Solving Image Puzzles with a Simple Quadratic Programming Formulation

Taubin, Gabriel; Goldenstein, Siome

doi:10.1109/sibgrapi.2012.18

Cited by 23 publications

(20 citation statements)

References 12 publications

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“…An overview of puzzle tasks and strategies is well described in [8]. Our puzzle solver is in a line with recent works [2,4,13,19,6,8,15] which solve non-overlapping square-piece jigsaw puzzles. Even though Demaine et al [5] discover that puzzle assembly is an NP-hard problem if the dissimilarity metric is unreliable, the literature has seen empirical performance increases in Type 1 puzzles by using better compatibility metrics and proposing novel assembly strategies such as greedy methods [13], particle filtering [19], genetic algorithms [15], and Markov Random Field formulations solved by belief propagation [4].…”

Section: Related Workmentioning

confidence: 83%

“…-Our solver significantly outperforms state-of-the-art methods [4,13,19,6,8,15] with the standard data sets [4,12,13]. For the more challenging Type 2 puzzle setup, we reduce the error rate by up to 70% from the most accurate prior work [8].…”

Section: Introductionmentioning

confidence: 89%

“…This paper proposes a computational puzzle solver for non-overlapping squarepiece jigsaw puzzles. Many prior puzzle assembly algorithms [4,13,19,6,15] assume that the orientation of each puzzle piece is known and only the location of each piece is unknown. These are called "Type 1" puzzles.…”

Section: Introductionmentioning

confidence: 99%

“…If the loops of 4 pieces still geometrically conflict with the dominant structures, we remove them as an another form of outlier rejection. We test our method with various Type 1 and Type 2 puzzle datasets to verify that our solver outperforms state-of-the-art methods [4,13,19,6,8,15]. Our contributions are summarized below:…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solving Square Jigsaw Puzzles with Loop Constraints

Son

Hays

Cooper

2014

Computer Vision – ECCV 2014

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Abstract. We present a novel algorithm based on "loop constraints" for assembling non-overlapping square-piece jigsaw puzzles where the rotation and the position of each piece are unknown. Our algorithm finds small loops of puzzle pieces which form consistent cycles. These small loops are in turn aggregated into higher order "loops of loops" in a bottom-up fashion. In contrast to previous puzzle solvers which avoid or ignore puzzle cycles, we specifically seek out and exploit these loops as a form of outlier rejection. Our algorithm significantly outperforms state-of-the-art algorithms in puzzle reconstruction accuracy. For the most challenging type of image puzzles with unknown piece rotation we reduce the reconstruction error by up to 70%. We determine an upper bound on reconstruction accuracy for various data sets and show that, in some cases, our algorithm nearly matches the upper bound.

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Section: Related Workmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving Square Jigsaw Puzzles with Loop Constraints

Son

Hays

Cooper

2014

Computer Vision – ECCV 2014

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“…While [39] solves only for translations in the plane, [20] and [43] allow pieces to rotate and solve for their orientations as well. Methods based on non-convex constrained quadratic programming [1] as well as genetic algorithms [41] have shown competitive results. All of these approaches however, rely on the statistics of natural images (either explicitly or implicitly).…”

mentioning

confidence: 99%

A Global Approach for Solving Edge-Matching Puzzles

Kovalsky¹,

Gläsner²

2015

SIAM J. Imaging Sci.

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Abstract. We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations.Key words. edge-matching puzzles, convex optimization, relaxation, polynomial systems AMS subject classifications.1. Introduction. Jigsaw puzzles [47], dating back to the 1760s, are among the most popular single-player puzzles. The edge-matching puzzle, introduced in the 1890s, is a variation of the jigsaw puzzle in which the goal is to arrange a given collection of tiles with colored edges so that the colors match up along the edges of adjacent tiles. An example of an edge-matching puzzle with 36 square pieces is shown in Figure 1.1. Edge-matching puzzles are challenging compared to standard jigsaw puzzles as only an entire solution guarantees the correctness of any local part of the solution.Despite recent breakthroughs in algorithmic solutions for pictorial puzzles [39,20,43], in which one aims to reorganize a scrambled image, relatively little attention had been given to apictorial edge-matching puzzles. This NP-hard problem [17] sparked a lot of interest following the launch of the Eternity puzzle challenges. The challenge posed in the "Eternity I" puzzle was to tile a large dodecagon with 209 irregularly shaped smaller polygonal pieces; it was marketed as being practically unsolvable, but was solved within a year, for a prize of 1 million pounds. For the "Eternity II" puzzle one must correctly place 256 square pieces, whose edges are marked with different patterns, into a 16×16 grid. The puzzle was launched in 2007 and to date remains unsolved. A prize of 2 million dollars was on offer up until December 2010.In this paper we propose a novel representation for apictorial edge-matching puzzle games in terms of algebraic varieties, i.e., as solutions of systems of polynomial equations derived from the pieces of the puzzle. We explain how to generate systems of polynomial equations which are satisfied by puzzle solutions. We refer to systems for which the converse also holds, that is, any solution of the system is a solution of the puzzle, as complete representations. We characterize and prove the completeness of representations for 2-dimensional translation only puzzles (i.e.puzzles with 2-dimensional pieces which can be translated but not rotated).Using our algebraic representation we devise new algorithms for solving edge-matching puzzles. We show that approximate solutions can be generated by solving a sequence of continuous and global convex optimization problems. Our motivation for seek...

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A Genetic Algorithm-Based Solver for Small-Scale Jigsaw Puzzles

Guo

Wei

Zhang

et al. 2020

Lecture Notes in Computer Science

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In this paper, we present a genetic algorithm-based puzzle solver, which is mainly used to solve small-scale puzzle problems. We introduce a new measurement function that improves its accuracy by normalizing the Mahalanobis distance and the Euclidean distance between two puzzle pieces. By calculating the difference between edges of two puzzle pieces and using the genetic algorithm to assemble pieces correctly, two “parent” solutions are merged into one improved “child” solution. Using the idea of local search, it avoids the problem of local optimum solutions brought by the genetic algorithm, which greatly improves the accuracy of the puzzle.

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Solving Image Puzzles with a Simple Quadratic Programming Formulation

Cited by 23 publications

References 12 publications

Solving Square Jigsaw Puzzles with Loop Constraints

Solving Square Jigsaw Puzzles with Loop Constraints

A Global Approach for Solving Edge-Matching Puzzles

A Genetic Algorithm-Based Solver for Small-Scale Jigsaw Puzzles

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