Genetic-based square jigsaw puzzle solver using the combined color+texture compatibility criterion

Journal of Universal Computer Science, vol. 31, no. 13 (2025), 1564-1580 submitted: 14/6/2024, accepted: 2/6/2025, appeared: 28/11/2025 CC BY 4.0 Genetic-based square jigsaw puzzle solver using the combined color+texture compatibility criterion Atefeh Parvin (University of Sistan and Baluchestan, Electrical and Computer Faculty, Department of Communications Engineering, Zahedan, Iran, https://orcid.org/0009-0004-9994-9239, [email protected]) Farahnaz Mohanna (University of Sistan and Baluchestan, Electrical and Computer Faculty, Department of Communications Engineering, Zahedan, Iran https://orcid.org/0000-0002-8689-6201, f_mohan[email protected]) Masoumeh Rezaei (University of Sistan and Baluchestan, Electrical and Computer Faculty, Department of Computer Engineering, Zahedan, Iran https://orcid.org/0000-0001-8184-1773, [email protected]) Abstract: When reconstructing jigsaw puzzles, the state-of-the-art algorithms struggle to distinguish between identically colored pieces that belong to different objects. This limitation significantly impacts the accuracy of puzzle solvers, especially in complex images with repetitive colors or textures. To address this issue, we propose a new GA-based square jigsaw puzzle solver. A combined color and texture discriminator is incorporated into the proposed solver to prevent pieces that have the same color but come from distinct objects from being joined together incorrectly. Color and texture features are extracted separately using the sum of square distances and Gabor filter. To evaluate the performance of the proposed solver, we used a dataset consisting 66 images: 20 puzzles with 432 pieces from the MIT collection, 20 puzzles with 540 pieces, and 20 puzzles with 805 pieces from the McGill collection, and 3 puzzles with 2360 pieces, and 3 puzzles with 3300 pieces from the Pomeranz collection. For the direct, neighbor, and largest component comparisons, the proposed method’s accuracy is 92.91%, 96.66%, and 90.83%, respectively. The proposed method demonstrates an improvement of 11.9%, and 3.65% in accuracy based on direct and neighbor comparison criteria, on the database images when compared to current state-of-the-art GA-based square jigsaw puzzle solver. Keywords: Jigsaw puzzle solving, Genetic algorithm, Compatibility criterion, Texture feature, Gabor filter Categories: H.3.1, H.3.2, H.3.3, H.3.7, H.5.1 DOI: 10.3897/jucs.129768 1 Introduction A square jigsaw puzzle consists of an image divided into N non-overlapping square pieces. By disrupting the order of these pieces, a player attempts to reconstruct the image using the shape and color information of each piece. Solving a square jigsaw puzzle is essential for restoration of ancient pieces, recovery of shredded photos, and repairing of fractured bones [Sholomon, 13]. Various methods have been used to solve 1565 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... a square jigsaw puzzle by computer. Among them, approaches utilizing Genetic Algorithm (GA) are more feasible to apply in the real world scenarios [Sholomon, 13, Sholomon, 14]. GA-based techniques outperform deep neural network-based techniques [Chen, 23a, Chen, 23b, Chen, 24a, Chen, 24b, Chen, 24c] in terms of accuracy for puzzles with more pieces [Sholomon, 14], and do not require a vast number of database images. Two essential components for reconstructing a jigsaw puzzle based on the GA are a compatibility criterion, which measures how well a pair of pieces fit together, and the reconstruction strategy [Sholomon, 14]. Jigsaw puzzles are generally classified into four types based on specific characteristics. In type 1 puzzles, the direction of each piece is known, but their locations are not. Therefore, a pair of pieces can be joined in only four different ways. In type 2 puzzles, neither the location nor the direction of any piece is known, resulting in 16 possible ways to fit a pair of pieces together. In type 3 puzzles, each piece location is known, but its direction is not. Finally, in type 4 puzzles, neither the location nor the direction of any piece is known, and a piece may need to be flipped, meaning the face of the puzzle is