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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 finds the most economical way to transport one probability measure to another, and it plays an important role in geometric modeling and processing. In this paper, we propose a moving mesh method to generate adaptive meshes by optimal transport. Given an initial mesh and a scalar density function defined on the mesh domain, our method will redistribute the mesh nodes such that they are adapted to the density function. Based on the Brenier theorem, solving an optimal transportation problem is reduced to solving a MongeAmp\`ere equation, which is difficult to compute due to the high nonlinearity. On the other hand, the optimal transportation problem is equivalent to the Alexandrov problem, which can finally induce an effective variational algorithm. Experiments show that our proposed method finds the adaptive mesh quickly and efficiently.more » « less

null (Ed.)An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a MongeAmp\`ere equation. Due to the highly nonlinear nature, the computation of optimal transportation maps in large scale is very challenging. This work proposes a simple but powerful method, the FFTOT algorithm, to tackle this difficulty based on three key ideas. First, solving MongeAmp\`ere equation is converted to a fixed point problem; Second, the obliqueness property of optimal transportation maps are reformulated as Neumann boundary conditions on rectangular domains; Third, FFT is applied in each iteration to solve a Poisson equation in order to improve the efficiency. Experiments on surfaces captured from 3D scanning and reconstructed from medical imaging are conducted, and compared with other existing methods. Our experimental results show that the proposed FFTOT algorithm is simple, general and scalable with high efficiency and accuracy.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.)Biomarkers play an important role in early detection and intervention in Alzheimer’s disease (AD). However, obtaining effective biomarkers for AD is still a big challenge. In this work, we propose to use the worst transportation cost as a univariate biomarker to index cortical morphometry for tracking AD progression. The worst transportation (WT) aims to find the least economical way to transport one measure to the other, which contrasts to the optimal transportation (OT) that finds the most economical way between measures. To compute the WT cost, we generalize the Brenier theorem for the OT map to the WT map, and show that the WT map is the gradient of a concave function satisfying the MongeAmpere equation. We also develop an efficient algorithm to compute the WT map based on computational geometry. We apply the algorithm to analyze cortical shape difference between dementia due to AD and normal aging individuals. The experimental results reveal the effectiveness of our proposed method which yields better statistical performance than other competiting methods including the OT.more » « less

null (Ed.)Shape analysis has been playing an important role in early diagnosis and prognosis of neurodegenerative diseases such as Alzheimer's diseases (AD). However, obtaining effective shape representations remains challenging. This paper proposes to use the Alexandrov polyhedra as surfacebased shape signatures for cortical morphometry analysis. Given a closed genus0 surface, its Alexandrov polyhedron is a convex representation that encodes its intrinsic geometry information. We propose to compute the polyhedra via a novel spherical optimal transport (OT) computation. In our experiments, we observe that the Alexandrov polyhedra of cortical surfaces between pathologyconfirmed AD and cognitively unimpaired individuals are significantly different. Moreover, we propose a visualization method by comparing local geometry differences across cortical surfaces. We show that the proposed method is effective in pinpointing regional cortical structural changes impacted by AD.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