In this work, we present a translation of the complete pipeline for variational shape approximation (VSA) to the setting of point sets. First, we describe an explicit example for the theoretically known non-convergence of the currently available VSA approaches. The example motivates us to introduce an alternate version of VSA based on a switch operation for which we prove convergence. Second, we discuss how two operations—split and merge—can be included in a fully automatic pipeline that is in turn independent of the placement and number of initial seeds. Third and finally, we present two approaches how to obtain a simplified mesh from the output of the VSA procedure. This simplification is either based on simple plane intersection or based on a variational optimization problem. Several qualitative and quantitative results prove the relevance of our approach.

在这项工作中，我们提出了完整管道的转换，以将变分形状近似（VSA）转换为点集。首先，我们为当前可用的VSA方法的理论上已知的不收敛性描述一个明确的示例。该示例激励我们基于切换操作引入VSA的替代版本，我们证明了该版本的收敛性。其次，我们讨论两个操作如何拆分和合并-可以包含在全自动管道中，该管道又与初始种子的放置和数量无关。第三，也是最后，我们提出了两种方法，如何从VSA过程的输出中获得简化的网格。这种简化是基于简单的平面相交，还是基于变分优化问题。几个定性和定量结果证明了我们方法的相关性。