  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.
