The Reeb space is a fundamental data structure in computational topology that represents the fiber topology of a multi-field (or multiple scalar fields), extending the level set topology of a scalar field. Efficient algorithms have been designed for computing Reeb graphs, however, computing correct Reeb spaces for PL bivariate fields, is a challenging open problem. There are only a few implementable algorithms in the literature for computing Reeb space or its approximation, via range quantization or by computing a Jacobi fiber surface, which are computationally expensive or have correctness issues, i.e., the computed Reeb space may not be topologically equivalent or homeomorphic to the actual Reeb space. In the current paper, we propose a novel algorithm for fast and correct computation of the Reeb space corresponding to a generic PL bivariate field defined on a triangulation $\mathbb{M}$ of a $3$-manifold without boundary, leveraging the fast algorithms for computing Reeb graphs in the literature. Our algorithm is based on the computation of a Multi-Dimensional Reeb Graph (MDRG) which is first proved to be homeomorphic with the Reeb space. For the correct computation of the MDRG, we compute the Jacobi set of the PL bivariate field and its projection into the Reeb space, called the Jacobi structure. Finally, the correct Reeb space is obtained by computing a net-like structure embedded in the Reeb space and then computing its $2$-sheets in the net-like structure. The time complexity of our algorithm is $\mathcal{O}(n^2 + n\, c_{int}\, \log n + nc_L^2)$, where $n$ is the total number of simplices in $\mathbb{M}$, $c_{int}$ is the number of intersection points of the projections of the non-adjacent Jacobi set edges on the range of the bivariate field and $c_L$ is the upper bound on the number of simplices in the link of an edge of $\mathbb{M}$.

翻译：Reeb空间是计算拓扑学中的一种基础数据结构，用于表示多场（或多标量场）的纤维拓扑结构，它扩展了标量场的水平集拓扑。尽管已有高效算法用于计算Reeb图，但为分段线性双变量场计算正确的Reeb空间仍是一个具有挑战性的开放性问题。现有文献中仅有少数可实现的算法通过值域量化或计算Jacobi纤维曲面来近似计算Reeb空间，这些方法计算代价高昂或存在正确性问题——即计算结果可能与实际Reeb空间缺乏拓扑等价性或同胚性。本文提出一种新颖算法，能够快速精确地计算定义在无边界3-流形三角剖分$\mathbb{M}$上的通用分段线性双变量场所对应的Reeb空间，该算法充分利用了现有文献中计算Reeb图的快速算法。我们的算法基于多维Reeb图的计算，首先证明了该结构与Reeb空间同胚。为确保MDRG的正确计算，我们计算了分段线性双变量场的Jacobi集及其在Reeb空间中的投影（称为Jacobi结构）。最终通过计算嵌入Reeb空间的网状结构及其二维片层，获得正确的Reeb空间。算法时间复杂度为$\mathcal{O}(n^2 + n\, c_{int}\, \log n + nc_L^2)$，其中$n$为$\mathbb{M}$中单纯形的总数，$c_{int}$表示非相邻Jacobi集边在双变量场值域上投影的交点数量，$c_L$为$\mathbb{M}$中边链环内单纯形数量的上界。