Density-equalizing maps for simply-connected open surfaces

  title={Density-equalizing maps for simply-connected open surfaces},
  author={Gary Pui-Tung Choi and Chris H. Rycroft},
In this paper, we are concerned with the problem of creating flattening maps of simply connected open surfaces in $\mathbb{R}^3$. Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing maps with any prescribed density distribution. By varying the initial density distribution, a large variety of flattening maps with different properties can be achieved. For instance, area-preserving parameterizations of simply connected open… 

A Constructive Algorithm for Disk Conformal Parameterizations

An efficient constructive algorithm for the computation of disk conformal parameterizations of simply connected open surfaces is proposed by combining the spherical harmonic mapping of the doubly covered surface and the geodesic algorithm.


A novel energy minimization algorithm for volume-preserving parameterizations of simply connected three-manifolds with a single boundary is developed and can also be applied to compute spherical angle and area- Preserving parameterization of genus zero closed surfaces, respectively.

Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory

  • G. Choi
  • Computer Science
    J. Sci. Comput.
  • 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.

Volumetric Density-Equalizing Reference Map with Applications

By exploiting the time-dependent nature of the proposed method, applications to shape morphing can be easily achieved and medical and sociological data can be visualized via deformations of 3D objects.

Efficient bijective parameterizations

A novel method to efficiently compute bijective parameterizations with low distortion on disk topology meshes relies on a second-order solver and develops a coarse shell to substantially reduce the number of collision constraints that are used to guarantee overlap-free boundaries.

Parallelizable global quasi-conformal parameterization of multiply-connected surfaces via partial welding

A novel parallelizable algorithm for computing the global conformal and quasi-conformal parameterization of multiply-connected surfaces onto a 2D circular domain using variants of the partial welding method and the Koebe’s iteration is proposed.

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.

Optimized surface parameterizations with applications to Chinese virtual broadcasting

Comparisons of the BEM algorithm with the angle and the area- Preserving parameterizations show that the angular distortion is close to that of the angle-preserving parameterization while the area distortion is significantly improved.

A unifying framework for n-dimensional quasi-conformal mappings

A variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems is proposed.



Fast Disk Conformal Parameterization of Simply-Connected Open Surfaces

A novel algorithm is proposed for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity.

A linear formulation for disk conformal parameterization of simply-connected open surfaces

An efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces by a composition of quasi-conformal mappings and applications to texture mapping are presented for illustrating the effectiveness of the proposed algorithm.

Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing

A fast algorithm for computing a spherical parameterization of the surface that satisfies the prescribed distortion is proposed and can be effectively applied to adaptive surface remeshing for improving the visualization in computer graphics and animations.

Authalic Parameterization of General Surfaces Using Lie Advection

A novel area-preserving surface parameterization method which is rigorous in theory, moderate in computation, yet easily extendable to surfaces of non-disc and closed-boundary topologies and provides a competitive alternative to the existing parameterization techniques for better surface-based analysis in various scenarios.

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

Boundary First Flattening

BFF opens the door to real-time editing or fast optimization of high-resolution maps, with direct control over boundary length or angle, and it is shown how this method can be used to construct maps with sharp corners, cone singularities, minimal area distortion, and uniformization over the unit disk.

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.

Optimal global conformal surface parameterization

This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces that relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories.

Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology

This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes and designs and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology.

Weighted Triangulations for Geometry Processing

It is demonstrated how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing.