Boundary First Flattening

@article{Sawhney2017BoundaryFF,
  title={Boundary First Flattening},
  author={Rohan Sawhney and Keenan Crane},
  journal={ACM Transactions on Graphics (TOG)},
  year={2017},
  volume={37},
  pages={1 - 14}
}
A conformal flattening maps a curved surface to the plane without distorting angles—such maps have become a fundamental building block for problems in geometry processing, numerical simulation, and computational design. Yet existing methods provide little direct control over the shape of the flattened domain, or else demand expensive nonlinear optimization. Boundary first flattening (BFF) is a linear method for conformal parameterization that is faster than traditional linear methods, yet… 

Optimal cone singularities for conformal flattening

A strategy that minimizes total area distortion among all possible cone configurations that have no more than a fixed total cone angle is developed, demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.

Variational surface cutting

A global variational approach to cutting curved surfaces so that they can be flattened into the plane with low metric distortion, and develops an Eulerian numerical integrator on triangulated surfaces that does not restrict cuts to mesh edges and can incorporate user-defined data such as importance or occlusion.

Globally Injective Flattening via a Reduced Harmonic Subspace

This work presents a highly efficient-and-robust method for free-boundary flattening of disk-like triangle meshes in a globally injective manner, and shows that by restricting the solution to a low-dimensional subspace of harmonic maps, it can dramatically accelerate the process while obtaining a high-distortion result.

Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills

In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the

Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory

  • G. Choi
  • Computer Science
    Journal of Scientific Computing
  • 2021
This work proposes two novel algorithms for computing the conformal parameterization of multiply-connected surfaces using quasi-conformal theory and develops an efficient method for conformally parameterizing an open surface with one hole to an annulus on the plane.

Free-Boundary Conformal Parameterization of Point Clouds

This work proposes a novel approximation scheme of the Laplace--Beltrami operator on point clouds and utilizes it for developing a free-boundary conformal parameterization method for disk-type point clouds, which shows that high-quality point cloud meshing can be easily achieved.

Efficient and robust discrete conformal equivalence with boundary

An efficient algorithm to compute a discrete metric with prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary of a mesh and derive continuous maps from the discrete metrics is described.

Seamless Parametrization with Arbitrarily Prescribed Cones

It is demonstrated that for arbitrary given sets of topologically admissible parametric cones with prescribed curvature, a global seamless parametrization always exists (with the exception of one well-known case).

Computing sparse integer-constrained cones for conformal parameterizations

A novel method to generate sparse integer-constrained cone singularities with low distortion constraints for conformal parameterizations is proposed and a novel solver is developed to optimize the ℓ0-norm without any approximation.

Conformal geometry of simplicial surfaces

What information about a surface is encoded by angles, but not lengths? This question encapsulates the basic viewpoint of conformal geometry, which studies holomorphic or (loosely speaking) angleand
...

References

SHOWING 1-10 OF 64 REFERENCES

Iterative Closest Conformal Maps between Planar Domains

This work proposes an alternating minimization algorithm for finding a boundary‐approximating conformal map given only an initial global alignment of the two input domains, and utilizes the Cauchy‐Green complex barycentric coordinates to parameterize the space of conformal maps from the source domain, to compute a continuous map without requiring the discretization of the domain, and without mapping to intermediate domains.

Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening

A new method to compute planar triangulations of faceted surfaces for surface parameterization that defines the flattening problem as a constrained optimization problem in terms of angles (only), which makes the method more stable and robust than previous approaches, which used node locations in their formulations.

Beyond developable

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, which can handle non-trivial topology and non-local dependencies inherent in auxetic material.

Conformal Geometry Processing

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in the theory and yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data.

Conformal Flattening by Curvature Prescription and Metric Scaling

This work presents an efficient method to conformally parameterize 3D mesh data sets to the plane by concentrating all the 3D curvature at a small number of select mesh vertices, called cone singularities, and cutting the mesh through those singular vertices to obtain disk topology.

Controllable conformal maps for shape deformation and interpolation

This work describes a novel 2D shape deformation system which generates conformal maps, yet provides the user a large degree of control over the result, and provides the ability to interpolate between shapes in a very natural way, such that also the intermediate deformations are conformal.

ABF++: fast and robust angle based flattening

ABF++ robustly parameterizes mesh models of hundreds of thousands and millions of triangles within minutes, and is extremely suitable for robustly and efficiently parameterizing models forgeometry-processing applications.

Intrinsic Parameterizations of Surface Meshes

This paper presents new theoretical and practical results on the parameterization of triangulated surface patches and proposes robust, efficient and tunable tools to obtain least‐distorted parameterizations automatically.

Least squares conformal maps for automatic texture atlas generation

This paper introduces a new quasi-conformal parameterization method, based on a least-squares approximation of the Cauchy-Riemann equations, which can parameterize large charts with complex borders, and introduces segmentation methods to decompose the model into charts with natural shapes, and a new packing algorithm to gather them in texture space.

Bijective parameterization with free boundaries

We present a fully automatic method for generating guaranteed bijective surface parameterizations from triangulated 3D surfaces partitioned into charts. We do so by using a distortion metric that
...