Automatic assembly of apictorial jigsaw puzzles presents a classic curve matching problem, fundamentally challenged by discrete and noisy contour data obtained from digitization. Conventional smoothing methods, which are required to process these data, often distort the curvature-based criteria used for matching and cause a loss of critical information. This paper proposes a method to overcome these issues, demonstrated on the automatic reconstruction of a 54-piece puzzle. We reconstruct each piece's contour using a novel corotational beam spline, which models the boundary as a flexible beam with compliant spring supports at the measured data points. A distinctive feature is the dynamic re-indexing of these points; as their calculated positions are refined, they are re-numbered based on their projection onto the computed contour. Another contribution is a method for determining spring compliance in proportion to the distance between the point projections. This approach uniquely ensures a uniform degree of smoothing for corresponding curves, making the matching process robust to variations in point density and dependent only on measurement accuracy. Practical computations and the successful automatic reconstruction of the puzzle demonstrate the proposed method's effectiveness.

翻译：无图案拼图的自动组装呈现了一个经典的曲线匹配问题，其根本挑战在于数字化获取的离散且含噪声的轮廓数据。处理这些数据所需的传统平滑方法通常会扭曲用于匹配的基于曲率的准则，并导致关键信息的丢失。本文提出了一种克服这些问题的方法，并在一个54片拼图的自动重构上进行了演示。我们使用一种新颖的共旋转梁样条来重构每个拼图块的轮廓，该样条将边界建模为一条柔性梁，并在测量数据点处设有顺应性弹簧支撑。一个显著特点是这些数据点的动态重新索引：随着其计算位置的细化，它们会根据其投影到计算轮廓上的位置进行重新编号。另一项贡献是提出了一种根据点投影之间的距离成比例确定弹簧顺应性的方法。该方法独特地确保了对应曲线具有均匀的平滑度，使得匹配过程对点密度的变化具有鲁棒性，并且仅依赖于测量精度。实际计算以及拼图的成功自动重构证明了所提方法的有效性。