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式形式（即隐式化）在几何建模及相关应用中一直是一个热门但具有挑战性的问题。然而，现有方法大多存在几何特征保持困难以及合理隐式阶数选择的问题。本文包含两项贡献：首先，针对多项式曲线和非多项式曲线引入了一种新的正则化约束（称为弱梯度约束），该约束能有效保持形状；其次，分别针对多项式曲线和非多项式曲线提出了两种自适应近似隐式化算法，这些算法基于弱梯度约束的行为寻找“最优”隐式阶数。更确切地说，其思路是逐渐增加隐式阶数，直至输出的弱梯度损失无明显改善为止。实验结果证明了所提方法的有效性和高质量。