also unknown [Sholomon, 14]. According to the previous work, the average accuracy of methods for solving the type 1 jigsaw puzzles, with 432 pieces to 22834 pieces, has been reported 97.12%, and 98.36% based on direct, and neighbor comparison criteria, respectively. Also, the average accuracy of methods solving the type 2 jigsaw puzzles, with 432 pieces to 22755 pieces, has been reported 96.16% based on the neighbor comparison criterion. The direct comparison criterion measures the number of pieces in the correct position in a solved puzzle [Sholomon, 14]. The neighbor comparison criterion counts the number of correctly positioned adjacent pairs of pieces in a solved puzzle [Sholomon, 14]. The largest component comparison identifies the largest contiguous area of correctly assembled pieces, regardless of their overall position in the puzzle [Sholomon, 14]. Additionally, the complete reconstruction criterion assesses whether every piece in a solved puzzle is in the correct position and orientation [Sholomon, 14]. In addition, the compatibility criterion of Sum of Squared Distances (SSD) [Sholomon, 13] [Sholomon, 14] [Sholomon, 16], Mahalanobis Gradient Compatibility (MGC) [Sholomon, 16], SSD+MGC [Guo, 20], and 𝐿! ""compatibility (LPQ) [Bezulj, 18] have been used as the compatibility measures for adjoining a pair of pieces in solving a square jigsaw puzzle. A three-phase procedure has also been employed as the crossover operator in the most of the GA-based puzzles solvers [Sholomon, 13] [Sholomon, 14]. A review of previous GA-based square jigsaw puzzle solvers reveals that these solvers primarily used color criteria to measure the compatibility of puzzle pieces. However, relying solely on color criteria can lead to inaccuracies, particularly when two pieces from different objects have similar colors. This limitation reduces the accuracy of these solvers. In this paper, we propose a new compatibility criterion that combines color and texture features. By using the joint color and texture compatibility criterion, the proposed GA-based square jigsaw puzzle solver (from now on, we call it the proposed GA-based solver) has achieved a higher accuracy on the database including 66 selected square jigsaw puzzles. The selected square jigsaw puzzles consist of, 20 puzzles with 432 pieces of the MIT dataset [Cho, 10], 20 puzzles with 540 pieces, and 20 puzzles with 805 pieces from the McGill dataset [Olmos, 15], three puzzles with 2360 pieces, 1566 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... and three puzzles with 3300 pieces from the Pomeranz dataset [Pomeranz, 11]. For this paper, the main contributions are as follows: 1. The proposed GA-based solver achieves higher accuracy compared to the other similar GA-based solvers. 2. It can easily be implemented in the real world. 3. The proposed solver introduces the first hybrid color+texture discriminator for GA-based puzzle solving. 4. Unlike deep neural network-based solvers, the proposed solver does not require a large number of puzzles. It offers a data-efficient, training-free alternative to neural networks. 5. The proposed method is capable of solving any type 1, 2, and 3 square jigsaw puzzles. It also is size-independent. However, the proposed solver has some drawbacks: 1It is a moderate time consuming for puzzles in the RGB color space. 2It cannot reconstruct the type 4 square jigsaw puzzles. 3It cannot reconstruct the non-square jigsaw puzzles. In continue, the related work is presented in section 2. The proposed GA-based solver is introduced in section 3. The results, comparisons, and evaluation are reported in section 4. The paper concluded in section 5. 