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

TL;DR

This work tackles automatic assembly of apictorial 2D jigsaw puzzles from densely sampled, noisy contour data by introducing corotational beam splines (CBS) to reconstruct piece boundaries with controlled smoothing. CBS models each contour as a planar beam with adaptive elastic supports, uses dynamic projection-based renumbering of contour points, and computes curvature-driven energies for matching, specifically $E = \int_0^l \kappa^2(s) ds$ and $E_{a,b}=\int_0^l (\kappa_1(s)-\kappa_2(s))^2 ds$. The approach demonstrates robust reconstruction on a 54-piece puzzle, showing that uniform smoothing across varying point densities can be achieved via adaptive rigidity $D_i = L_i/h^4$ and a carefully tuned smoothing bandwidth $h$, with a greedy assembly guided by curvature-energy comparisons. These results suggest broad applicability of CBS for high-noise contour matching tasks in archaeology, document restoration, and pattern recognition, where preserving curvature information during smoothing is critical.

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.

Figures (9)

• Figure 1: Model of discrete beams on elastic supports
• Figure 2: (a) Puzzle piece mask and (b) $\mathbf{B}_i$ points in a puzzle piece
• Figure 3: Contour after the first iteration. The black dots are the $\mathbf{B}_i$ points. The purple dots are the silhouette points (every 20th measured point). The orange line is the initial polygon, and the blue contour shows the calculated curve after 1 iteration. The red dashed lines show the correspondence between every silhouette point and the points on the smoothed contour, i.e., their corresponding $\mathbf{A}_i$ points
• Figure 4: Projections of the $\mathbf{B}_i$ points onto the calculated contour and their renumbering
• Figure 5: The puzzle under consideration