Temporal sequences of terrains arise in various application areas. To analyze them efficiently, one generally needs a suitable abstraction of the data as well as a method to compare and match them over time. In this paper we consider merge trees as a topological descriptor for terrains and the interleaving distance as a method to match and compare them. An interleaving between two merge trees consists of two maps, one in each direction. These maps must satisfy ancestor relations and hence introduce a ''shift'' between points and their image. An optimal interleaving minimizes the maximum shift; the interleaving distance is the value of this shift. However, to study the evolution of merge trees over time, we need not only a number but also a meaningful matching between the two trees. The two maps of an optimal interleaving induce a matching, but due to the bottleneck nature of the interleaving distance, this matching fails to capture local similarities between the trees. In this paper we hence propose a notion of local optimality for interleavings. To do so, we define the residual interleaving distance, a generalization of the interleaving distance that allows additional constraints on the maps. This allows us to define locally correct interleavings, which use a range of shifts across the two merge trees that reflect the local similarity well. We give a constructive proof that a locally correct interleaving always exists.

翻译：地形的时间序列在多个应用领域中涌现。为高效分析这些序列，通常需要合适的数据抽象方法以及随时间比较和匹配它们的技术。本文以合并树作为地形的拓扑描述符，并采用交错距离作为匹配与比较的手段。两个合并树之间的交错由两个映射构成，每个方向各一个。这些映射必须满足祖先关系，从而在点与其像之间引入“偏移”。最优交错最小化最大偏移；交错距离即为此偏移值。然而，为研究合并树随时间的演化，我们不仅需要数值度量，还需两树间有意义的匹配。最优交错的两个映射可导出匹配，但由于交错距离的瓶颈特性，该匹配无法捕捉树间的局部相似性。为此，本文提出交错局部最优性的概念。通过定义残差交错距离——一种允许对映射施加额外约束的交错距离推广形式，我们得以定义局部正确交错，其利用跨越两合并树的一系列偏移，从而良好反映局部相似性。我们通过构造性证明，局部正确交错总是存在。