1. CAD’12, CanadaDepartment of Engineering Design, IIT Madras P. Jiju and M. Ramanathan Department of Engineering Design Indian Institute of Technology 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 λ p

6. Minimal Perimeter Simple Polygon  Concave hull of set of n points in plane is the minimal perimeter simple 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 CAD’11, TaipeiDepartment of Engineering Design, IIT Madras L4 L3 L2 L1

8. Algorithm

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

10. 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 Alpha hull(α=10) 1.Presence of holes 2.Perimeter Length

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

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

16. Comparison Sl. No attributesConcave Hull χ-shapeA-shaper-shapeS-shape 1Connected ness √Not always 2Uniqueness√xxxx 3Presence of holes x x√√√ 4Enclosure√Not always √ 5External parameter x√ (l)√ (t)√ (s)√ (ε) 6ApplicationReconstr uction GISGenericDigital domain 7Complexity of algorithm O(n 4 )O(nlogn)-O(n)

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 knearest neighbours 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.

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