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Jigsaw Puzzles with Pieces of Unknown Orientation
Andrew C. GallagherEastman Kodak Research Laboratories
Rochester, New York
Outline
• Introduction• Solving Puzzles– Measuring Pairwise Compatibility– Tree-Based Reassembly for Types 1 and 2– An MRF for Solving Type 3 Puzzles
• Experiments• Conclusion
Introduction
• Solving puzzleassembling the pieces of a jigsaw puzzle into a complete picture
Introduction
Introduction
• Puzzle type– Type 1: known Rotation, unknown Location– Type 2: Unknown Rotation and Location– Type 3: Unknown Rotation, known Location
Solving Puzzles
• a measure of jigsaw piece compatibility• puzzle assembly
Solving Puzzles
• Measuring Pairwise Compatibility– describes the local gradients near the boundary of
a puzzle piece– Mahalanobis distance
Solving Puzzles
• Measuring Pairwise Compatibilitycompute the compatibility DLR(xi, xj) of a jigsaw piece xj on the right side of piece xi
mean distribution:
also compute the covariance matrix SiL
Solving Puzzles
• Measuring Pairwise Compatibilitycompute the gradient from the right side of piece xi to the left side of piece xj
then, Mahalanobis distance:
Solving Puzzles
• Measuring Pairwise Compatibilitymodified the above equations to compute DRL(xj, xi), then get the symmetric compatibility measure CLR(xi, xj)
store the confidence ratio in the 3D array S(xi, xj, r)
Solving Puzzles
• Evaluation in Puzzle Assembly
Similarity performance for types 1 and 2
Solving Puzzles
• Tree-Based Reassembly for Types 1 and 2– a greedy assembly algorithm inspired by Kruskal’s
algorithm for finding a minimal spanning tree– three stages:
• constrained tree stage• Trimming• Filling
Solving Puzzles
• Tree-Based Reassembly for Types 1 and 2– The constrained tree stage
• nothing prevents the MST from being a graph that results in an assembled puzzle that overlaps onto itself
• If a collision has occurred then the edge is discarded without merging the forests
Solving Puzzles
• Tree-Based Reassembly for Types 1 and 2– The constrained tree stage
Solving Puzzles
• Tree-Based Reassembly for Types 1 and 2– Trimming and Filling
Solving Puzzles
• An MRF for Solving Type 3 PuzzlesAn natural function to minimize is the total sum of the cost across the boundaries of any two pieces
Experiments
• Four measures– Direct comparison– Neighbor comparison– Largest Component– Perfect Reconstruction
Experiments
• Type 1 Puzzles
• Type 2 Puzzles
Experiments
• Type 3 Puzzles– orientation accuracy is 97.2% when considering puzzles
with 432 pieces each with 28 × 28 pixels
• Result
Experiments
• Mixed-Bag Puzzles
Conclusion
• a new class of square piece jigsaw puzzles that having pieces with unknown orientations
• a new measure (MGC) for the compatibility of a potential jigsaw piece matches
• a tree-based reassembly that greedily merges components
• a pair-wise MRF where each node represents a jigsaw piece’s orientation