Improved Approximation Bounds for Improved Approximation Bounds for Planar Point Pattern MatchingPlanar Point Pattern Matching

(under rigid motions)(under rigid motions)

Minkyoung Cho Department of Computer ScienceUniversity of Maryland

Joint work with David M. Mount

ExampleExample

Problem DefinitionProblem 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 …

Directed Hausdorff distanceDirected 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

)q,p(d minmax)Q,P(hQqPp

1q

2q

QP

4q

3q1p

2p3p

)q,p(d 22

)q,p(d 43

Previous ResultPrevious Result

Recall |P| = m, |Q| = nΔ : the ratio of the distances between the farthest and closest pairs of pointsσ : an upper bound on the Hausdorff distance given by user or computed by usi

ng binary search

Running time Approximation Bound

Chew, et. al [CGH+97]

O(m3 n2 log2mn) optimal

Goodrich, Mitchell, Orletsky [GMO94]

O(n2 m logn) 4

Indyk, Motwani, Venkatasubramanian [IMV99]

O(n4/3 Δ 1/3 logn) (1 + ε)σ

Cardoze, Schulman [CS98] O(n2 logn + logO(1) Δ) (1 + ε)σ

Our resultOur result

• Improved Alignment-Based Algorithm of GMO.

• Approximation factor is always ≤ 3.13, 

• Approximation factor ≤ 3 + 1/(√3ρ).      ρ = ½ diam(P)/hopt 

      where diam(P) denote the diametric distance of P and                  hopt denote the optimal Hausdorff distance.

• Lower bound ≥3 + 1/(10ρ2)     – we present an example                      

Talk OverviewTalk Overview

- Serial alignment algorithm (GMO’94)- Symmetric alignment algorithm (ours)- Analysis of the approximation factor for symmetric

alignment - translation - rotation - Lower bound- Future work & conclusion

Serial Alignment Algorithm [GMO94]Serial Alignment Algorithm [GMO94]

1. Pick a diametrical pair (p1, p2) in P2. For all possible pairs (qi, qj) in Q,            translate p1 to qi            rotate to align p1p2 with qiqj            compute Hausdorff distance3.   Return the transformation with minimum Hausdorff distance

iq

jq

1p

2p

1p

2p

Simple ExampleSimple Example

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

optimaloptimal

2 x optimal2 x optimal1q 2q

1p

2p

1q 2q

1p 2p

1p 2p

1q 2q

Symmetric alignment AlgorithmSymmetric alignment Algorithm

1. Pick a diametric pair (p1, p2) in P

2. 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 distance3.   Return the transformation with minimum Hausdorff distance

iq

jq

1p

2p

1p

2p

ComparisonComparison

Serial alignment Symmetric alignment

>

3q

2q1q

3p

1p 2p

3q

2q1q

3p

1p 2p

<

3q

2q1q

3p

1p 2p

3q

2q1q

3p

1p 2p

Main TheoremMain Theorem

Theorem. Consider two planar point sets P and Q whose optimal Hausdorff distance under rigid transformations is hopt. Recall that

      ρ = ½ diam(P)/hopt, where diam(P) denotes the diameter of P. Then the for all ρ > 0, the approximation ratio of symmetric alignment satisfies:

124

3

13 2 ρρ,

ρmin)ρ(Asym

Talk OverviewTalk Overview

- Serial alignment algorithm (GMO’94)- Symmetric alignment algorithm (ours)- Analysis of the approximation factor for symmetric

alignment - translation - rotation - Lower bound- Future work & conclusion

Outline of the ProofOutline of the Proof

1. 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. 

Illustration of displacement distance

hopt = 1pm

2q

qm

1p

2p

1q

1. Initial Displacement2. Translation Displacement3. Rotation Displacement

disp

hoptt r

opthdisp

Appx

Why can we assume an optimal placement?Why can we assume an optimal placement?

OptimalOptimal ArbitraryArbitrary

Algorithm’s result is independent of initial placementAlgorithm’s result is independent of initial placement

Basic Set-UpBasic Set-Up

Our approximation factor is sensitive to a geometric parameter ρ.

 hopt :The optimal Hausdorff distance ρ : half ratio of the diametric distance of P and hopt α : the acute angle between line segment p1p2 and q1q2 Assume hopt = 1 and ρ > hopt

hopt = 1

pm

1q

2q

qm

1p

2pα

ρ

Translation DisplacementTranslation Displacement

21 1 )hαsinρ( |s|

22 1 )hαsinρ( |s|

),(mp 00

hopt = 1

)h,sαcosρ (q 11 )h,sαcosρ(q 22

h,ss

mq 221

pm

1q

2q

qm1s

2s

h

1p 2pρ

αcosρ

α

Translation Displacement (Con’t)Translation Displacement (Con’t)

21 1 )hαsinρ( |s| 2

2 1 )hαsinρ( |s|

h,ss

mq 221

22 h ssss

h ss

mm T2

21212

212qp

222

222

αsinρhss 21 22222

12

pmhopt = 1

1q

2q

qm

1s

2s1p 2pρ

α

1q

2q

qm

1s

2s

Rotation DisplacementRotation Displacement

A rotation  displacement distance depends on angle(α) and distance(x) from center of rotation.

And, the maximum rotational distance will be | R | ≤ 2√3ρ sin(α/2).

hopt

pm

1q

2q

qm

1p2p

α

ρ

Rotation and Distance from Center of RotationRotation and Distance from Center of Rotation

22

αsinx

x

ρ3ρ

α

The distances from The distances from the rotation centerthe rotation centerare at most are at most √√33ρρ

The rotation distance withThe rotation distance withside length x and angle side length x and angle αα

is is 2x sin2x sinαα/2/2..

1p 2p

pm

x

pm

1p 2p

Approximation factor: Putting it TogetherApproximation factor: Putting it Together

Approximation factor = Translation + Rotation + Initial Displacement = |T + R| + 1 ≤ |T| + |R| + 1

12

321 22 α

sinραsinρ

131

12

321

12

321

2

22

22

xx

αραρ

αsinραsinρ

32

3 appx ,αsinρραx

Due to the restriction of time & space, we just show the case, Due to the restriction of time & space, we just show the case, ρρ -> ∞ -> ∞

Recall… hRecall… hoptopt = 1 = 1

ρ31

3

limρ

Talk OverviewTalk Overview

- Serial alignment algorithm (GMO’94)- Symmetric alignment algorithm (ours)- Analysis of the approximation factor for symmetric

alignment - translation - rotation - Lower bound- Future work & conclusion

Lower Bound ExampleLower Bound Example

hopt = 1pm

2qqm

1p 2p

1q

ρarcsinα

2

3

ρ3

ρ

210

13

ρ

Future WorkFuture Work

• Does there exist a factor 3 approximation based on simple point alignments

• Improve running time & robustness?

ThanksThanks

ResultResult

Plot of approximation factor as function of Plot of approximation factor as function of ρρ

ρρ < 1 < 1

ρ3

ρ1q 2q

1p 2phopt = 1

xx.ρ,ρρ)ρ()ρ( 1342131 222

One fixingOne 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 ρ, 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 ) 

1q 2q

1p 2p

1q1p 2p

Minor error for proof of GMO’94Minor error for proof of GMO’94

1q

2q

1p

2p

For High DimensionFor High Dimension

GMO algorithm works for all dimension. How about symmetric alignment? If we match d-points symmetrically, it’s 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.