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

Example

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 …

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

Previous Result Recall |P| = m, |Q| = n Δ : the ratio of the distances between the farthest and closest pairs of points σ : an upper bound on the Hausdorff distance given by user or computed by using binary search Running timeApproximation Bound Chew, et. al [CGH+97] O(m 3 n 2 log 2 mn)optimal Goodrich, Mitchell, Orletsky [GMO94] O(n 2 m logn)4 Indyk, Motwani, Venkatasubramanian [IMV99] O(n 4/3 Δ 1/3 logn)(1 + ε)σ Cardoze, Schulman [CS98]O(n 2 logn + log O(1) Δ)(1 + ε)σ

Our result Improved Alignment-Based Algorithm of GMO. Approximation factor is always ≤ 3.13, Approximation factor ≤ 3 + 1/(√3ρ). ρ = ½ diam(P)/h opt where diam(P) denote the diametric distance of P and h opt denote the optimal Hausdorff distance. Lower bound ≥3 + 1/(10ρ 2 ) – we present an example

Talk 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] 1.Pick a diametrical pair (p 1, p 2 ) in P 2.For all possible pairs (q i, q j ) in Q, translate p 1 to q i rotate to align p 1 p 2 with q i q j compute Hausdorff distance 3. Return the transformation with minimum Hausdorff distance

Simple Example For unique transformation between two planar point sets, we need at least two points from each set. optimal 2 x optimal

Symmetric alignment Algorithm 1.Pick a diametric pair (p 1, p 2 ) in P 2.For all possible pairs (q i, q j ) in Q, translate the midpoint of p 1 & p 2 to the midpoint of q i & q j rotate to align p 1 p 2 with q i q j compute Hausdorff distance 3. Return the transformation with minimum Hausdorff distance

Comparison Serial alignmentSymmetric alignment > <

Main Theorem Theorem. Consider two planar point sets P and Q whose optimal Hausdorff distance under rigid transformations is h opt. Recall that ρ = ½ diam(P)/h opt, where diam(P) denotes the diameter of P. Then the for all ρ > 0, the approximation ratio of symmetric alignment satisfies:

Talk 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 Proof 1.Suppose that we know the optimal transformation T* between P and Q. ( i.e. h(T*(P), Q) = h opt and each point in P has initial displacement distance ≤ h opt ) 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 h opt = 1 1.Initial Displacement 2.Translation Displacement 3.Rotation Displacement disp h opt t r

Why can we assume an optimal placement? Optimal Arbitrary Algorithm’s result is independent of initial placement

Basic Set-Up Our approximation factor is sensitive to a geometric parameter ρ. h opt :The optimal Hausdorff distance ρ : half ratio of the diametric distance of P and h opt α : the acute angle between line segment p 1 p 2 and q 1 q 2 Assume h opt = 1 and ρ > h opt h opt = 1

Translation Displacement h opt = 1

Translation Displacement (Con’t) h opt = 1

Rotation 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). h opt

Rotation and Distance from Center of Rotation The distances from the rotation center are at most √3ρ The rotation distance with side length x and angle α is 2x sinα/2.

Approximation factor: Putting it Together Approximation factor = Translation + Rotation + Initial Displacement = |T + R| + 1 ≤ |T| + |R| + 1 Due to the restriction of time & space, we just show the case, ρ -> ∞ Recall… h opt = 1

Talk 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 Example h opt = 1

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

Thanks

Result Plot of approximation factor as function of ρ

ρ < 1 h opt = 1

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 ρ, we return any transformation which the midpoint m p 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 )

Minor error for proof of GMO’94

For 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.