Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees

Abstract

This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on a volumetric mesh M in R3. We introduce a procedure called "loop surgery" that transforms M into a mesh M' by a sequence of cuts and guarantees the Reeb graph of f(M') to be loop free. Therefore, loop surgery reduces Reeb graph computation to the simpler problem of computing a contour tree, for which well-known algorithms exist that are theoretically efficient (O(n log n)) and fast in practice. Inverse cuts reconstruct the loops removed at the beginning. The time complexity of our algorithm is that of a contour tree computation plus a loop surgery overhead, which depends on the number of handles of the mesh. Our systematic experiments confirm that for real-life data, this overhead is comparable to the computation of the contour tree, demonstrating virtually linear scalability on meshes ranging from 70 thousand to 3.5 million tetrahedra. Performance numbers show that our algorithm, although restricted to volumetric data, has an average speedup factor of 6,500 over the previous fastest techniques, handling larger and more complex data-sets.We demonstrate the versatility of our approach by extending fast topologically clean iso surface extraction to non simply-connected domains. We apply this technique in the context of pressure analysis for mechanical design. In this case, our technique produces results in matter of seconds even for the largest meshes. For the same models, previous Reeb graph techniques do not produce a result.

Horizontal decomposition of triangulated solids for the simulation of dip-coating processes

“…The resulting meshes were decomposed into 310 flow volumes within 0.5 to 5 seconds; see also Section 8.2. Thus, our algorithm compares favorably with existing algorithms for the computation of Reeb graphs of 3-manifolds with boundary, although the algorithms are not directly comparable since they use different input (surface/volumetric mesh) and produce different output (Reeb graph of interior/exterior of the solid): We process the mesh of 3 million triangles within 5 seconds, while the state-of-the-art algorithm presented in [35] processes an object of similar complexity, given as mesh of 3.5 million tetrahedra, in 7.8 seconds.…”

“…These algorithms use volumetric descriptions of the object, e.g. tetrahedral meshes [9,26,15,16,35], or voxel based representations [30], and they are able to deal with general Morse functions. In the case of flow volumes, where the Morse function is the height function, using a boundary representation is sufficient and simplifies the computation.…”

A randomized O ( m log m ) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

“…We compare the performance of our algorithm with that of two state-of-the-art algorithms: the output-sensitive algorithm proposed in [9], denoted by OS, and the loop-surgery algorithm proposed in [26], denoted by LS. Algorithm OS can handle arbitrary simplicial complexes.…”

“…The same algorithm can handle arbitrary simplicial complexes, but the bound of O(m log m + L) will then not hold. Very recently in [26], Tierny et al proposed an algorithm that computes the Reeb graph for a 3-manifold with boundary embedded in IR 3 in time O(m log m + hm), where h is number of independent loops in the Reeb graph. Their approach leverages the efficient contour tree algorithm in [5] by using a novel surgery idea to first cut the input simplicial complex so that its Reeb graph is loop-free.…”

Feature-Based Uncertainty Visualization

“…A contour tree [4] is a loop free case of a Reeb graph [31] which describes the hierarchical relationship of contours. A MScomplex decomposes a scalar field into quadrangular cells with uniform gradient flow behavior [32], [33].…”