On solving 2D and 3D puzzles using curve matching
01 Dec 2001-Vol. 2, pp 583-590
TL;DR: R ridges of 3D fragments scanned using a laser range finder are detected using a dynamic programming method and a pair of ridges are matched using a generalization of the 2D curve matching approach to space curves by using an energy solution involving curvature and torsion.
Abstract: We approach the problem of 2D and 3D puzzle solving by matching the geometric features of puzzle pieces three at a time. First, we define an affinity measure for a pair of pieces in two stages, one based on a coarse-scale representation of curves and one based on a fine-scale elastic curve matching method. This re-examination of the top coarse-scale matches at the fine scale results in an optimal relative pose as well as a matching cost which is used as the affinity measure for a pair of pieces. Pairings with overlapping boundaries are impossible and are removed from further consideration, resulting in a set of top valid candidate pairs. Second, triples arising from generic junctions are formed from this rank-ordered list of pairs. The puzzle is solved by a recursive grouping of triples using a best-first search strategy, with backtracking in the case of overlapping pieces. We also generalize aspects of this approach to matching of 3D pieces. Specifically, ridges of 3D fragments scanned using a laser range finder are detected using a dynamic programming method. A pair of ridges are matched using a generalization of the 2D curve matching approach to space curves by using an energy solution involving curvature and torsion, which are computed using a novel robust numerical method. The reconstruction of map fragments and broken tiles using this method is illustrated.
Citations
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01 Jul 2006
TL;DR: This work develops several new techniques in the area of geometry processing, including the novel integral invariants for computing multi-scale surface characteristics, registration based on forward search techniques and surface consistency, and a non-penetrating iterated closest point algorithm.
Abstract: We present a system for automatic reassembly of broken 3D solids. Given as input 3D digital models of the broken fragments, we analyze the geometry of the fracture surfaces to find a globally consistent reconstruction of the original object. Our reconstruction pipeline consists of a graph-cuts based segmentation algorithm for identifying potential fracture surfaces, feature-based robust global registration for pairwise matching of fragments, and simultaneous constrained local registration of multiple fragments. We develop several new techniques in the area of geometry processing, including the novel integral invariants for computing multi-scale surface characteristics, registration based on forward search techniques and surface consistency, and a non-penetrating iterated closest point algorithm. We illustrate the performance of our algorithms on a number of real-world examples.
318 citations
Cites background from "On solving 2D and 3D puzzles using ..."
...…Objects by Geometric Matching I 2D jigsaw puzzles [Freeman and Gardner 1964, Goldberg et al. 2004] I 2D puzzles with arbitrary boundary curves [Kong and Kimia 2001, da Gama Leitão and Stolfi 2002] I 3D puzzles with fragments of rotational pots [Willis and Cooper 2004, Kampel and Sablatnig…...
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TL;DR: It is shown that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete.
Abstract: We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type.
241 citations
Cites background from "On solving 2D and 3D puzzles using ..."
...The shape-matching community has studied computational solutions to jigsaw puzzles extensively [12,19,23,30,40]....
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13 Jun 2010
TL;DR: In this article, the problem of reconstructing an image from a bag of square, non-overlapping image patches, the jigsaw puzzle problem, is considered and a graphical model is developed to solve it.
Abstract: We explore the problem of reconstructing an image from a bag of square, non-overlapping image patches, the jigsaw puzzle problem. Completing jigsaw puzzles is challenging and requires expertise even for humans, and is known to be NP-complete. We depart from previous methods that treat the problem as a constraint satisfaction problem and develop a graphical model to solve it. Each patch location is a node and each patch is a label at nodes in the graph. A graphical model requires a pairwise compatibility term, which measures an affinity between two neighboring patches, and a local evidence term, which we lack. This paper discusses ways to obtain these terms for the jigsaw puzzle problem. We evaluate several patch compatibility metrics, including the natural image statistics measure, and experimentally show that the dissimilarity-based compatibility – measuring the sum-of-squared color difference along the abutting boundary – gives the best results. We compare two forms of local evidence for the graphical model: a sparse-and-accurate evidence and a dense-and-noisy evidence. We show that the sparse-and-accurate evidence, fixing as few as 4 – 6 patches at their correct locations, is enough to reconstruct images consisting of over 400 patches. To the best of our knowledge, this is the largest puzzle solved in the literature. We also show that one can coarsely estimate the low resolution image from a bag of patches, suggesting that a bag of image patches encodes some geometric information about the original image.