2 Related Work The GA-based type 1 square jigsaw puzzles solver was introduced in [Sholomon, 13] based on the compatibility criterion of Sum of Squared Distances (SSD) in the L*a*b color space, and employs a three-phase procedure as the crossover operator. In the three-phase procedure for piece selection, and assignment, first, the agreed pieces, second, the best-buddy pieces, and finally the most compatible pieces available were placed. The average accuracy of this solver with 10 runs on the database was reported 87.89%, and 93.46% according to the direct, and neighbor comparisons criteria, respectively. The GA-based type 2 square jigsaw puzzles solver introduced in [Sholomon, 14] uses the SSD compatibility criterion in the L*a*b color space, and employs the Prim algorithm as the crossover operator. The average accuracy of this solver with 30 runs on the database was reported 84.23%, and 93.00% according to the direct, and neighbor comparison criteria, respectively. The GA-based type 2 square jigsaw puzzles solver was presented in [Sholomon, 16] to solve the square jigsaw puzzles type 2 up to 30745 pieces. This solver employs both the SSD, and Mahalanobis Gradient Compatibility (MGC) fitness functions, and uses the three-phase procedure as the crossover operator. The average accuracy of this solver on 20 puzzles with 432 pieces was reported 96.16%, and 96.03 according to the neighbor comparison using the SSD, and MGC compatibility criteria, respectively, demonstrating the superiority of SSD over MGC. The GA-based jigsaw puzzles solver was presented in [Guo, 20] to solve the smallscale puzzles. The fitness function was based on the SSD+MGC compatibility criterion, and the crossover operator was the three-phase procedure. Additionally, two new mutation operators were introduced. The database included 25 square jigsaw puzzles 1567 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... with 256, 646, and 1196 pieces. The average accuracy of this solver was reported 94.31% based on the direct comparison. The GA-based type 2 jigsaw puzzles solver was presented in [Bezulj, 18] giving a weight to each GA solution to use in the roulette wheel. The fitness function was the LPQ compatibility and the crossover operator was the three-phase procedure without its first phase. The accuracy of this solver on 20 MIT puzzles with 432 pieces was 92.6% according to the direct comparison. An image super-resolution reconstruction method was introduced in [Chen, 24a] leveraging the multi-level information compensation and U-Net network. First, the UNet network performed multi-level feature extraction and channel compression on input features. Next, correlation features from different channels were extracted and integrated. A multi-level information compensation model was then devised to address information loss during the compression process. The image inpainting network using multi-scale feature module was introduced in [Chen, 24b] and enhanced attention module. First, a multi-scale fusion module was presented based on dilated convolution to reduce information loss. To ensure consistency in image inpainting details and style, style loss and perceptual loss functions were incorporated. A lightweight method presented in [Chen, 23a] integrates group convolution and attention mechanism to enhance and replace the traditional convolution module. The image inpainting method was introduced in [Chen, 24c] according to partial multi-scale channel attention mechanism and deep neural networks to process on images with large missing sections. The method employed a Res-U-Net module as its generator, where the U-Net backbone handled the encoding and decoding steps for damaged images. The residual network enhanced the network’s capability to extract the features from the damaged images. The image restoration method was introduced in [Chen, 23b] consisting of the semantic priors, deep attention residual group, and full-scale skip connection. The semantic priors network learned comprehensive semantic information about visual elements in missing regions to facilitate completion. The deep attention residual group enabled the generator to focus on both missing regions and adaptively learn channel features. The full-scale skip connection combined the low-level feature maps containing image boundaries with the high-level feature maps containing image textures and details to repair the missing regions. In [Markaki, 23], the problem of jigsaw puzzle solvers has been thoroughly discussed, including their evaluation and applications. The jigsaw puzzles solver introduced in [Li, 22] based on the Generative Adversarial Network (GAN). The multi-task pipeline was designed including a classification, and the GAN branches that were connected by the flow-based warp module. The average accuracy of this solver was reported 