Optimal cone singularities for conformal flattening

Abstract

Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles ( e.g. , positive, or within a given range), and sophisticated boundary conditions ( e.g. , convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.

2D material programming for 3D shaping

“…The idea is to first map a 3D shape on a cone surface ( K = 0) and then unfold the cone surface to the 2D plane (without further area distortion) by cutting the surface through cone points, rather than directly mapping a shape to the plane (as shown in Fig. 2 ) 19 , 24 . This approach can significantly reduce Ω r , as a 3D shape can be conformally mapped to a cone surface with lower area distortion than to the plane and the cone surface can be isometrically flattened to the plane.…”

Voting for Distortion Points in Geometric Processing

“…When using default values, some important vertices are missed (see Fig. 15 in [20]). In our method, the tradeoff between the number of distortion points and the level of isometric distortion is controlled by a parameter that represents the range of influence of distortion points.…”

Panoramic Image Generation Using Centerline- Constrained Mesh Parameterization for Arbitrarily Shaped Tunnel Lining

“…Reference [32] used curvature to find the cone singular point, and obtained the conformal parameterized scaling factor by solving the Yamabe equation, adding new singular points incrementally each time until the difference of the scaling factor is at a given threshold. Reference [13], [14] studied the problem of how to find the optimal cone singularity from different angles. However, their optimal cone singularity method is affected by some parameters that are not interpreted intuitively.…”

“…Reference [12] proposed a nonlinear local/global algorithm (ARAP) that can preserve the shape of the tunnel mesh, but the algorithm is time consuming for meshes with hundreds of millions of points. There are algorithms that find a cone singularity based on the conformal mapping [13], [14] to reduce distortion, and other algorithms for improving algorithm efficiency of mesh parameterization [15]- [17]. But the above mesh parameterization method is usually applied for a manifold surface, and the engineering data are often non-manifold and multipletiles data, which poses a challenge to the unwrapping of the tunnel mesh.…”