Mathematical framework for large-scale brain network modeling in The Virtual Brain.

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Discrete Surface Ricci Flow

We propose an equivalence of the partition functions of two different 3d gauge theories. On one side of the correspondence we consider the partition function of 3d $ {\text{SL}}\left( {2,\mathbb{R}} \right) $
 Chern-Simons theory on a 3-manifold, obtained as a punctured Riemann surface times an interval. On the other side we have a partition function of a 3d $ \mathcal{N} = 2 $
 superconformal field theory on S
 3, which is realized as a duality domain wall in a 4d gauge theory on S
 4. We sketch the proof of this conjecture using connections with quantum Liouville theory and quantum Teichmuller theory, and study in detail the example of the once-punctured torus. Motivated by these results we advocate a direct Chern-Simons interpretation of the ingredients of (a generalization of) the Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as well as on possible generalizations of our proposals.

SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls

Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simple topology, are subsumed by the Ricci flow-based method, which handles surfaces with arbitrary topology. We present a general framework for the computation of Ricci flow, which can design any Riemannian metric by user-defined curvature. The solution to Ricci flow is unique and robust to noise. We provide implementation details for Ricci flow on discrete surfaces of either Euclidean or hyperbolic background geometry. Our Ricci flow-based method can convert all 3D problems into 2D domains and offers a general framework for 3D shape analysis. We demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration, and shape indexing. Surfaces with large nonrigid anisotropic deformations can be registered using Ricci flow with constraints of feature points and curves. We show how conformal equivalence can be used to index shapes in a 3D surface shape space with the use of Teichmuller space coordinates. Experimental results are shown on 3D face data sets with large expression deformations and on dynamic heart data.

Ricci Flow for 3D Shape Analysis

We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications.

/pdf/beyond-developable-computational-design-and-fabrication-with-2ogs6dsq70.pdf

Beyond developable: computational design and fabrication with auxetic materials

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

https://www.ams.org/tran/2011-363-09/S0002-9947-2011-05239-6/S0002-9947-2011-05239-6.pdf

Local rigidity of inversive distance circle packing

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper This discrete conformality is shown to be computable It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss-Bonnet formula Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric

A discrete uniformization theorem for polyhedral surfaces II

Surface Ricci flow is a powerful tool to design Riemannian metrics by user defined curvatures. Discrete surface Ricci flow has been broadly applied for surface parameterization, shape analysis, and computational topology. Conventional discrete Ricci flow has limitations. For meshes with low quality triangulations, if high conformality is required, the flow may get stuck at the local optimum of the Ricci energy. If convergence to the global optimum is enforced, the conformality may be sacrificed.



This work introduces a novel method to generalize the traditional discrete Ricci flow. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. Conventional method is based on circle packing, which requires two circles on an edge intersect each other at an acute angle. Generalized method allows the two circles either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. Furthermore, the generalized Ricci flow preserves the convexity of the Ricci energy, this ensures the uniqueness of the global optimum. Therefore the algorithm won't get stuck at the local optimum.



Generalized discrete Ricci flow algorithms are explained in details for triangle meshes with both Euclidean and hyperbolic background geometries. Its advantages are demonstrated by theoretic proofs and practical applications in graphics, especially surface parameterization.

/pdf/generalized-discrete-ricci-flow-2p7h2gw8lg.pdf

Generalized Discrete Ricci Flow

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete surface Ricci flow, including all the common schemes: tangential circle packing, Thurston's circle packing, inversive distance circle packing and discrete Yamabe flow. Furthermore, this work also introduces a novel schemes, virtual radius circle packing and the mixed type schemes, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Experimental results show the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.

https://ir.library.oregonstate.edu/downloads/6q182m89b

The unified discrete surface Ricci flow

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson in [Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc. (2004)] as a generalization of Andreev and Thurston’s circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [Trans. Amer. Math. Soc. 363 (2011) 4757–4776] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [arXiv 0612.5714], verifying a conjecture in [arXiv 0612.5714]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.

/pdf/rigidity-of-polyhedral-surfaces-ii-215qa73wp8.pdf

Ren Guo

Papers

Local rigidity of inversive distance circle packing

A discrete uniformization theorem for polyhedral surfaces II

Generalized Discrete Ricci Flow

The unified discrete surface Ricci flow

Rigidity of polyhedral surfaces, II