0
Research Papers

Circle-Point Containment, Monte Carlo Method for Shape Matching Based on Feature Points Using the Technique of Sparse Uniform Grids

[+] Author and Article Information
Xiangzhi Wei

Institute of Intelligent Manufacturing
and Information Engineering,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: antonwei@sjtu.edu.cn

Jie Zhao

Department of Physics and Astronomy,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: carrotzhao@sjtu.edu.cn

Siqi Qiu

Institute of Intelligent Manufacturing
and Information Engineering,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: siqiqiu@sjtu.edu.cn

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received January 18, 2017; final manuscript received January 4, 2018; published online January 31, 2018. Assoc. Editor: Monica Bordegoni.

J. Comput. Inf. Sci. Eng 18(1), 011008 (Jan 31, 2018) (8 pages) Paper No: JCISE-17-1012; doi: 10.1115/1.4038968 History: Received January 18, 2017; Revised January 04, 2018

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

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. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Heffernan, P. J. , and Schirra, S. , 1994, “Approximate Decision Algorithms for Point Set Congruence,” Comp. Geom. Theor. Appl., 4(3), pp. 137–156. [CrossRef]
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
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. [CrossRef]
Indyk, P. , and Venkatasubramanian, S. , 2003, “Approximate Congruence in Nearly Linear Time,” Comp. Geom., 24(2), pp. 115–128. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Efrat, A. , Itai, A. , and Katz, M. J. , 2001, “Geometry Helps in Bottleneck Matching and Related Problems,” Algorithmica, 31(1), pp. 1–28. [CrossRef]
Wang, X. , and Zhang, X. , 2012, “Point Pattern Matching Algorithm for Planar Point Sets Under Euclidean Transform,” J. Appl. Math., 2012, p. 139014.
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. [CrossRef]
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. [CrossRef]
Takacs, B. , 1998, “Comparing Face Images Using the Modified Hausdorff Distance,” Pattern Recognit., 31(12), pp. 1873–1881. [CrossRef]
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.
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.
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.
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.
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. [CrossRef]
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
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. [CrossRef]
Mount, D. M. , Netanyahu, N. S. , and Moigne, J. L. , 1999, “Efficient Algorithms for Robust Feature Matching,” Pattern Recognit., 32(1), pp. 17–38. [CrossRef]
Cho, M. Y. , and Mount, D. M. , 2008, “Improved Approximation Bounds for Planar Point Pattern Matching,” Algorithmica, 50(2), pp. 175–207. [CrossRef]
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
Chazelle, B. , 1983, “The Polygon Placement Problem,” Adv. Comput. Res., 1, pp. 1–33.
Kirkpatrick, D. , 1983, “Optimal Search in Planar Subdivisions,” SIAM J. Comput., 12(1), pp. 28–35. [CrossRef]
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. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Illustration of stable placements: (a) A stable placement of a dashed polygon with respect to the polygon represented by solid lines, the three contacts points are marked by disks; (b) A stable circle-point placement of A with respect to OB with three points of A being fixed on three circles ofOB

Grahic Jump Location
Fig. 1

Illustration of a laser head that requires a precise matching for the centers of the holes within a tolerance of 20μm

Grahic Jump Location
Fig. 3

A grid G with side length 2ε. The indices given along the x- and y-axes are the integer times of 2ε. Points of A (shown as dots) and points of B (shown as crosses) are divided by the cells.

Grahic Jump Location
Fig. 4

Each cell of G can contain at most two points of B in its interior (a) and at most four points of B on its boundary ((b) and (c))

Grahic Jump Location
Fig. 5

Illustration of the scope of rotation angle

Grahic Jump Location
Fig. 6

The video measuring machine used to scan planar models

Grahic Jump Location
Fig. 7

Models used for shape matching experiments: (a) the CAD model of a cycloidal gear with feature points (including the centers of the circles), (b) model I: the cycloidal gear produced by Object500 Connex (layer thickness: 16 μm), (c) model II: the cycloidal gear produced by Object30pro (layer thickness: 16 μm), and (d) model III: the cycloidal gear produced by Fortus450mc (layer thickness: 0.127 mm)

Grahic Jump Location
Fig. 8

A process of iteratively filtering out the outliers and fitting the point cloud with a polynomial curve. The unit for the x and y coordinates is millimeter.

Grahic Jump Location
Fig. 9

Matching the feature points of the cycloidal gear with the feature points of the gear's CAD model, a triangle in each figure indicates the triplet of points that is used to fix the placement of A: (a) the matching result for model I, two points of A are not matched, (b) the matching result for model II, one point of A (the uppermost one in the right figure) is not matched, and (c) the matching result for model III, all points are matched properly

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In