A graph theoretic approach for the construction of concave hull in r 2
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CAD’12, CanadaDepartment of Engineering Design, IIT Madras
A GRAPH THEORETIC APPROACH FOR THE CONSTRUCTION OF CONCAVE HULL IN R2
P. Jiju and M. RamanathanDepartment of Engineering DesignIndian Institute of Technology Madras
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Outline
Introduction Related Works Algorithm Implementation & Results Conclusion References
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IntroductionConvex hull-minimal Area convex enclosureLimitations
Region occupied by trees in a forestBoundary of a city
Applications of non-convex shapes GIS Image processing Reconstruction Protein structure Data classification
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Related WorksPapers 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 setsα-shape, A-shape, S-shape, r-shape, chi-
shape[1,2,6,7]
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Limitations
lacks a standard definition non-uniqueDepends on external parameterApplication specific
χ –shape for different λp
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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 TSPNP Complete Problem
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Minimal Perimeter Simple Polygon
Asymmetric point set Vs Symmetric Point set
CAD’11, TaipeiDepartment of Engineering Design, IIT Madras
L4 L3
L2
L1
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Algorithm
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Path Improvement
Original path
Path after a local move
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Path Improvement
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Path Improvement
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Implementation & Results
Used Concorde TSP solver-LKH Heuristic[8]
Point sets used were st70, krod100 and pr299 from TSPLIB
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Implementation & Results-ST70
points Concave hull
Alpha hull(α=10)
1.Presence of holes
2.Perimeter Length
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Implementation & Results-KROD100
Alpha hull(α=175)
Concave hull
3. Enclosure4. Connectedness
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Implementation & results-PR299
Points Concave hull
Alpha hull(α=150)
5. Points spanned
6. Uniqueness
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ComparisonSl. No
attributes Concave Hull
χ-shape A-shape r-shape S-shape
1 Connectedness
√ Not always
Not always
Not always
Not always
2 Uniqueness √ x x x x
3 Presence of holes
x
x √ √ √
4 Enclosure √ Not always
√ Not always
Not always
5 External parameter
x √ (l) √ (t) √ (s) √ (ε)
6 Application Reconstruction
GIS Generic Digital domain
Digital domain
7 Complexity of algorithm
O(n4) O(nlogn) - O(n) O(n)
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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
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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.
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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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