Abstract
Unless the targeted mesh is developable, metric distortion is inevitable during the process of surface mesh parameterization, thus one important objective of all involved parametric studies is to reduce the metric distortion. In order to further reduce area and angle distortion, a novel method of boundary-free mesh parameterization is presented in the paper. Firstly, the initial boundary-fixed conformal parameterization from 3D surface mesh patch to a plane is performed in the method. Then, based on the initial parameterization, the iterations of boundary-free quasi-harmonic parameterization are developed, where the tensor field is updated in each iterative step and the principal curvature direction is utilized to terminate the iteration. The solution of the novel method is convenient to calculate since it involves a series of linear systems. In our novel parameterization method, lower metric distortion and considerable efficiency have been obtained in experiments.
Highlights
Mesh parameterization involves computing the mapping between a triangulated mesh surface and certain parametric domain
We represent the total area distortion of the surface mesh by (Γγ)m ((Γ/γ)m) which is the average value of area distortion parameterized on all the triangles of the mesh, for reason that the average value is just a determinant of the measurement distortion
To describe the metric distortion of a mesh patch as a whole, we utilized the average value of area distortion of parameterization to represent the metric distortion, while not considering the variance and the maximum of distortion on all the triangles which only have local significance
Summary
Mesh parameterization involves computing the mapping between a triangulated mesh surface and certain parametric domain. It is well known that, unless the targeted surface mesh is developable, mesh parameterization inevitably incurs some metric distortions in both angle and area. The goal of the paper is to calculate the boundary-free parameterization which maintains angle-preserving and area-preserving (i.e., isometric) as much as possible. To better understand metric distortion in parameterization, let us see what happens to the surface point f(u,v) as we move a tiny little away from (u, v) in the parameter domain. If we denote this infinitesimal parameter displacement by(Δu,Δv), the new surface point f(Δu,Δv) is approximately given by the first order Taylor expansion f of f around (u,v),
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