Boundaries Constrained Low-Distortion Parameterization

Computer-aided geometric design is the foundation of modern computer modeling and design.It is an important technology in the fields of manufacturing,industrial Internet,game design,and film and television production.As discrete data such as point clouds and meshes become easier to obtain,the large amount of data promotes the development of reverse modeling design.In the inverse modeling design,Many problems can be described as the boundaries constrained low-distortion parameterization.In these problems,the two complex issues of boundary constraints and low-distortion parameterization must be considered simultaneously.To solve such problem,we avoid constructing the boundary constraints directly.Instead,the computation of the boundary constraints is hidden in the optimization process of low-distortion parameterization.We study plane parameterization,volume parameterization,and polycube parameterization of boundary constraints and their specific applications.First,we propose a new boundary-constrained plane parameterization cut generation problem.This problem is to find the optimal cut lines on spherical so that the unfolded the shape of the cut spherical is similar to the input plane shape.The problem is from an art form called Citrus Peeling Art.Some artists peel citrus fruits into a variety of elegant 2D shapes,depicting animals,plants,and cartoons.This art form follows the conservation principle,i.e.,each shape must be created using one entire peel.Our key insight is that instead of solving the difficult cut generation problem,we map a designed input shape onto a citrus in an attempt to cover the entire citrus and use the mapped boundary to generate the cut paths.As an application of this problem,we present a computational method for citrus peeling art designs.Sometimes,a mapped shape is unable to completely cover a citrus.Consequently,we have developed five customized ways of interaction that are used to rectify the input shape so that it is suitable for citrus peeling art.The mapping process and user interactions are iteratively conducted to satisfy a user’s design intentions.A large number of experiments,including a formative user study,demonstrate the capability and practicability of our method.Next,we focus on bijective volumetric parameterization for isogeometric analysis.We present a novel method to compute bijective volumetric domain parameterizations with low distortion for isogeometric analysis.Central to this approach is a simultaneous optimization of the interior and the boundary mappings.The simultaneous optimization is formulated as a constrained optimization problem.The objective is a combination of a distortion metric of the volumetric domain parameterization and similarity error between the mapped parametric domain and the computational domain.Bijection is the constraint.The success of our method to solve this very challenging problem relies on two key components:(1)a tet-to-spline optimization strategy that mitigates the difficulty to handle the similarity error and(2)a three-step initialization procedure that enables effective distortion reduction.We demonstrate the efficacy of our method on various complex models.Compared to state-of-the-art methods,our method achieves bijective volumetric domain parameterizations with much lower distortion.Finally,we propose a method for computing low-distortion spline fits arbitrary topological inputs by boundary-constrained polycube parameterization.This method uses the unstructured T-splines and polycube structures to fit arbitrary topologies input.Central to this method is solves spline distortion optimization as a boundaryconstrained polycube re-parameterization.By performing re-parameterization based on the former fitting result,the correspondences between the data points and the surface points are optimized and the distortion of the spline is reduced.To solve this boundaryconstrained problem,The tet-to-spline optimization strategy is used again to incorporate the bounded fitting error constraints into the optimization,and finally,the adaptive subdivision is performed to make the fitting error be bounded.This method can generate low-distortion spline fittings for arbitrarily complex inputs while satisfying the fitting accuracy.Compared with the state-of-the-art methods,the fitting results have smaller distortion.