Greedy Cut Construction for Parameterizations

Abstract

We present a novel method to construct short cuts for parameterizations with low isometric distortion. The algorithm contains two steps: (i) detect feature points, where the distortion is usually concentrated; and (ii) construct a cut by connecting the detected feature points. Central to each step is a greedy method. After generating a redundant feature point set, a greedy filtering process is performed to identify the feature points required for low isometric distortion parameterizations. This filtering process discards the feature points that are useless for distortion reduction while still enabling us to obtain low isometric distortion. Next, we formulate the process of connecting the detected feature points as a Steiner tree problem. To find an approximate solution, we first successively and greedily produce a collection of auxiliary points. Then, a cut is constructed by connecting the feature points and auxiliary points. In the 26,299 test cases in which an exact solution to the Steiner tree problem is available, the length of the cut obtained by our method is on average 0.17% longer than optimal. Compared to state‐of‐the‐art cut construction methods, our method is one order of magnitude faster and generates shorter cuts while achieving similar isometric distortion.

Inversion-free geometric mapping construction: A survey

“…As shown in Ref. [147], the greedy algorithm outperforms MST and SPH, and approximate the optimal solution better in the sense of relative error.…”

“…The local operations lead to early entrapment by local minimum, thereby resulting in long cuts, as shown in Ref. [147]. Besides, local operations also cause a high computational cost.…”

“…Similar to Ref. [148], Zhu et al [147] also use planar parameterizations to generate proxy metrics. To detect necessary feature points to achieve low isometric distortion and prevent too many feature points, a greedy filtering process is proposed [147].…”

“…[144][145][146]. A greedy algorithm [147] is proposed to compute an approximate solution driven by auxiliary points. As shown in Ref.…”

“…• Optimization-based methods simultaneously optimize the parameterization distortion and the cut length [142,143]. • Point-to-cut methods first detect feature points where the distortion is usually concentrated and then connect these feature points to construct cuts [144][145][146][147][148]. Since the segmentation-based methods do not explicitly minimize cut lengths, we discuss other two methods in details.…”