Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Abstract

Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution better resemble smooth surfaces, they pose computational difficulties for the existing parameterization algorithms. In this work, we propose a novel parallelizable algorithm for computing the global conformal parameterization of simply-connected surfaces via partial welding maps. A given simply-connected surface is first partitioned into smaller subdomains. The local conformal parameterizations of all subdomains are then computed in parallel. The boundaries of the parameterized subdomains are subsequently integrated consistently using a novel technique called partial welding, which is developed based on conformal welding theory. Finally, by solving the Laplace equation for each subdomain using the updated boundary conditions, we obtain a global conformal parameterization of the given surface, with bijectivity guaranteed by quasi-conformal theory. By including additional shape constraints, our method can be easily extended to achieve disk conformal parameterization for simply-connected open surfaces and spherical conformal parameterization for genus-0 closed surfaces. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. When compared to the state-of-the-art conformal parameterization methods, our method achieves a significant improvement in both computational time and accuracy.

References in corpus (9)

• Boundary First Flattening
• TEMPO: Feature-Endowed Teichmüller Extremal Mappings of Point Clouds
• Fast Disk Conformal Parameterization of Simply-connected Open Surfaces
• Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing
• Tooth morphometry using quasi-conformal theory
• A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces
• Density-equalizing maps for simply-connected open surfaces
• Area-preserving mapping of 3D ultrasound carotid artery images using density-equalizing reference map
• Fast Spherical Quasiconformal Parameterization of Genus-0 Closed Surfaces with Application to Adaptive Remeshing

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• Spherical Density-Equalizing Map for Genus-0 Closed Surfaces
• Spherical Cap Harmonic Analysis (SCHA) for Characterising the Morphology of Rough Surface Patches
• Shape analysis via inconsistent surface registration
• Hemispheroidal parameterization and harmonic decomposition of simply connected open surfaces
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• Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces