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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