Abstract
The relative convex hull of a simple polygon A, contained in a second simple polygon B, is known to be the minimum perimeter polygon (MPP). Digital geometry studies a special case: A is the inner and B the outer polygon of a component in an image, and the MPP is called minimum length polygon (MLP). The MPP or MLP, or the relative convex hull, are uniquely defined. The paper recalls properties and algorithms related to the relative convex hull, and proposes a (recursive) algorithm for calculating the relative convex hull. The input may be simple polygons A and B in general, or inner and outer polygonal shapes in 2D digital imaging. The new algorithm is easy to understand, and is explained here for the general case. Let N be the number of vertices of A and B; the worst case time complexity is \({\cal O}(N^2)\), but it runs for “typical” (as in image analysis) inputs in linear time.
Chapter PDF
Similar content being viewed by others
Keywords
References
Klette, R., Kovalevsky, V.V., Yip, B.: Length estimation of digital curves. Vision Geometry, SPIE 3811, 117–129 (1999)
Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisco (2004)
Klette, R., Zunic, J.: Multigrid convergence of calculated features in image analysis. J. Mathematical Imaging Vision 13, 173–191 (2000)
Melkman, A.: On-line construction of the convex hull of a simple polygon. Information Processing Letters 25, 11–12 (1987)
Provençal, X., Lachaud, J.-O.: Two linear-time algorithms for computing the minimum length polygon of a digital contour. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 104–117. Springer, Heidelberg (2009)
Sklansky, J.: Measuring cavity on a rectangular mosaic. IEEE Trans. Computing 21, 1355–1364 (1972)
Sklansky, J., Kibler, D.F.: A theory of nonuniformly digitized binary pictures. IEEE Trans. Systems, Man, and Cybernetics 6, 637–647 (1976)
Sloboda, F., Stoer, J.: On piecewise linear approximation of planar Jordan curves. J. Comput. Appl. Math. 55, 369–383 (1994)
Sloboda, F., Zatko, B., Stoer, J.: On approximation of planar one-dimensional continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160 (1998)
Tarjan, R.E., Van Wyk, C.J.: An \({\cal O}(n \log \log n)\) algorithm for triangulating a simple polygon. SIAM J. Computing 17, 143–178 (1988)
Toussaint, G.T.: An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In: EURASIP, Signal processing lll: Theories and Applications, Part 2, pp. 853–856. North-Holland, Amsterdam (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Klette, G. (2011). Recursive Calculation of Relative Convex Hulls. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2011. Lecture Notes in Computer Science, vol 6607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19867-0_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-19867-0_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19866-3
Online ISBN: 978-3-642-19867-0
eBook Packages: Computer ScienceComputer Science (R0)