Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological summaries cannot and 2) to still be robust to perturbations in the dataset. The combination of these two properties, known respectively as stability and discriminativity, has led to theoretical distances which are either thought to be or shown to be computationally complex and thus their implementations have been scarce. In order to design similarity measures on merge trees which are computationally feasible for more complex merge trees, many researchers have elected to loosen the restrictions on at least one of these two properties. The question still remains, however, if there are practical situations where trading these desirable properties is necessary. Here we construct a distance between merge trees which is designed to retain both discriminativity and stability. While our approach can be expensive for large merge trees, we illustrate its use in a setting where the number of nodes is small. This setting can be made more practical since we also provide a proof that persistence simplification increases the outputted distance by at most half of the simplified value. We demonstrate our distance measure on applications in shape comparison and on detection of periodicity in the von K\'arm\'an vortex street.
翻译:合并树上的距离有助于对斜体字段的收藏进行视觉比较。 这些距离展示的两种可取的特性是:(1) 能够辨别出其他不那么复杂的地形摘要所不能区分的斜体字段,(2) 仍然能够对数据集的扰动保持稳健。这两种特性的结合,分别称为稳定性和歧视性,导致理论距离,被认为或显示为计算复杂,因此其执行很少。为了设计合并树的相似性措施,这些措施在计算上对于更复杂的合并树是可行的,许多研究人员选择至少放宽对这两种特性之一的限制。然而,问题仍然存在,如果在实际情况下有必要进行这些可取特性的交易。我们在这里在合并树之间建造一段距离,目的是既保持歧视性又保持稳定性。虽然我们的方法对于大合并树来说可能很昂贵,但我们可以用它来说明在偏小的地方使用。这个环境可以更实际化,因为我们还提供了一个证据,即持续的简化会增加产出距离,而最接近简化值的一半。我们展示了我们应用的距离尺度,用于形状对比和趋势的周期。