Counterexamples to Proofs for Volumetric Parameterization of Topological Sweeps

TL;DR

This work questions the validity of bijectivity guarantees for volumetric parameterizations produced by sweep-based harmonic maps in 3D. By constructing a concrete geometry homeomorphic to $Γ_0\times[0,1]$ and computing a discrete sweep harmonic function that satisfies the discrete maximum principle, the authors demonstrate interior critical points (a 1-saddle and a 2-saddle) that induce topology changes in level sets, contradicting claims in prior proofs. They dissect the gaps in Lin 2015 and Xia 2010, showing that saddle-induced topology changes and potential handle cancellations can occur, which invalidates universal bijectivity guarantees for these sweeps. The result clarifies the limitations of swept-domain parameterizations and provides guidance for their use and the development of more robust guarantees in higher dimensions.

Abstract

Harmonic maps are important in generating parameterizations for various domains, particularly in two and three dimensions. General extensions of two-dimensional harmonic parameterizations for volumetric parameterizations are known to fail in a variety of contexts, though more specialized volumetric parameterizations have been proposed. This work provides and contextualizes a counterexample to various proposed proofs that employ harmonic maps to sweep a parameterization from a base surface, $Γ_0$, to the entire domain of a geometry that is homeomorphic to $Γ_0\times[0,1]$ or $Γ_0\times S^1$. While this does not negate the potential value of such topological sweep parameterizations, it does clarify that these swept parameterizations come with no inherent guarantees of bijectivity, as they may in two dimensions.

Figures (4)

• Figure 1: The local effects of 1-saddles and 2-saddles on the level sets of a function for a critical point with value $a$.
• Figure 2: (a) The geometry of the computational counterexample. The geometry is symmetrical in two directions, and not in the third. $\Gamma_0$ is indicated in red, and $\Gamma_1$ in blue. (b) Level sets of the discrete harmonic function at values of 0.0, 0.85, and 1.0. Also shown are the critical points, with an index 1 critical point in red, and an index 2 singularity in blue. The level set $\hat{f}^{-1}(0.85)$, shown in green, has nonzero Betti numbers of $\beta_0=1$, $\beta_1=2$, and $\beta_2=0$, while both $\Gamma_0$ and $\Gamma_1$ have $\beta_0=1$, $\beta_1=0$, and $\beta_2=0$.
• Figure 3: Cancellation of a 1-handle and a 2-handle attached to a 3-manifold. Attaching regions are filled with hash lines, The attaching sphere of each handle is shown in purple, and the belt sphere of each handle is shown in green. (a) Attaching a 1-handle to the manifold. (b) Attaching a 2-handle to the manifold and one-handle. Notice that the attaching sphere of the 2-handle intersects the belt sphere of the 1-handle at a single point. (c) The result of the handle attachment is homeomorphic to the original manifold.
• Figure 4: Sublevel sets of the discrete harmonic function on the counterexample. Compare to Fig. \ref{['fig:handle_cancellation']}. (a) $\hat{f}^{-1}([0,0.7])$. (b) $\hat{f}^{-1}([0,0.85])$. (c) $\hat{f}^{-1}([0,1])=\hat{M}$.