79.0% on the database including 7639 puzzles. The type 2 large square jigsaw puzzles solver introduced in [Huroyan, 20] utilizes the MGC to match puzzle pieces. Additionally, a graph connection Laplacian is employed to recover the puzzle pieces rotation. The average accuracy of this solver was reported 89.1%, 91.4%, and 90.62% respectively based on the direct, neighbor, and largest component comparisons on the database. The pairwise compatibility measure was presented in [Guerroui, 18] using the Gist and color distance to solve the jigsaw puzzles. The rotation-based strategy was 1568 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... presented to work on the multiple parts, and solve puzzles from the local matching candidates. The average accuracy of this solver on 20 MIT type 2 square jigsaw puzzles with 432 pieces was reported 93%, and 96% respectively based on the direct, and neighbor comparisons. For 20 MIT type 1 square jigsaw puzzles, the reported average accuracy was 94.16%, and 96.44% based on direct and neighbor comparisons respectively. The deep learning model was introduced in [Ahmad, 22] to determine the different states of jigsaw puzzle. The task was represented as a classification problem, where each state of puzzle was considered as a class. The model can also serve as a fitness function for other GA-based jigsaw puzzles solvers. The design was introduced in [Chen, 22] to enhance the search throughput by evaluating newly generated inputs with the JIT-compiled path constraints. The model achieved three orders of magnitude higher search throughput than the existing fuzzers and could scale to the multiple cores. The computational puzzles solver presented in [Son, 18] aims to maximize geometric consensus among hierarchical piece loops for square jigsaw puzzles without prior information. Initially, loops of four pieces are searched, and then aggregated into higher-level piece loops. The average accuracy of this solver was reported as follows: for type 1 square jigsaw puzzles, 94.9%, 95.7%, and 93.42% based on direct, neighbor, and largest component comparisons respectively, and for four type 2 square jigsaw puzzles, 93.46%, and 95.3% based on direct, neighbor comparisons respectively. The square jigsaw puzzles solver presented in [McGill-Smith, 19] utilizes the ORB feature detector in conjunction the FLANN-based matcher. Initially, an image of the square jigsaw puzzle is segmented into regions of interest using contour detection. These regions undergo a feature matching process, followed by filtering using Lowe's ratio test to refine matches. Outliers are subsequently identified and removed using the homography transformation matrix. The accuracy of the solver was reported 81.2% on a jigsaw puzzle with 24 pieces. The square jigsaw puzzles solver introduced in [Paikin, 15] employs a greedy approach for comparing the compatibility of puzzle pieces. However, the greedy solvers depended on the initial placement of puzzle pieces. This solver is capable of reconstructing puzzles with missing pieces and handling puzzles contained multiple puzzles pieces. A jigsaw puzzle-solving task as a self-supervised pre-training mechanism was presented in [Peng, 20] for fine-grained sketch-based image retrieval (FG-SBIR). However, it primarily focused on small-grid puzzles (e.g., 2×2 or 3×3 pieces) rather than large-scale puzzles. Table 1 showed the summary results of the state-of-the-art GA-based square jigsaw solver so far. Time Crossove r Standard Deviation Average Accuracy piece s Data Sets Typ e Fitnes s Ref. 48.73 sec. Three phase 0.3 4 2.62 95.7 0 82.9 4 432 MIT & Pomeran z & McGill 1 SSD [Sholomon , 13] 64.04 sec. 0.4 0 0.65 95.3 8 91.6 5 540 116.1 8 sec. 0.3 1 0.62 95.8 5 93.6 3 805 17.60 min. 0.3 8 0.86 88.0 0 84.6 2 2360 1569 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... 