Shape matching using their critical feature points is useful in mechanical processes such as precision measure of manufactured parts and automatic assembly of parts. In this paper, we present a practical algorithm for measuring the similarity of two point sets A and B: Given an allowable tolerance ε, our target is to determine the feasibility of placing A with respect to B such that the maximum of the minimum distance from each point of A to its corresponding matched point in B is no larger than ε. For sparse and small point sets, an improved algorithm is achieved based on a sparse grid, which is used as an auxiliary structure for building the correspondence relationship between A and B. For large point sets, allowing a trade-off between efficiency and accuracy, we approximate the problem as computing the directed Hausdorff distance from A to B, and provide a two-phase nested Monte Carlo method for solving the problem. Experimental results are presented to validate the proposed algorithms.

References

References
1.
Bhowmick
,
P.
,
Pradhan
,
R. K.
, and
Bhattacharya
,
B. B.
,
2009
, “
Approximate Matching of Digital Point Sets Using a Novel Angular Tree
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
31
(
5
), pp.
769
782
.
2.
Li
,
H.
,
Shen
,
T.
, and
Huang
,
X.
,
2011
, “
Approximately Global Optimization for Robust Alignment of Generalized Shapes
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
33
(
6
), pp.
1116
1131
.
3.
Heffernan
,
P. J.
, and
Schirra
,
S.
,
1994
, “
Approximate Decision Algorithms for Point Set Congruence
,”
Comp. Geom. Theor. Appl.
,
4
(
3
), pp.
137
156
.
4.
Goodrich
,
M. T.
,
Mitchell
,
J. S. B.
, and
Orletsky
,
M. W.
,
1994
, “
Practical Methods for Approximate Geometric Pattern Matching Under Rigid Motion
,”
Tenth Annual ACM Symposium on Computational Geometry
(
SCG
), Stony Brook, NY, June 6–8, pp. 103–112.https://dl.acm.org/citation.cfm?id=177424.177572
5.
Gavrilov
,
M.
,
Indyk
,
P.
,
Motwani
,
R.
, and
Venkatasubramanian
,
S.
,
2004
, “
Combinatorial and Experimental Methods for Approximate Point Pattern Matching
,”
Algorithmica
,
38
(
1
), pp.
59
90
.
6.
Indyk
,
P.
, and
Venkatasubramanian
,
S.
,
2003
, “
Approximate Congruence in Nearly Linear Time
,”
Comp. Geom.
,
24
(
2
), pp.
115
128
.
7.
Arkin
,
E. M.
,
Kedem
,
K.
,
Mitchell
,
J. S. B.
,
Sprinzak
,
J.
, and
Werman
,
M.
,
1992
, “
Matching Points Into Pairwise-Disjoint Noise Regions: Combinatorial Bounds and Algorithms
,”
ORSA J. Comput.
,
4
(
4
), pp.
375
386
.
8.
Alt
,
H.
,
Mehlhorn
,
K.
,
Wagener
,
H.
, and
Welzl
,
E.
,
1988
, “
Congruence, Similarity, and Symmetries of Geometric Objects
,”
Discrete. Comput. Geom.
,
3
(
1
), pp.
237
256
.
9.
Efrat
,
A.
,
Itai
,
A.
, and
Katz
,
M. J.
,
2001
, “
Geometry Helps in Bottleneck Matching and Related Problems
,”
Algorithmica
,
31
(
1
), pp.
1
28
.
10.
Wang
,
X.
, and
Zhang
,
X.
,
2012
, “
Point Pattern Matching Algorithm for Planar Point Sets Under Euclidean Transform
,”
J. Appl. Math.
,
2012
, p.
139014
.
11.
Aiger
,
D.
, and
Kedem
,
K.
,
2009
, “
Geometric Pattern Matching for Point Sets in the Plane Under Similarity Transformations
,”
Inf. Process. Lett.
,
109
(
16
), pp.
935
940
.
12.
Huttenlocher
,
D. P.
,
Klanderman
,
G. A.
, and
Rucklidge
,
W. J.
,
1993
, “
Comparing Images Using the Hausdorff Distance
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
15
(
9
), pp.
