Robust Solvers for Square Jigsaw Puzzles

Debajyoti MondalYang Wang Stephane Durocher

Department of Computer ScienceUniversity of Manitoba

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What are Jigsaw Puzzles?

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Square Jigsaw Puzzles

24×18 = 432 puzzle pieces

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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/

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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

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Our Robust Jigsaw Solver (Noise and Missing Boundary)

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Our Robust Jigsaw Solver (Noise and Missing Boundary)

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How to Solve a Puzzle?

Xi Xj

Xi Xk

Xi Xj

Xi Xk

Xi Xj

Xi Xk

Xi Xj

Xi Xk

Xi Xj

Xi Xk

Xi Xj

Xi Xk

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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)

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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

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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

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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!

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Selectively Weighted MGC (wMGC)

MGCScoring matrix

3

2 2

3

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Selectively Weighted MGC (wMGC)

MGCScoring matrix

A bijection with optimum weight

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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.

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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.

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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

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Gallagher’s Reassembly [CVPR 2012]

Scoring Matrix

Construct Spanning Tree Trimming Filling

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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

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Future Research

Image Filtering?

How much does it help if we know the image category?

Robust functions for compatibility scoring.

Thank You