Apictorial Jigsaw Puzzle Reconstruction Based on Curve Matching via a Corotational Beam Spline

Abstract: 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.