Research on shape reconstruction based on optimization

首先，将曲线重构问题描述为图论问题，并提出了解决这些问题的方法。从理论上证明了传统阈值算法、贪婪算法、精确算法的性能依次提高。另一方面，发现曲面重建问题的公式是困难的。因此，我们探索了另外两种方法。在第一种方法中，将未知参数放置在样本点上，将形状重建问题表示为“起伏”最小化问题。在第二种方法中，曲线被认为是一对带有距离函数的中轴。未来的研究将包括完成这两种方法。

At first, the curve reconstruction problem was formulated as a graph-theoretical problem, and methods for solving these problems were developed. It was proved theoretically that the performances of the conventional thresholding method, greedy algorithm, and exact algorithm become better in this order. On the other hand, a formulation for surface reconstruction problem was found to be hard. Therefore, two other approaches were explored. In the first approach, unknown parameters are put at sample points, and the shape reconstruction problem is formulated as an "undulation" minimization problem. In the second approach, a curve is considered as a pair of a medial axis with a distance function. Future research will include to complete these two approaches.