condition by integer programming; compute \textcolor{red}{a} flat Riemannian metric with cone singularities at the divisor by Ricci flow; \textcolor{red}{isometrically} immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a TMesh. In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the Tmesh. The solution provides a flat metric with integral deck transformations, which leads to the final quadmesh. The proposed method is rigorous and practical. The Tmesh and quadmesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries.more » « less

null (Ed.)Optimal transportation (OT) finds the most economical way to transport one measure to another and plays an important role in geometric modeling and processing. Based on the Brenier theorem, the OT problem is equivalent to the Alexandrov problem, which is the dual to the Pogorelov problem. Although solving the Alexandrov/Pogorelov problem are both equivalent to solving the MongeAmp\`{e}re equation, the former requires second type boundary condition and the latter requires much simpler Dirichlet boundary condition. Hence, we propose to use the Pogorelov map to approximate the OT map. The Pogorelov problem can be solved by a convex geometric optimization framework, in which we need to ensure the searching inside the admissible space. In this work, we prove the discrete Alexandrov maximum principle, which gives an apriori estimate of the searching. Our experimental results demonstrate that the Pogorelov map does approximate the OT map well with much more efficient computation.more » « less

null (Ed.)This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quadmesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quadmesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quadmesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quadmesh. Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quadmesh generation using algebraic geometric approach.more » « less

Generative adversarial networks (GANs) have attracted huge attention due to its capability to generate visual realistic images. However, most of the existing models suffer from the mode collapse or mode mixture problems. In this work, we give a theoretic explanation of the both problems by Figalli’s regularity theory of optimal transportation maps. Basically, the generator compute the transportation maps between the white noise distributions and the data distributions, which are in general discontinuous. However, DNNs can only represent continuous maps. This intrinsic conflict induces mode collapse and mode mixture. In order to tackle the both problems, we explicitly separate the manifold embedding and the optimal transportation; the first part is carried out using an autoencoder to map the images onto the latent space; the second part is accomplished using a GPUbased convex optimization to find the discontinuous transportation maps. Composing the extended OT map and the decoder, we can finally generate new images from the white noise. This AEOT model avoids representing discontinuous maps by DNNs, therefore effectively prevents mode collapse and mode mixture.more » « less