A plane-probing algorithm computes the normal vector of a digital plane from a starting point and a predicate “Is a point \({x}\) in the digital plane?”. This predicate is used to probe the digital plane as locally as possible and decide on the fly the next points to consider. However, several existing plane-probing algorithms return the correct normal vector only for some specific starting points and an approximation otherwise, e.g., the H- and R-algorithm proposed in Lachaud et al. (J Math Imaging Vis 59(1):23–39, 2017). In this paper, we present a general framework for these plane-probing algorithms that provides a way of retrieving the correct normal vector from any starting point, while keeping their main features. There are \(O(\omega \log \omega )\) calls to the predicate in the worst-case scenario, where \(\omega \) is the thickness of the underlying digital plane, but far fewer calls are experimentally observed on average. In the context of digital surface analysis, the resulting algorithm is expected to be of great interest for normal estimation and shape reconstruction.

中文翻译：

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平面探测算法的优化框架

平面探测算法从起点和谓词“数字平面中的点\（{x} \）是？”计算数字平面的法线向量。该谓词用于尽可能局部地探查数字平面，并即时确定要考虑的下一点。但是，现有的几种平面探测算法仅针对某些特定起点返回正确的法向矢量，否则仅返回近似值，例如Lachaud等人提出的H和R算法。（J Math Imaging Vis 59（1）：23–39，2017）。在本文中，我们为这些平面探测算法提供了一个通用框架，该框架提供了一种从任何起点检索正确法线向量的方法，同时保留了它们的主要特征。有\（O（\ omega \ log \ omega）\）在最坏情况下对谓词的调用，其中\（\ omega \）是基础数字平面的厚度，但是从实验上平均观察到的调用要少得多。在数字表面分析的背景下，预期所得的算法将对法线估计和形状重构非常感兴趣。