181 citations
01 Aug 2008
TL;DR: This work presents an inexpensive system for acquiring all three types of information, and associated metadata, for small objects such as fragments of wall paintings, and presents a novel 3-D matching algorithm that efficiently searches for matching fragments using the scanned geometry.
Abstract: Although mature technologies exist for acquiring images, geometry, and normals of small objects, they remain cumbersome and time-consuming for non-experts to employ on a large scale. In an archaeological setting, a practical acquisition system for routine use on every artifact and fragment would open new possibilities for archiving, analysis, and dissemination. We present an inexpensive system for acquiring all three types of information, and associated metadata, for small objects such as fragments of wall paintings. The acquisition system requires minimal supervision, so that a single, non-expert user can scan at least 10 fragments per hour. To achieve this performance, we introduce new algorithms to robustly and automatically align range scans, register 2-D scans to 3-D geometry, and compute normals from 2-D scans. As an illustrative application, we present a novel 3-D matching algorithm that efficiently searches for matching fragments using the scanned geometry.
161 citations
05 Jun 2002
TL;DR: The overall strategy follows that of previous algorithms but applies a number of new ideas, such as robust fiducial points, "highest- confidence-first" search, and frequent global reoptimization of partial solutions.
Abstract: We present a new algorithm for automatically solving jigsaw puzzles by shape alone. The algorithm can solve more difficult puzzles than could be solved before, without the use of backtracking or branch-and-bound. The algorithm can handle puzzles in which pieces border more than four neighbors, and puzzles with as many as 200 pieces. Our overall strategy follows that of previous algorithms but applies a number of new ideas, such as robust fiducial points, "highest- confidence-first" search, and frequent global reoptimization of partial solutions.
151 citations
Cites background from "On solving 2D and 3D puzzles using ..."
...Related problems include reconstructing archeological artifacts [8, 10, 11, 12, 13], mating surface patches of scanned objects [14], and even fitting a protein with known amino acid sequence to a 3D electron density map [19]....
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...Yet more possibilities include a string matching approach [3] or a dynamic programming energy minimization approach [8, 17]....
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References
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TL;DR: Two methods of sharpening contact discontinuities-the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework-are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations.
Abstract: In this paper we extend our earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes. We provide a new simplified expression for the ENO construction procedure based again on numerical fluxes rather than cell-averages. We also consider two improvements which we label ENO-LLF (local Lax-Friedrichs) and ENO-Roe, which yield sharper shock transitions, improved overall accuracy, for lower computational cost than previous implementation of the ENO schemes. Two methods of sharpening contact discontinuities-the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors originally used in the cell average framework-are applied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations. The implementation for nonlinear systems and multi-dimensions is given. Finally, many numerical examples, including a compressible shock turbulence interaction flow calculation, are presented.
5,292 citations
TL;DR: This work extends earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes by providing a new simplified expression for the ENO constructio...
Abstract: In this paper we extend our earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock-capturing schemes. We provide a new simplified expression for the ENO constructio...
3,688 citations
TL;DR: An hierarchy of uniformly high-order accurate schemes is presented which generalizes Godunov's scheme and its second- order accurate MUSCL extension to an arbitrary order of accuracy.
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01 Jan 1966
TL;DR: In this paper, the authors introduce the geometry of curves and surfaces and provide an introduction to the theory of differential geometrical geometry of surfaces and curves, as well as their properties.
Abstract: Elementary Differential Geometry, Second Edition provides an introduction to the geometry of curves and surfaces.
1,108 citations
"On solving 2D and 3D puzzles using ..." refers background in this paper
...From the local theory of curves in differential geometry [10], we know that two different curves parameterized by their arc length, have the same torsion and curvature function, (s) and (s), if and only if they are the same curves (up to a rigid motion, i....
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01 Jan 2001
TL;DR: The first fundamental form of surfaces is the curvature of surfaces as discussed by the authors, which is defined by Gauss' Theorema Egregium and the Gauss-Bonnet theorem.
Abstract: Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of surfaces.- Gaussian, mean and principal curvatures.- Geodesics.- Gauss' Theorema Egregium.- Hyperbolic geometry.- Minimal surfaces.- The Gauss-Bonnet theorem.
761 citations