30.24 min. 0.2 7 7.19 92.3 7 86.6 2 3300 61.06 min. 0.1 1 1.74 95.0 6 92.0 4 5015 3.21 hours 0.0 8 0.45 98.3 6 97.1 2 10375 13.19 hours 0.0 5 0.28 96.2 2 91.7 4 22834 8 sec. Prim 0.3 9 10.4 7 94.4 4 81.5 1 432 MIT & Pomeran z 2 SSD [Sholomon , 14] 11.90 sec. 0.4 9 12.4 9 91.3 3 74.5 5 540 22.10 sec. 0.4 8 10.5 6 91.4 5 79.2 0 805 3.67 min. 0.1 8 0.90 97.3 3 92.7 5 2360 4.82 min. 0.7 9 1.14 94.9 3 93.1 4 3300 Threephase 96.1 6 432 MIT 2 SSD [Sholomon , 16] 96.0 3 MGC Three phase 0.61 99.3 7 256 2 SSD+ MGC [Guo, 20] 0.68 98.4 2 646 2.72 84.9 4 1196 Threephase without its first phase 92.6 0 432 MIT 2 LPQ [Bezulj, 18] Table 1: The summary results of all the state-of-art GA-based square jigsaw solver 3 The Proposed GA-based Solver The block diagram of the proposed method is shown in Fig. 1, and the pseudocode of the proposed GA-based solver is shown in Table 2. 1570 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... Figure 1: The block diagram of the proposed GA-based square jigsaw puzzles solver First, the puzzle pieces are randomly assembled to form an initial population of chromosomes. Second, the fitness score for each chromosomes is calculated using a combination of the SSD compatibility criterion and the Gabor texture, referred to as the SSD+Gabor compatibility criterion. Third, some of the best chromosomes are selected as parents to generate a new population. Fourth, the crossover operation is performed on two selected parents to create a new child chromosome. Fifth, the children chromosomes, along with the parent chromosomes, form a new generation, which replaces the old generation. If the number of runs has not yet been completed, the proposed GA-based solver returns to step 2 and continues the process. Otherwise, if the number of runs is completed, the best chromosome at this stage represents the solved puzzle. It is mentioned, the fitness function for the proposed method combines the SSD compatibility criterion between each pair of puzzle pieces with the extracted Gabor features from the same two pieces. The color dissimilarity between two puzzle pieces of 𝑥#, and 𝑥$ is computed by (1) [Sholomon, 14]. Algorithm 1: population ← generate 1000 random chromosomes 2: for generation number = 1 → 20 do 3: evaluate all chromosomes using the SSD+Gabor fitness function 4: new population ← NULL 5: copy 4 best chromosomes to new population 6: while size (new population) ≤ 1000 do 7: parent1 ←select chromosome 8: parent2 ←select chromosome 9: child ← Three phase procedure (parent1, parent2) 10: add child to new population 11: end while 12: population ← new population 1571 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... 13: end for 14: solution ← best population Table 2: The pseudocode of the proposed GA-based solver 𝐷(𝑥#,𝑥$,𝑟)=*∑ ∑ (𝑥#(𝑘,𝐾,𝑏)−𝑥$(𝑘,1,𝑏))% & '() * +() (1) where, K is a piece dimension, b is a piece component, r means right and (𝑥#,𝑥$,𝑟) indicates piece of 𝑥$is in the right side of piece of 𝑥# in the puzzle. D means the dissimilarity and 𝐷(𝑥#,𝑥$,𝑟) indicates the dissimilarity between two pieces of 𝑥#, and 𝑥$, that 𝑥$ is in the right side of piece of 𝑥#. In the proposed solver, the RGB color space is used, so the number of color components (b) is 3, and each piece dimension is considered 28×28, so K= 28×28. The sum of dissimilarity on all a puzzle pieces by considering all the neighbor pieces in the right and down directions in the puzzle is computed by (2) [Sholomon, 14]. ∑ ∑ (𝐷(𝑥#,$,𝑥#,$-),𝑟)) ./) $() 0 #() +∑ ∑ (𝐷(𝑥#,$,𝑥#-),$,𝑑)) . $() 0/) #() (2) where, 𝑟and 𝑑 denote the right and down directions, respectively. 𝑀×𝑁is the chromosome size. 𝑥#,$is a single puzzle piece at coordinates(𝑥,𝑦). 𝐷is the dissimilarity between two pieces. To extract the Gabor texture from each puzzle piece, the Gabor filter is applied as descriptor in equations (3) to (6) [Otomo, 21]. 𝐺(𝑥,𝑦)=𝑒𝑥𝑝:−) %;1! " 2# "+3! " 2$ "<𝑐𝑜𝑠@%41! 5+𝜓BC (3) 𝑥6=𝑥𝑐𝑜𝑠(𝜃)+𝑦𝑠𝑖𝑛(𝜃) (4) 𝑦6=−𝑥𝑠𝑖𝑛(𝜃)+𝑦𝑐𝑜𝑠(𝜃) (5) 𝜎1=𝜎,𝜎3=2 ϒ (6) where, the output of Gabor filter is 𝐺(𝑥,𝑦).(𝑥,𝑦) is the spatial coordinates.𝜎"is the standard deviation of the Gaussian distribution."𝜃 is the rotation angle."𝜆 is the sinusoidal operator wavelength.