Solving Jigsaw Puzzles

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Solving Jigsaw Puzzles

• Presentation by
• Perouz Taslakian
• McGill University
• COMP644 Winter 2005

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Presented Paper

• A Global Approach to Automatic Solution of
  Jigsaw Puzzles
• by
• David Goldberg
• Christopher Malon
• Marshall Bern
• Proc of the 18th Annual ACM Symposium on
  Computational
• Geometry 2002 June 5-7 Barcelona Spain. NY
  ACM
• 2002 82-87.

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Introduction

• Solutions for jigsaw puzzles by shape alone date
  back to 1964
• Two main difficulties
• Combinatorial Too many ways in which pieces can
  be assembled
• Geometric difficult to detect if a pair of
  pieces really match.

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Standard Rules

• Rectangular outside border
• Each interior piece has 4 primary neighbors
• Pieces interlock by tabs (indent outdent)
• Each piece has no neighbors except its primary
  neighbors

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Standard Rules

• The primary neighbors of piece P are the pieces
  A, B, C, D

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Algorithm Overview

• Find the ordering of the border pieces
• Embed the border pieces in the plane
• Find all the pockets
• Fill in the pockets
• Optimize
• Repeat steps 3 5 until no more pieces are left

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Data Acquisition

• Use color copier to copy pieces and then scan the
  copies (at 300 dpi)
• Extract pieces by computing the connected
  components of the complement of the background
• Use morphological operations to smooth the pieces
• Take every other pixel on the boundary and do
  Gaussian smoothing

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Finding Tabs

• Find indents
• Mark the boundary near the indents
• Find outdents along the unmarked parts of the
  boundary

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Ordering the Border

• s(A,B) how well the right side of A fits the
  left side of piece B
• Use an asymmetric TSP heuristic to find an
  ordering of the border pieces

B
A
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Computing s(A, B)

• s(A, B)
• Best case x1 x2 x5 ? s(A,B) 0

A
B
x1
x5
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Embedding the Border in the Plane

• For each piece A we know its right left
  neighbor
• To align adjacent pieces, we define 1 fiducial
  point per tab (indents and outdents)
• fiducial point the center of a fitted ellipse
  passing through the two inflection points

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Embedding the Border in the Plane

• Using the fiducial points, pick points along the
  ellipse which are equiangular w.r.t. its center
• For two pieces A and B to fit, the corresponding
  tab points of each piece has to fit.

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Embedding the Border in the Plane

• To align A and B, use a least-squares fit to find
  the best rigid motion that minimizes the sum of
  the squared distances between corresponding points

B
A
a
a
c
c
b
b
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Problem!
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Global Relaxation

• Distribute error equally among pieces
• Pick k 30 pairs of corresponding points along
  each common border of two pieces, and minimize
  the sum of the squares of all intra-pair
  distances.
• Min

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Placing Interior Pieces

• At each step we find all the eligible pockets

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Placing Interior Pieces

• For each eligible pocket P, and for every piece A
  we compute
• The position of A if it really belongs to P
• The score of A using this position

A
P
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Placing Interior Pieces

• The position of A is the one that minimizes the
  sum of squares of distances between corresponding
  fiducial points of A and P

Fiducial pt.
A
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Placing Interior Pieces

• The score of A is the average distance between a
  boundary vertex of A its closest pocket point
  in P

tangent pt.
A
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Highest-Confidence Placement

• For each pocket Pi
• Ai 1st best fit (min score)
• Bi 2nd best fit
• Fill the Pi with the
• Minimum ratio
• Ai / Bi

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Highest-Confidence Placement

• For each pocket Pi
• Ai 1st best fit (min score)
• Bi 2nd best fit
• Fill the Pi with the
• Minimum ratio
• Ai / Bi

P1
P4
P21
P22
P3
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Experimental Results

• Using a Sun Ultra-60 workstation, it took
• 3 minutes to solve a 100-piece puzzle
• 20 minutes to solve a 204-piece puzzle
• Techniques applicable to other problems, for
  example shattered glass reconstruction.

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Thank You