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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
Outline
Introduction Related Works Algorithm Implementation & Results Conclusion References
Introduction
Convex 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
Related Works
Papers on concave hullω-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]
Limitations
lacks a standard definition non-uniqueDepends on external parameterApplication specific
χ –shape for different λp
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
Minimal Perimeter Simple Polygon
Asymmetric point set Vs Symmetric Point set
CAD’11, TaipeiDepartment of Engineering Design, IIT Madras
L4 L3
L2
L1
Algorithm
Path Improvement
Original path
Path after a local move
Path Improvement
Path Improvement
Implementation & Results
Used Concorde TSP solver-LKH Heuristic[8]
Point sets used were st70, krod100 and pr299 from TSPLIB
Implementation & Results-ST70
points Concave hull
Alpha hull(α=10)
1.Presence of holes
2.Perimeter Length
Implementation & Results-KROD100
Alpha hull(α=175)
Concave hull
3. Enclosure4. Connectedness
Implementation & results-PR299
PointsConcave hull
Alpha hull(α=150)
5. Points spanned
6. Uniqueness
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)
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[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.
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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