850
863
.
13.
Takacs
,
B.
,
1998
, “
Comparing Face Images Using the Modified Hausdorff Distance
,”
Pattern Recognit.
,
31
(
12
), pp.
1873
1881
.
14.
Jesorsky
,
O.
,
Kirchberg
,
K.
, and
Frischholz
,
R.
,
2001
, “
Robust Face Detection Using the Hausdorff Distance
,”
Third International Conference on Audio- and Video-Based Biometric Person Authentication (AVBPA), Halmstad
, Sweden, June 6–8, pp. 90–95.
15.
Srisuk
,
S.
, and
Kurutach
,
W.
,
2001
, “
New Robust Hausdorff Distance-Based Face Detection
,”
International Conference on Image Processing
(
ICIP
), Thessaloniki, Greece, Oct. 7–10, pp. 1022–1025.
16.
Lu
,
Y.
,
Tan
,
C. L.
,
Huang
,
W.
, and
Fan
,
L.
,
2001
, “
An Approach to Word Image Matching Based on Weighted Hausdorff Distance
,”
Sixth International Conference on Document Analysis and Recognition
(
ICDAR
), Seattle, WA, Sept. 10–13, pp. 10–13.
17.
Dubuisson
,
M. P.
, and
Jain
,
A. K.
,
1994
, “
A Modified Hausdorff Distance for Object Matching
,”
12th International Conference on Pattern Recognition
(
ICPR
), Jerusalem, Israel, Oct. 9–13, pp. 566–568.
18.
Huttenlocher
,
D. P.
,
Kedem
,
K.
, and
Sharir
,
M.
,
1993
, “
The Upper Envelope of Voronoi Surfaces and Its Applications
,”
Discrete Comput. Geom.
,
9
(
3
), pp.
267
291
.
19.
Huttenlocher
,
D. P.
,
Kedem
,
K.
, and
Kleinberg
,
J. M.
,
1992
, “
On Dynamic Voronoi Diagrams and the Minimum Hausdorff Distance for Point Sets Under Euclidean Motion in the Plane
,”
Eighth Annual ACM Symposium on Computational Geometry
(
SCG
), Berlin, June 10–12, pp. 110–119.https://dl.acm.org/citation.cfm?id=142700
20.
Chew
,
L. P.
,
Goodrich
,
M. T.
,
Huttenlocher
,
D. P.
,
Kedem
,
K.
,
Kleinberg
,
J. M.
, and
Kravets
,
D.
,
1997
, “
Geometric Pattern Matching Under Euclidean Motion
,”
Comp. Geom. Theory. Appl.
,
7
(
1–2
), pp.
113
124
.
21.
Mount
,
D. M.
,
Netanyahu
,
N. S.
, and
Moigne
,
J. L.
,
1999
, “
Efficient Algorithms for Robust Feature Matching
,”
Pattern Recognit.
,
32
(
1
), pp.
17
38
.
22.
Cho
,
M. Y.
, and
Mount
,
D. M.
,
2008
, “
Improved Approximation Bounds for Planar Point Pattern Matching
,”
Algorithmica
,
50
(
2
), pp.
175
207
.
23.
Alt
,
H.
,
Aichholzer
,
O.
, and
Rote
,
G.
,
1994
, “
Matching Shapes With a Reference Point
,”
Tenth Annual ACM Symposium on Computational Geometry
(
SCG
), Stony Brook, NY, June 6–8, pp. 85–91.https://dl.acm.org/citation.cfm?id=177555
24.
Chazelle
,
B.
,
1983
, “
The Polygon Placement Problem
,”
Adv. Comput. Res.
,
1
, pp.
1
33
.
25.
Kirkpatrick
,
D.
,
1983
, “
Optimal Search in Planar Subdivisions
,”
SIAM J. Comput.
,
12
(
1
), pp.
28
35
.
26.
Tang
,
M.
,
Leey
,
M.
, and
Kim
,
Y. J.
,
2009
, “
Interactive Hausdorff Distance Computation for General Polygonal Models
,”
ACM Trans. Graph.
,
28
(
74
), pp.
1
9
.
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