1 / 20

A graph theoretic approach for the construction of concave hull in r 2

P. Jiju and M. Ramanathan Department of Engineering Design Indian Institute of Technology Madras. A graph theoretic approach for the construction of concave hull in r 2. Outline. Introduction Related Works Algorithm Implementation & Results Conclusion References. Introduction.

gefjun
Download Presentation

A graph theoretic approach for the construction of concave hull in r 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. P. Jiju and M. Ramanathan Department of Engineering Design Indian Institute of Technology Madras A graph theoretic approach for the construction of concave hull in r2 Department of Engineering Design, IIT Madras

  2. Outline • Introduction • Related Works • Algorithm • Implementation & Results • Conclusion • References

  3. Introduction • Convex hull-minimal Area convex enclosure • Limitations • Region occupied by trees in a forest • Boundary of a city • Applications of non-convex shapes • GIS • Image processing • Reconstruction • Protein structure • Data classification

  4. Related Works • Papers on concave hull • ω-concave hull algorithm[5] • K-nearest neighbor algorithm[4] • Swinging arm algorithm[3] • Concave hull[11] • Different shapes proposed for point sets • α-shape, A-shape, S-shape, r-shape, chi-shape[1,2,6,7]

  5. Limitations • lacks a standard definition • non-unique • Depends on external parameter • Application specific χ –shape for different λp

  6. Minimal Perimeter Simple Polygon • Concave hull of set of n points in plane is the minimal perimetersimple polygon which passes through all the n points • An algorithm based on Euclidean TSP • NP Complete Problem

  7. Minimal Perimeter Simple Polygon • Asymmetric point set Vs Symmetric Point set L2 L4 L3 L1 Department of Engineering Design, IIT Madras

  8. Algorithm

  9. Path Improvement • Original path • Path after a local move

  10. Path Improvement

  11. Path Improvement

  12. Implementation & Results • Used Concorde TSP solver-LKH Heuristic[8] • Point sets used were st70, krod100 and pr299 from TSPLIB

  13. Implementation & Results-ST70 points Concave hull • Presence of holes • Perimeter Length Alpha hull(α=10)

  14. Implementation & Results-KROD100 Concave hull 3. Enclosure 4. Connectedness Alpha hull(α=175)

  15. Implementation & results-PR299 Points Concave hull 5. Points spanned 6. Uniqueness Alpha hull(α=150)

  16. Comparison

  17. Conclusion & Future Work • An attempt to relate concave hull to minimum perimeter simple polygon. • Compared the concave hull with other shapes • The idea can be extended to 3-dimension • Some methodology to tackle symmetric point set

  18. Reference [1].A. R. Chaudhuri, B. B. Chaudhuri, and S. K. Parui. A novel approach to computation of the shape of a dot pattern and extraction of its perceptual border. Comput. Vis. Image Underst., 68:257–275, December 1997. [2]. H. Edelsbrunner, D. Kirkpatrick, and R. Seidel. On the shape of a set of points in the plane. Information Theory, IEEE Transactions on, 29(4):551 – 559, jul 1983. [3]. A. Galton and M. Duckham. What is the region occupied by a set of points? In M. Raubal, H. Miller, A. Frank, and M. Goodchild, editors, Geographic Information Science, volume 4197 of Lecture Notes in Computer Science, pages 81–98. Springer Berlin / Heidelberg,2006. 10.1007/118639396. [4].A. J. C. Moreira and M. Y. Santos. Concave hull: A knearestneighbours approach for the computation of the region occupied by a set of points. In GRAPP (GM/R), pages 61–68, 2007. [5]. J. Xu, Y. Feng, Z. Zheng, and X. Qing. A concave hull algorithm for scattered data and its applications. In Image and Signal Processing (CISP), 2010 3rd International Congress on, volume 5, pages 2430 –2433, oct.2010.

  19. Reference [6]. M. Melkemi and M. Djebali. Computing the shape of a planar points set. Pattern Recognition, 33(9):1423 –1436, 2000. [7]. M. Duckham, L. Kulik, M. Worboys, and A. Galton.Efficient generation of simple polygons for characterizingthe shape of a set of points in the plane. Pattern Recogn., 41:3224–3236, October 2008. [8]. D. Karapetyan and G. Gutin. Lin-Kernighan Heuristic Adaptations for the Generalized Traveling Salesman Problem. ArXiv e-prints, Mar. 2010. [9]. K. Helsgaun. An effective implementation of the linkernighan traveling salesman heuristic. European Journal of Operational Research, 126:106–130, 2000. [10]. Jin-Seo Park and Se-Jong Oh, A New Concave Hull Algorithm and Concaveness Measure for n-dimensional Datasets, Journal of Information Science and Engineering, 2011.

  20. THANK YOUQUESTIONS?

More Related