结构化离散形状逼近：理论复杂性与实用算法
Structured Discrete Shape Approximation: Theoretical Complexity and Practical Algorithm
摘要：
我们考虑的问题，近似的二维形状轮廓（或曲线段）使用离散装配系统，允许建立基于有限的节点和边缘类型的边缘长度和方向的限制集的几何结构。我们表明，已经决定这种近似问题的可行性是NP难的，即使是非常简单的设置仍然是棘手的。然后，我们设计了一个算法框架，结合形状采样与确切的基数最小化，以获得良好的近似使用几个组件。作为一个特定的应用程序和展示的例子，我们讨论近似形状轮廓使用经典的Zometool建设工具包，并提供有前途的计算结果，表明我们的算法是能够获得良好的形状表示在合理的时间内，尽管问题的一般棘手。最后，我们展望了可能的扩展所开发的方法，特别是关于3D形状近似任务的文件。
英文摘要：
We consider the problem of approximating a two-dimensional shape contour (or curve segment) using discrete assembly systems, which allow to build geometric structures based on limited sets of node and edge types subject to edge length and orientation restrictions. We show that already deciding feasibility of such approximation problems is NP-hard, and remains intractable even for very simple setups. We then devise an algorithmic framework that combines shape sampling with exact cardinality-minimization to obtain good approximations using few components. As a particular application and showcase example, we discuss approximating shape contours using the classical Zometool construction kit and provide promising computational results, demonstrating that our algorithm is capable of obtaining good shape representations within reasonable time, in spite of the problem's general intractability. We conclude the paper with an outlook on possible extensions of the developed methodology, in particular regarding 3D shape approximation tasks.
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