Optimum solution and evaluation of rectangular jigsaw puzzles based on branch and bound method and combinatorial accuracy

Abstract

Automatic solution of rectangular jigsaw puzzles can be broken down into two separate steps of calculating pairwise compatibility metrics and jigsaw assembling algorithm. This work discriminates between two different sources of errors, each corresponding to one of these two steps. In this regard, type I error is defined as the imperfection of the used compatibility metric, and type II error is reserved to measure the imperfection of jigsaw assembling algorithm. Differentiating between these two types of error allows us to tweak and optimize different parts of the algorithm to achieve the best performance. Based upon these defined terms, this study argues that current jigsaw assembling algorithms mainly rely on either greedy methods or metaheuristic algorithms, which may impose a considerable amount of type II error to the final solution. This paper demonstrates that a powerful and perfect (i.e., type II error-free) jigsaw assembling algorithm is achievable by combining branch and bound technique with graph theory. This perfect jigsaw assembling algorithm is then utilized to measure the performances of various compatibility metrics and color models. The superiority of red-green-blue (RGB) color model and Mahalanobis gradient compatibility (MGC) metric in solving rectangular jigsaw puzzles is shown by providing conclusive evidence. Additionally, a mean opinion score (MOS) test is conducted to examine the accuracy of the existing metrics. According to the results from MOS test, we argue that the existing performance criteria are not concise and accurate; thus, a new accuracy metric is proposed on the basis of comparing different sub-blocks of solutions. Finally, the efficiency of jigsaw assembling algorithm is measured by proposing a new performance criterion.