debajyoti mondalyang wang stephane durocher department of computer science university of manitoba
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Robust Solvers for Square Jigsaw Puzzles
Debajyoti MondalYang Wang Stephane Durocher
Department of Computer ScienceUniversity of Manitoba
CRV 2013 231/05/2013
What are Jigsaw Puzzles?
CRV 2013 331/05/2013
Square Jigsaw Puzzles
24×18 = 432 puzzle pieces
CRV 2013 431/05/2013
State-of-Art Solvers
Pomeranz, Shemesh and Ben-Shahar
CVPR 2011
Cho, Avidan and Freeman
CVPR 2010
CVPR 2012
Andrew Gallagher
Solved puzzles with 432 pieces
Average 10% accuracy on432 piece puzzles
Solved puzzles with 3300 pieces
Average 94% accuracy on432 piece puzzles
Solved puzzles with 9600 pieces
Average 95% accuracy on432 piece puzzles
http://www.cs.bgu.ac.il/faculty/person/dolevp.html http://www.cs.bgu.ac.il/faculty/person/shemeshm.html http://www.cs.bgu.ac.il/~ben-shahar/
http://www.eng.tau.ac.il/~avidan/ http://people.csail.mit.edu/taegsang/ http://people.csail.mit.edu/billf/
http://chenlab.ece.cornell.edu/people/Andy/
CRV 2013 531/05/2013
Why Solving Jigsaw Puzzles ?
Restore Torn Apart Documents
http://www.bouldercitysocial.com/wp-content/uploads/2011/04/paperShredding.jpg
Fossil Reconstructionhttp://www.aim.uzh.ch/morpho/wiki//CAP/3-2
Ancient art/document reassemblyhttp://www.edgarlowen.com/n1/b7300.jpg
CRV 2013 631/05/2013
Our Robust Jigsaw Solver (Noise and Missing Boundary)
CRV 2013 731/05/2013
Our Robust Jigsaw Solver (Noise and Missing Boundary)
CRV 2013 831/05/2013
How to Solve a Puzzle?
XiXj
XiXk
XiXj
XiXk
XiXj
XiXk
XiXj
XiXk
XiXj
XiXk
XiXj
XiXk
CRV 2013 931/05/2013
Successful Strategies
Pomeranz et. al. [CVPR 2011]Sum of Squared Distance (SSD)
Gallagher [CVPR 2012]Mahalanobis Gradient Compatibility (MGC)
SSD ( xi , xj ) = DLR ( xi , xj ) MGC ( xi , xj ) = f (μi , Gij)
CRV 2013 1031/05/2013
Our Approach: M+S
(M+S) Compatibility Score = MGC( xi , xj ) SSD( xi , xj )1/q.
MGC
SSD
M+S,4
M+S,5
M+S,6
M,S,7MGC
SSD
M+S
20 images, each with 432 Puzzle Pieces of size 28×28×3
CRV 2013 1131/05/2013
Further Refinements
MGCScoring matrix
(M+S) Compatibility Score = MGC( xi , xj ) SSD( xi , xj ) 1/q.
5 3 9 1 6 7 2 4 8
| MGC(3,1) - MGC(3,2) | < σ
Row 3
CRV 2013 1231/05/2013
How to Refine this further?
MGCScoring matrix
(M+S) Compatibility Score = MGC( xi , xj ) SSD( xi , xj ) 1/q.
5 3 9 1 6 7 2 4 8
| MGC(3,1) - MGC(3,2) | < σ
Row 3
Greedy choice!No global Agreement!
CRV 2013 1331/05/2013
Selectively Weighted MGC (wMGC)
MGCScoring matrix
3
2 2
3
CRV 2013 1431/05/2013
Selectively Weighted MGC (wMGC)
MGCScoring matrix
A bijection with optimum weight
CRV 2013 1531/05/2013
Selectively Weighted MGC (wMGC)
5
2
MGCScoring matrix
5 3 9 1 6 7 2 4 8
Row 2
wMGC (xi , xj) =
Column 4
(M+S) Score, if ‘Conflict’
MGC Score, otherwise.
CRV 2013 1631/05/2013
Selectively Weighted MGC (wMGC)
5
2
MGCScoring matrix
5 3 9 1 6 7 2 4 8
Row 2
wMGC (xi , xj) =
Column 4
(M+S) Score, if ‘Conflict’
MGC Score, otherwise.
CRV 2013 1731/05/2013
Experimental ResultsMGC
SSD
M+S
wMGC,4
(M+S) Compatibility Score = MGC( xi , xj ) SSD( xi , xj )1/q.
wMGC (xi , xj)
=(M+S) Score, if ‘Conflict’
MGC Score, otherwise.
20 images, each with 432 Puzzle Pieces of size 28×28×3
MGC
SSD
M+S
wMGC
CRV 2013 1831/05/2013
Gallagher’s Reassembly [CVPR 2012]
Scoring Matrix
Construct Spanning Tree Trimming Filling
CRV 2013 1931/05/2013
Results
Perfect Noisy Cropped 0
50
100
150
200
250
300
350
Forest
SSDMGCOurs
Perfect Noisy Cropped 0
50
100
150
200
250
300
350
City
SSDMGCOurs
MIT scene database, 328 images of forest, 308 images of city81 pieces per puzzle, each piece of size 28×28×3
CRV 2013 2031/05/2013
Future Research
Image Filtering?
How much does it help if we know the image category?
Robust functions for compatibility scoring.
Thank You
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