Converting a parametric curve into the implicit form, which is called implicitization, has always been a popular but challenging problem in geometric modeling and related applications. However, the existing methods mostly suffer from the problems of maintaining geometric features and choosing a reasonable implicit degree. The present paper has two contributions. We first introduce a new regularization constraint(called the weak gradient constraint) for both polynomial and non-polynomial curves, which efficiently possesses shape preserving. We then propose two adaptive algorithms of approximate implicitization for polynomial and non-polynomial curves respectively, which find the ``optimal'' implicit degree based on the behavior of the weak gradient constraint. More precisely, the idea is gradually increasing the implicit degree, until there is no obvious improvement in the weak gradient loss of the outputs. Experimental results have shown the effectiveness and high quality of our proposed methods.
翻译:将参数曲线转换为隐含形式,即所谓的隐含性,在几何建模和相关应用中始终是一个普遍但具有挑战性的问题;然而,现有方法大多存在保持几何特征和选择合理隐含程度的问题;本文件有两种贡献;我们首先对多元曲线和非多极曲线实行新的正规化限制(称为微弱梯度限制),这种限制有效地保持了形状;然后我们提出了两种适应性算法,即对多元曲线和非多极性曲线的近似隐含性,这两类算法基于薄弱梯度限制的行为,发现“最佳”的隐含程度;更确切地说,这种设想正在逐渐增加隐含程度,直到产出的微梯度损失没有明显改善。实验结果表明我们拟议方法的有效性和高质量。