Surface-based computation of the Euler characteristic in the cubical grid

IF 2.5 4区计算机科学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Graphical Models Pub Date : 2020-11-01 DOI:10.1016/j.gmod.2020.101093

Lidija Čomić , Paola Magillo

引用次数: 3

Abstract

For well-composed (manifold) objects in the 3D cubical grid, the Euler characteristic is equal to half of the Euler characteristic of the object boundary, which in turn is equal to the number of boundary vertices minus the number of boundary faces. We extend this formula to arbitrary objects, not necessarily well-composed, by adjusting the count of boundary cells both for vertex- and for face-adjacency. We prove the correctness of our approach by constructing two well-composed polyhedral complexes homotopy equivalent to the given object with the two adjacencies. The proposed formulas for the computation of the Euler characteristic are simple, easy to implement and efficient. Experiments show that our formulas are faster to evaluate than the volume-based ones on realistic inputs, and are faster than the classical surface-based formulas.

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基于曲面的立方体网格欧拉特性计算

对于三维立方体网格中组成良好的(流形)物体，欧拉特征等于物体边界欧拉特征的一半，而欧拉特征又等于边界顶点的数量减去边界面的数量。我们将这个公式扩展到任意对象，不一定是很好的组合，通过调整顶点和面邻接的边界细胞的计数。通过构造两个构造良好的多面体配合物同伦，证明了该方法的正确性。所提出的欧拉特性计算公式简单、易于实现、效率高。实验表明，在实际输入条件下，我们的计算公式比基于体积的计算公式更快，比经典的基于表面的计算公式更快。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Graphical Models 工程技术-计算机：软件工程

CiteScore

3.60

自引率

5.90%

发文量

审稿时长

47 days

期刊介绍： Graphical Models is recognized internationally as a highly rated, top tier journal and is focused on the creation, geometric processing, animation, and visualization of graphical models and on their applications in engineering, science, culture, and entertainment. GMOD provides its readers with thoroughly reviewed and carefully selected papers that disseminate exciting innovations, that teach rigorous theoretical foundations, that propose robust and efficient solutions, or that describe ambitious systems or applications in a variety of topics. We invite papers in five categories: research (contributions of novel theoretical or practical approaches or solutions), survey (opinionated views of the state-of-the-art and challenges in a specific topic), system (the architecture and implementation details of an innovative architecture for a complete system that supports model/animation design, acquisition, analysis, visualization?), application (description of a novel application of know techniques and evaluation of its impact), or lecture (an elegant and inspiring perspective on previously published results that clarifies them and teaches them in a new way). GMOD offers its authors an accelerated review, feedback from experts in the field, immediate online publication of accepted papers, no restriction on color and length (when justified by the content) in the online version, and a broad promotion of published papers. A prestigious group of editors selected from among the premier international researchers in their fields oversees the review process.