ϒ"is the spatial dimension ratio. ψ is the phase offset. In the proposed solver, values ψ=0.2, ϴ=0.67, σ=0.65, λ=0.57, and ϒ=0.35 were chosen based on extensive experiments conducted on the puzzles database. In the proposed solver, after computing the SSD and Gabor values for each puzzle pieces individually, these two values are added and listed for each puzzle piece. Then, for each chromosome, a fitness score is computed using equation (2) based on the combined SSD and Gabor (SSD+Gabor) values. Following this, two chromosomes are selected using the Roulette wheel selection method to participate in the crossover operator, which generates a child chromosome. The new generation, comprising the children and selected parent chr0mosomes, replaces the old generation. This procedure iterates until the predetermined number of iterations is completed. The three-phase crossover operator [Sholomon, 13] begins by selecting the first puzzle randomly. It then checks if there is an edge where both parents agree, meaning both contain the piece in the same orientation. If such a piece exists, it is placed in the correct position. If the parents agree on two or more edges, one of these edges is randomly selected, and the corresponding piece is assigned. If no common edge is 1572 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... found, the operator enters the second phase and searches for the best matching piece according to equations (7) and (8). ∀𝑥+∈𝑃𝑖𝑒𝑐𝑒𝑠.𝐶(𝑥#,𝑥$,𝑅))≥𝐶(𝑥#,𝑥+,𝑅)) (7) ∀𝑥"∈𝑃𝑖𝑒𝑐𝑒𝑠.𝐶(𝑥$,𝑥#,𝑅%)≥𝐶(𝑥$,𝑥",𝑅%) (8) where 𝑃𝑖𝑒𝑐𝑒𝑠. is a set of all the pieces of a puzzle and 𝑅) and 𝑅% are “complementary” spatial relationships (for example, if 𝑅)=right, 𝑅%=left and vice versa) and 𝐶(𝑥$,𝑥#,𝑅) is the adjacency probability of two pieces in the 𝑅 direction. In the second phase, the operator checks whether one of the parents has a piece in the spatial relation 𝑅𝑅 to 𝑃𝑃, which is also the best match for 𝑃𝑃 in that relation. If such a piece is found, it is selected and assigned. If there are several best matches, one is randomly selected. If a best match is found but has already been assigned to 𝑃𝑃, it is ignored, and the search for other best matches continues. If no best match is found, the operator proceeds to the third phase. In the third phase, the operator follows Prim’s algorithm to find a Minimum Spanning Tree (MST). The algorithm starts with an empty MST and maintains two sets of vertices (puzzle pieces). The first set contains the vertices already included in the MST, while the other set contains the vertices not yet included. At each step, the algorithm considers all edges connecting the two sets and picks the minimum weight edge. After selecting the edges, it moves the other endpoint of the edge to the set containing the MST. Thus, at every step of Prim’s algorithm, a cut is selected, and the corresponding vertex is included in the MST. To select the Gabor compatibility, we investigated alternative methods such as the ORB descriptor [Chen, 22], SIFT descriptor [Rublee, 11], Co-occurrence matrix [Humeau-Heurtier, 19], Tamura texture [Humeau-Heurtier, 19], and Variogram texture [Humeau-Heurtier, 19]. However, results indicated that descriptors invariant to the rotation and scaling are unsuitable for the compatibility criterion. This is because areas of the same color in two adjacent pieces should not be considered adjacent if they differ in direction and size. The Co-occurrence matrix, Tamura, and Varigram texture extraction algorithms, which are well-known texture extractors, have also failed to improve the accuracy of the proposed solver either alone or when combined with the SSD criterion. 4 Simulation Results, Comparisons, and Evaluation The proposed GA-based solver simulation has been performed on a system with GPU Quadro P400 with 8GB memory, and CPU Core i7-8700 3.2GHz 12 processors. The initial generation comprised 1000 randomly selected chromosomes. To obtain results, the proposed solver was executed 10 times on the database puzzles, then the average, best, and worst accuracies, along with their standard deviation, were reported based on criteria comparisons of the direct, neighbor, largest component, and complete reconstruction. Furthermore, to ensure a fair comparison of the proposed solver’s performance with similar solvers [Sholomon, 13], [Sholomon, 14], [Sholomon, 16], [Guo, 20], and [Bezulj, 18], we implemented all solvers with an initial random selection of 1000 chromosomes, and each solver was executed 10 times on the system. The accuracy comparisons of the proposed solver with the solver in [Sholomon, 14] are shown in Table 3, and Table 4. 1579 Parvin A., Mohanna F., Rezaei M.: Genetic-based square jigsaw puzzle ... 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