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

TL;DR

This paper tackles the problem of computing the Reeb space for a generic PL bivariate field on a triangulated 3-manifold, a 2D quotient capturing fiber topology that traditional range-quantization methods can misrepresent. It introduces the Multi-Dimensional Reeb Graph (MDRG), proves its homeomorphism to the Reeb space, and develops a pipeline that includes computing the Jacobi structure, augmenting RGs, embedding second-dimensional Reeb graphs via a net-like Jacobi skeleton, and reconstructing the Reeb space through 2-sheets without range quantization, achieving topological correctness. Key theoretical contributions include a proof of homeomorphism between the Reeb space and MDRG and a characterization of topology-change points on RGs, guiding four main algorithms: Jacobi structure, MDRG, net-like structure, and 2-sheet Reeb-space reconstruction, with a complexity bound of $O\bigl(n^2 + n c_{int} \log n + n c_L^2\bigr)$ where $n$ is the total number of simplices, $c_{int}$ the number of Jacobi-edge projections intersections, and $c_L$ the link-size bound. The approach provides a provably correct alternative to quantized methods, with practical impact for multivariate topology analysis and visualization across domains requiring accurate joint-field representations. Overall, the work establishes a rigorous, quantization-free framework for extracting fully correct Reeb spaces and offers a foundation for extending to higher-dimensional or more general PL multivariate fields.

Abstract

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 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 $\mathbb{M}$.

Figures (12)

• Figure 1: Example of a Jacobi fiber surface and the corresponding Reeb space near a double point of the Jacobi structure for a bivariate field, as illustrated in Fig. \ref{['fig:violation-second-Morse-condition']}. (a) The left-hand figure shows the inverse image of the Jacobi structure edges (in blue), along with two regular edges (in green) in the Reeb space. The Jacobi fiber surfaces associated with two disjoint Jacobi set edges (blue) intersect along a singular fiber (shown in red). (b) The right-hand figure depicts the projection of the Jacobi set edges—i.e., the Jacobi structure edges—into the Reeb space, where they intersect at a double point (red). The 2-sheets of the Reeb space (shown in different colors) meet along the intersecting Jacobi structure edges.
• Figure 2: A classification list of local structures of the Reeb space for the smooth stable map case. The horizontal direction corresponds to $pr_1\circ\mathbf{f}$ and the vertical direction to $pr_2\circ\mathbf{f}$, where $pr_i$ projects the range of $\mathbf{f}$ onto the range of $f_i$, for $i=1,2$. Red curves depict the Jacobi structure and the thick graphs on the left and the right-hand sides depict the corresponding Reeb graphs of $f_2$, restricted to contours of $f_1$. Each figure with the letter "C" contains the image of exactly one critical point of $f_1$. There are also up-side down or right-left reversed versions (see Kushner-1984, Levine-2006 for more details).
• Figure 3: (a) Each function corresponding to two topologically equivalent Reeb graphs has the same set of indices corresponding to its critical points, (b) Converse is not true: each function corresponding to two Reeb graphs has the same set of indices corresponding to its critical points, but the Reeb graphs are not topologically equivalent.
• Figure 4: Topological changes in the second-dimensional Reeb graphs of the MDRG for a bivariate field $\mathbf{f} = (f_1, f_2)$ due to saddle critical points of $f_1$. In (a) and (b), points along arcs of $\mathcal{RG}_{f_1}$ are shown on the left. On the right, the top row shows contours of $f_1$ colored based on the values of $f_2$, critical points of $f_2$ restricted to the contours of $f_1$, and the connectivity between the critical points based on the segments of the Jacobi set $\mathbb{J}_{\mathbf{f}}$. The middle row displays the corresponding second-dimensional Reeb graphs, while the Jacobi structure $\mathcal{J}_{\mathbf{f}}$ is presented in the bottom row. Dotted lines illustrate the relationship between the critical points, Reeb graph nodes, and the points in $\mathcal{J}_{\mathbf{f}}$. In both cases, a topological change in the second-dimensional Reeb graphs occurs at the node $p_3$ of $\mathcal{RG}_{f_1}$ due to a saddle critical point of $f_1$. In (a), this critical point causes a split of a contour into two, thereby making $p_3$ an up-fork. In (b), the critical point causes a change in the genus of the contour of $f_1$ due to the addition of a handle, making $p_3$ a degree-$2$ node of $\mathcal{RG}_{f_1}$.
• Figure 5: Topological changes in the second-dimensional Reeb graphs of the MDRG for a bivariate field $\mathbf{f} = (f_1, f_2)$ due to the violation of the first Morse condition. (a) Points along an arc of $\mathcal{RG}_{f_1}$. (b) and (c) depict the birth of an arc in the second-dimensional Reeb graphs: (b) involving a minimum and down-fork, and (c) involving an up-fork and maximum. In both (b) and (c), the top row shows contours of $f_1$ colored based on the values of $f_2$, critical points of $f_2$ restricted to the contours of $f_1$, and the connectivity between the critical points based on the segments of the Jacobi set $\mathbb{J}_{\mathbf{f}}$. The middle row displays the corresponding second-dimensional Reeb graphs, while the Jacobi structure $\mathcal{J}_{\mathbf{f}}$ is presented in the bottom row. Dotted lines illustrate the relationship between the critical points, Reeb graph nodes, and the points in $\mathcal{J}_{\mathbf{f}}$. In both cases, the birth event is captured by a minimum of $f_1$ restricted to $\mathbb{J}_{\mathbf{f}}$.

Theorems & Definitions (12)

• Lemma 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Proposition 3.3
• Definition 1
• Lemma 3.4
• proof
• Lemma 3.5
• proof