TY - JOUR
T1 - Geometric pattern matching for point sets in the plane under similarity transformations
AU - Aiger, Dror
AU - Kedem, Klara
N1 - Funding Information:
✩ This work was partly supported by the MAGNET program of the Israel Ministry of Industry and Trade (IMG4 consortium). * Corresponding author at: Department of Computer rion University, Be’er Sheva, Israel. E-mail address: [email protected] (D. Aiger).
PY - 2009/7/31
Y1 - 2009/7/31
N2 - We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ > 0, find a similarity transformation T (translation, rotation and scale) such that h (T (P), Q) < δ, where h (ṡ, ṡ) is the directional Hausdorff distance with L∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O (m n2 log n) time, where m and n are the numbers of points in P and Q respectively. If the L∞ distance between every pair of points in Q is at least cδ, for some c, 0 < c < 1, we present a randomized approximate solution with expected runtime O (n2 c- 4 ε- 8 log4 m n), where ε > 0 controls the approximation. Our approximation is on the size of the subset, B ⊆ P, such that h (T (B), Q) < δ and | B | > (1 - ε) | P | with high probability.
AB - We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ > 0, find a similarity transformation T (translation, rotation and scale) such that h (T (P), Q) < δ, where h (ṡ, ṡ) is the directional Hausdorff distance with L∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O (m n2 log n) time, where m and n are the numbers of points in P and Q respectively. If the L∞ distance between every pair of points in Q is at least cδ, for some c, 0 < c < 1, we present a randomized approximate solution with expected runtime O (n2 c- 4 ε- 8 log4 m n), where ε > 0 controls the approximation. Our approximation is on the size of the subset, B ⊆ P, such that h (T (B), Q) < δ and | B | > (1 - ε) | P | with high probability.
KW - Approximation algorithms
KW - Computational geometry
KW - Geometric pattern matching
KW - Hausdorff distance
KW - Randomized algorithms
UR - http://www.scopus.com/inward/record.url?scp=67649354233&partnerID=8YFLogxK
U2 - 10.1016/j.ipl.2009.04.021
DO - 10.1016/j.ipl.2009.04.021
M3 - Article
AN - SCOPUS:67649354233
SN - 0020-0190
VL - 109
SP - 935
EP - 940
JO - Information Processing Letters
JF - Information Processing Letters
IS - 16
ER -