Edge2Vec: A High Quality Embedding for the Jigsaw Puzzle Problem

Pairwise compatibility measure (CM) is a key component in solving the jigsaw puzzle problem (JPP) and many of its recently proposed variants. With the rapid rise of deep neural networks (DNNs), a trade-off between performance (i.e., accuracy) and computational efficiency has become a very significant issue. Whereas an end-to-end DNN-based CM model exhibits high performance, it becomes virtually infeasible on very large puzzles, due to its highly intensive computation. On the other hand, exploiting the concept of embeddings to alleviate significantly the computational efficiency, has resulted in degraded performance, according to recent studies. This paper derives an advanced CM model (based on modified embeddings and a new loss function, called hard batch triplet loss) for closing the above gap between speed and accuracy; namely a CM model that achieves SOTA results in terms of performance and efficiency combined. We evaluated our newly derived CM on three commonly used datasets, and obtained a reconstruction improvement of 5.8% and 19.5% for so-called Type-1 and Type-2 problem variants, respectively, compared to best known results due to previous CMs.

TEN: Twin Embedding Networks for the Jigsaw Puzzle Problem with Eroded Boundaries

The jigsaw puzzle problem (JPP) is a well-known research problem, which has been studied for many years. Solving this problem typically involves a two-stage scheme, consisting of the computation of a pairwise piece compatibility measure (CM), coupled with a subsequent puzzle reconstruction algorithm. Many effective CMs, which apply a simple distance measure, based merely on the information along the piece edges, have been proposed. However, the practicality of these classical methods is rather doubtful for problem instances harder than pure synthetic images. Specifically, these methods tend to break down in more realistic scenarios involving, e.g., monochromatic puzzles, eroded boundaries due to piece degradation over long time periods, missing pieces, etc. To overcome this significant deficiency, a few deep convolutional neural network (CNN)-based CMs have been recently introduced. Despite their promising accuracy, these models are very computationally intensive. Twin Embedding Networks (TEN), to represent a piece with respect to its boundary in a latent embedding space. Combining this latent representation with a simple distance measure, we then demonstrate a superior performance, in terms of accuracy, of our newly proposed pairwise CM, compared to that of various classical methods, for the problem domain of eroded tile boundaries, a testbed for a number of real-world JPP variants. Furthermore, we also demonstrate that TEN is faster by a few orders of magnitude, on average, than the recent NN models, i.e., it is as fast as the classical methods. In this regard, the paper makes a significant first attempt at bridging the gap between the relatively low accuracy (of classical methods) and the intensive computational complexity (of NN models), for practical, real-world puzzle-like problems.

A Novel Hybrid Scheme Using Genetic Algorithms and Deep Learning for the Reconstruction of Portuguese Tile Panels

This paper presents a novel scheme, based on a unique combination of genetic algorithms (GAs) and deep learning (DL), for the automatic reconstruction of Portuguese tile panels, a challenging real-world variant of the jigsaw puzzle problem (JPP) with important national heritage implications. Specifically, we introduce an enhanced GA-based puzzle solver, whose integration with a novel DL-based compatibility measure (DLCM) yields state-of-the-art performance, regarding the above application. Current compatibility measures consider typically (the chromatic information of) edge pixels (between adjacent tiles), and help achieve high accuracy for the synthetic JPP variant. However, such measures exhibit rather poor performance when applied to the Portuguese tile panels, which are susceptible to various real-world effects, e.g., monochromatic panels, non-squared tiles, edge degradation, etc. To overcome such difficulties, we have developed a novel DLCM to extract high-level texture/color statistics from the entire tile information. Integrating this measure with our enhanced GA-based puzzle solver, we have demonstrated, for the first time, how to deal most effectively with large-scale real-world problems, such as the Portuguese tile problem. Specifically, we have achieved 82% accuracy for the reconstruction of Portuguese tile panels with unknown piece rotation and puzzle dimension (compared to merely 3.5% average accuracy achieved by the best method known for solving this problem variant). The proposed method outperforms even human experts in several cases, correcting their mistakes in the manual tile assembly.

DNN-Buddies: A Deep Neural Network-Based Estimation Metric for the Jigsaw Puzzle Problem

This paper introduces the first deep neural network-based estimation metric for the jigsaw puzzle problem. Given two puzzle piece edges, the neural network predicts whether or not they should be adjacent in the correct assembly of the puzzle, using nothing but the pixels of each piece. The proposed metric exhibits an extremely high precision even though no manual feature extraction is performed. When incorporated into an existing puzzle solver, the solution's accuracy increases significantly, achieving thereby a new state-of-the-art standard.

A Genetic Algorithm-Based Solver for Very Large Jigsaw Puzzles

In this paper we propose the first effective automated, genetic algorithm (GA)-based jigsaw puzzle solver. We introduce a novel procedure of merging two "parent" solutions to an improved "child" solution by detecting, extracting, and combining correctly assembled puzzle segments. The solver proposed exhibits state-of-the-art performance solving previously attempted puzzles faster and far more accurately, and also puzzles of size never before attempted. Other contributions include the creation of a benchmark of large images, previously unavailable. We share the data sets and all of our results for future testing and comparative evaluation of jigsaw puzzle solvers.

A Generalized Genetic Algorithm-Based Solver for Very Large Jigsaw Puzzles of Complex Types

In this paper we introduce new types of square-piece jigsaw puzzles, where in addition to the unknown location and orientation of each piece, a piece might also need to be flipped. These puzzles, which are associated with a number of real world problems, are considerably harder, from a computational standpoint. Specifically, we present a novel generalized genetic algorithm (GA)-based solver that can handle puzzle pieces of unknown location and orientation (Type 2 puzzles) and (two-sided) puzzle pieces of unknown location, orientation, and face (Type 4 puzzles). To the best of our knowledge, our solver provides a new state-of-the-art, solving previously attempted puzzles faster and far more accurately, handling puzzle sizes that have never been attempted before, and assembling the newly introduced two-sided puzzles automatically and effectively. This paper also presents, among other results, the most extensive set of experimental results, compiled as of yet, on Type 2 puzzles.

An Automatic Solver for Very Large Jigsaw Puzzles Using Genetic Algorithms

In this paper we propose the first effective genetic algorithm (GA)-based jigsaw puzzle solver. We introduce a novel crossover procedure that merges two "parent" solutions to an improved "child" configuration by detecting, extracting, and combining correctly assembled puzzle segments. The solver proposed exhibits state-of-the-art performance, as far as handling previously attempted puzzles more accurately and efficiently, as well puzzle sizes that have not been attempted before. The extended experimental results provided in this paper include, among others, a thorough inspection of up to 30,745-piece puzzles (compared to previous attempts on 22,755-piece puzzles), using a considerably faster concurrent implementation of the algorithm. Furthermore, we explore the impact of different phases of the novel crossover operator by experimenting with several variants of the GA. Finally, we compare different fitness functions and their effect on the overall results of the GA-based solver.

Genetic Algorithm-Based Solver for Very Large Multiple Jigsaw Puzzles of Unknown Dimensions and Piece Orientation

In this paper we propose the first genetic algorithm (GA)-based solver for jigsaw puzzles of unknown puzzle dimensions and unknown piece location and orientation. Our solver uses a novel crossover technique, and sets a new state-of-the-art in terms of the puzzle sizes solved and the accuracy obtained. The results are significantly improved, even when compared to previous solvers assuming known puzzle dimensions. Moreover, the solver successfully contends with a mixed bag of multiple puzzle pieces, assembling simultaneously all puzzles.