An Algorithm for Fast and Correct Computation of Reeb Spaces for PL Bivariate Fields

Reeb space is an important tool (data-structure) for topological data analysis that captures the quotient space topology of a multi-field or multiple scalar fields. For piecewise-linear (PL) bivariate fields, the Reeb spaces are $2$-dimensional polyhedrons while for PL scalar fields, the Reeb graphs (or Reeb spaces) are of dimension $1$. Efficient algorithms have been designed for computing Reeb graphs, however, computing correct Reeb spaces for PL bivariate fields, is a challenging open problem. 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 intersections 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 $\mathbb{M}$. This complexity is comparable with the fastest algorithm available in the literature. Moreover, we claim to provide the first algorithm to compute the topologically correct Reeb space without using range quantization.