Improved Approximation Bounds for Planar Point Pattern Matching under rigid motions

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Improved Approximation Bounds for Planar Point
Pattern Matching (under rigid motions)

• Minkyoung Cho
• Department of Computer Science
• University of Maryland
• Joint work with David M. Mount

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Example
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Problem Definition

• Point Pattern Matching Given a pattern set P
  (size m) and a background set Q (size n), compute
  the transformation T that minimizes some distance
  measure from T(P) to Q.
• Transformation
• a. Translation
• b. Translation Rotation (Rigid
  Transformation)
• c. Translation Rotation Scale .
• Distance Measure
• a. Mean Squared Error
• b. (Bidirectional, Directed) Hausdorff
  distance
• c. Absolute distance
• d. Hamming distance

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Directed Hausdorff distance

•  
• Def maximum distance of a set(P) to the nearest
  point in the other set(Q)
• i.e.
• h(P, Q)
• h(Q, P)
• Property Not symmetric

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Previous Result

• Recall P m, Q n
• ? the ratio of the distances between the
  farthest and closest pairs of points
• s an upper bound on the Hausdorff distance
  given by user or computed by using binary search

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Our result

• Improved Alignment-Based Algorithm of GMO.
• Approximation factor is always 3.13,
• Approximation factor 3 1/(v3?).
• ? ½ diam(P)/hopt
• where diam(P) denote the diametric distance
  of P and
• hopt denote the optimal
  Hausdorff distance.
• Lower bound 3 1/(10?2)
• we present an example

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Talk Overview

• Serial alignment algorithm (GMO94)
• Symmetric alignment algorithm (ours)
• Analysis of the approximation factor for
  symmetric alignment
• - translation
• - rotation
• Lower bound
• Future work conclusion

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Serial Alignment Algorithm GMO94

• Pick a diametrical pair (p1, p2) in P
• For all possible pairs (qi, qj) in Q,
• translate p1 to qi
• rotate to align p1p2 with qiqj
• compute Hausdorff distance
• 3. Return the transformation with minimum
  Hausdorff distance

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Simple Example

• For unique transformation between two planar
  point sets, we need at least two points from each
  set.

optimal
2 x optimal
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Symmetric alignment Algorithm

• Pick a diametric pair (p1, p2) in P
• For all possible pairs (qi, qj) in Q,
• translate the midpoint of p1 p2 to
  the midpoint of qi qj
• rotate to align p1p2 with qiqj
• compute Hausdorff distance
• 3. Return the transformation with minimum
  Hausdorff distance

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Comparison
Serial alignment
Symmetric alignment
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Main Theorem

• Theorem. Consider two planar point sets P and Q
  whose optimal Hausdorff distance under rigid
  transformations is hopt. Recall that
• ? ½ diam(P)/hopt, where diam(P) denotes
  the diameter of P. Then the for all ? gt 0, the
  approximation ratio of symmetric alignment
  satisfies

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Talk Overview

• Serial alignment algorithm (GMO94)
• Symmetric alignment algorithm (ours)
• Analysis of the approximation factor for
  symmetric alignment
• - translation
• - rotation
• Lower bound
• Future work conclusion

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Outline of the Proof

• Suppose that we know the optimal transformation
  T between P and Q. ( i.e. h(T(P), Q) hopt and
  
• each point in P has initial
  displacement distance hopt)
• 2. Apply Symmetric alignment algorithm
  (translation rotation) to the optimal solution
• - compute the upper bound of translation
  displacement distance
• - compute the upper bound of rotation
  displacement distance
• 3. Add these three displacement distances. It
  will become the approximation factor.

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Illustration of displacement distance
disp

• Initial Displacement
• Translation Displacement
• Rotation Displacement

hopt
r
t
hopt 1
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Why can we assume an optimal placement?
Arbitrary
Optimal
Algorithms result is independent of initial
placement
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Basic Set-Up

• Our approximation factor is sensitive to a
  geometric parameter ?.
• hopt The optimal Hausdorff distance
• ? half ratio of the diametric distance of P
  and hopt
• a the acute angle between line segment p1p2
  and q1q2
• Assume hopt 1 and ? gt hopt

hopt 1
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Translation Displacement
hopt 1
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Translation Displacement (Cont)
hopt 1
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Rotation Displacement

• A rotation displacement distance depends on
  angle(a) and distance(x) from center of rotation.
• And, the maximum rotational distance will be R
  2v3? sin(a/2).

hopt
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Rotation and Distance from Center of Rotation
The distances from the rotation center are at
most v3?
The rotation distance with side length x and
angle a is 2x sina/2.
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Approximation factor Putting it Together

• Approximation factor Translation Rotation
  Initial Displacement
• T R 1
  T R 1

Recall hopt 1
Due to the restriction of time space, we just
show the case, ? -gt 8
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Talk Overview

• Serial alignment algorithm (GMO94)
• Symmetric alignment algorithm (ours)
• Analysis of the approximation factor for
  symmetric alignment
• - translation
• - rotation
• Lower bound
• Future work conclusion

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Lower Bound Example
hopt 1
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Future Work

• Does there exist a factor 3 approximation based
  on simple point alignments
• Improve running time robustness?

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Thanks
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Result
Plot of approximation factor as function of ?
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? lt 1
hopt 1
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One fixing

• We assumed that the diameter pair has a
  corresponding pair. (Even though it can be a
  point, not a pair)
• If the minimum Hausdorff distance from symmetric
  alignment is bigger than ?, we return any
  transformation which the midpoint mp matches with
  any point in Q.
• This is quite unrealistic case since the
  transformation is not uniquely decided and any
  point in Q can be matched with all points in P.(
  meaningless )

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Minor error for proof of GMO94
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For High Dimension

• GMO algorithm works for all dimension. How about
  symmetric alignment? If we match d-points
  symmetrically, its unbounded.
• However, if we follow GMO algorithm except
  matching midpoint rather than match one of the
  points, then our algorithm can be extend to all
  dimension with better approximation factor.