The Reeb space is a topological structure which is a generalization of the notion of the Reeb graph to multi-fields. Its effectiveness has been established in revealing topological features in data across diverse computational domains which cannot be identified using the Reeb graph or other scalar-topology-based methods. Approximations of Reeb spaces such as the Mapper and the Joint Contour Net have been developed based on quantization of the range. However, computing the topologically correct Reeb space dispensing the range-quantization is a challenging problem. In the current paper, we develop an algorithm for computing a correct net-like approximation corresponding to the Reeb space of a generic piecewise-linear (PL) bivariate field based on a multi-dimensional Reeb graph (MDRG). First, we prove that the Reeb space is homeomorphic to its MDRG. Subsequently, we introduce an algorithm for computing the MDRG of a generic PL bivariate field through the computation of its Jacobi set and Jacobi structure, a projection of the Jacobi set into the Reeb space. This marks the first algorithm for MDRG computation without requiring the quantization of bivariate fields. Following this, we compute a net-like structure embedded in the corresponding Reeb space using the MDRG and the Jacobi structure. We provide the proof of correctness and complexity analysis of our algorithm.

翻译：Reeb空间是一种拓扑结构，是对Reeb图概念向多场情形的推广。其有效性已在多个计算领域的数据拓扑特征揭示中得到验证，这些特征无法通过Reeb图或其他基于标量拓扑的方法识别。基于值域量化的Mapper和联合等值线网等Reeb空间近似方法已被提出。然而，在无需值域量化的前提下计算拓扑正确的Reeb空间仍是一项具有挑战性的问题。本文基于多维Reeb图（MDRG）提出了一种算法，用于计算一般分片线性（PL）双变量场对应Reeb空间的类网格正确近似。首先，我们证明了Reeb空间与其MDRG同胚。随后，通过计算一般PL双变量场的Jacobi集及Jacobi结构（Jacobi集在Reeb空间中的投影），引入了一种计算其MDRG的算法。这标志着首个无需对双变量场进行量化的MDRG计算算法。在此基础上，我们利用MDRG与Jacobi结构在对应Reeb空间中嵌入类网格结构。本文给出了算法的正确性证明与复杂度分析。