# CONCAVE HULL: A K-NEAREST NEIGHBOURS APPROACH FOR THE COMPUTATION OF THE REGION OCCUPIED BY A SET OF POINTS

### Adriano Moreira, Maribel Yasmina Santos

#### 2007

#### Abstract

This paper describes an algorithm to compute the envelope of a set of points in a plane, which generates convex or non-convex hulls that represent the area occupied by the given points. The proposed algorithm is based on a k-nearest neighbours approach, where the value of k, the only algorithm parameter, is used to control the “smoothness” of the final solution. The obtained results show that this algorithm is able to deal with arbitrary sets of points, and that the time to compute the polygons increases approximately linearly with the number of points.

#### References

- Graham, R.L., 1972, An efficient algorithm for determining the convex hull of a planar set, Information Processing Letters 1, 132-133
- Jarvis, R.A., 1973, On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters 2, 18-21
- Preparata, F.P., and Hong, S.J., 1977, Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20, 2 (Feb.), 87-93.
- Eddy, W.F., 1977, A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3, 4 (Dec.), 398-403.
- Edelsbrunner, H., Kirkpatrick D.G, and Seidel R., 1983, On the Shape of a Set of Points in the Plane, IEEE Transactions on Information Theory, Vol. IT-29, No. 4, July
- Galton, A, and Duckham, M., 2006, What is the Region Occupied by a Set of Points?, Proceedings of the Fourth International Conference on Geographic Information Science - GIScience 2006, Munich, Germany, September 20-23
- Edelsbrunner, H., 1992a, Weighted Alpha Shapes, Technical Report: UIUCDCS-R-92-1760
- Edelsbrunner, H., and Mucke, E.P., 1992b, Threedimensional Alpha Shapes, In Proceedings of the 1992 Workshop on Volume visualization, p.75-82, Boston, Massachusets, USA, October 19-20
- Amenta, N., Bern, M., Kamvysselis, M., 1998, A New Voronoi-Based Surface Reconstruction Algorithm, Proceedings of the 25th annual conference on Computer graphics and interactive techniques, p.415- 421, July
- LOCAL, 2006, http://get.dsi.uminho.pt/local, visited December 2006.
- Santos, M. Y., and Moreira A., 2006, Automatic Classification of Location Contexts with Decision Trees, Proceedings of the Conference on Mobile and Ubiquitous Systems - CSMU 2006, p. 79-88, Guimarães, Portugal, June 29-30
- Ertoz, L., Steinbach, M. and Kumar, V., 2003, Finding Clusters of Different Sizes, Shapes, and Densities in Noisy, High Dimensional Data. In Proceedings of the Second SIAM International Conference on Data Mining, San Francisco, CA, USA, May
- Mathematica, http://www.wolfram.com, visited October 2006.

#### Paper Citation

#### in Harvard Style

Moreira A. and Yasmina Santos M. (2007). **CONCAVE HULL: A K-NEAREST NEIGHBOURS APPROACH FOR THE COMPUTATION OF THE REGION OCCUPIED BY A SET OF POINTS** . In *Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,* ISBN 978-972-8865-71-9, pages 61-68. DOI: 10.5220/0002080800610068

#### in Bibtex Style

@conference{grapp07,

author={Adriano Moreira and Maribel Yasmina Santos},

title={CONCAVE HULL: A K-NEAREST NEIGHBOURS APPROACH FOR THE COMPUTATION OF THE REGION OCCUPIED BY A SET OF POINTS},

booktitle={Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},

year={2007},

pages={61-68},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002080800610068},

isbn={978-972-8865-71-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Second International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,

TI - CONCAVE HULL: A K-NEAREST NEIGHBOURS APPROACH FOR THE COMPUTATION OF THE REGION OCCUPIED BY A SET OF POINTS

SN - 978-972-8865-71-9

AU - Moreira A.

AU - Yasmina Santos M.

PY - 2007

SP - 61

EP - 68

DO - 10.5220/0002080800610068