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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.

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p jiju and m ramanathan department of engineering design indian institute of technology madras
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

outline
Outline
  • Introduction
  • Related Works
  • Algorithm
  • Implementation & Results
  • Conclusion
  • References
introduction
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
related works
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]
limitations
Limitations
      • lacks a standard definition
  • non-unique
      • Depends on external parameter
      • Application specific

χ –shape for different λp

minimal perimeter simple polygon
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
minimal perimeter simple polygon1
Minimal Perimeter Simple Polygon
  • Asymmetric point set Vs Symmetric Point set

L2

L4

L3

L1

Department of Engineering Design, IIT Madras

path improvement
Path Improvement
  • Original path
  • Path after a local move
implementation results
Implementation & Results
  • Used Concorde TSP solver-LKH Heuristic[8]
  • Point sets used were st70, krod100 and pr299 from TSPLIB
implementation results st70
Implementation & Results-ST70

points

Concave hull

  • Presence of holes
  • Perimeter Length

Alpha hull(α=10)

implementation results krod100
Implementation & Results-KROD100

Concave hull

3. Enclosure

4. Connectedness

Alpha hull(α=175)

implementation results pr299
Implementation & results-PR299

Points

Concave hull

5. Points spanned

6. Uniqueness

Alpha hull(α=150)

conclusion future work
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
reference
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.

reference1
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.