Objects interact with each other in various ways, including containment,
contact, or maintaining fixed distances. Ensuring these topological
interactions is crucial for accurate modeling in many scenarios. In this paper,
we propose a novel method to refine 3D object representations, ensuring that
their surfaces adhere to a topological prior. Our key observation is that the
object interaction can be observed via a stochastic approximation method: the
statistic of signed distances between a large number of random points to the
object surfaces reflect the interaction between them. Thus, the object
interaction can be indirectly manipulated by using choosing a set of points as
anchors to refine the object surfaces. In particular, we show that our method
can be used to enforce two objects to have a specific contact ratio while
having no surface intersection. The conducted experiments show that our
proposed method enables accurate 3D reconstruction of human hearts, ensuring
proper topological connectivity between components. Further, we show that our
proposed method can be used to simulate various ways a hand can interact with
an arbitrary object.

通过一种新方法，我们提出了一种改进 3D 对象表示的方法，保证其表面遵循拓扑先验性，以确保对象之间的拓扑相互作用。我们的关键观察是，可以通过随机逼近方法观察到对象之间的相互作用：大量随机点到对象表面的符号距离的统计数据反映了它们之间的相互作用。因此，可以通过选择一组点作为锚点以改进对象表面，间接操纵对象之间的相互作用。特别地，我们展示了我们的方法可以在两个对象之间强制实现特定的接触比例而没有表面交叉。进行的实验证明了我们的方法可以准确重建人类心脏，保证组件之间的正确拓扑连接。此外，我们展示了我们的方法可以用于模拟手与任意对象互动的各种方式。