Abstract
Unlike the cells of a Voronoi diagram in the real space the cells of a discrete Voronoi diagram in the discrete space has finite number of points. Hence in the literature we find a significant difference in the approach to the construction of the discrete voronoi diagram as compared to the construction of the Voronoi Diagram in the real space. In the discrete space the construction of discrete Voronoi diagram for a set of given sites is the process of assigning each pixel in the discrete space to its nearest site. Although there are many algorithms for the construction of discrete Voronoi diagram using the above mentioned approach, none of these algorithms found in the literature take into consideration the geometrical properties of the Voronoi Diagram while constructing it. We present a novel approach for the construction of discrete Voronoi Diagram for a given set of points based on a purely digital geometric approach taking into consideration its geometry. Since the circle is defined as the locus of a point that is equidistant from a given point, our algorithm constructs digital circles around each site, to assign the pixels that are nearest to that site using an iterative circle growing technique. The key idea of the proposed algorithm is that in the i-th iteration (initially \(i=1\)) we assign pixels which are at a distance given by the open interval \((i-\frac{1}{2}, i+\frac{1}{2})\) from each site, provided, they are not already assigned to any other site. Thus, at any instant of time a pixel, p is assigned to a site s if and only if s is its nearest site in the digital space. Our algorithm terminates once all pixels have been assigned to the nearest site.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aurenhammer, F.: Voronoi diagrams-a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991). https://doi.org/10.1145/116873.116880
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann Publishers Inc., San Francisco (2004)
Wein, R., van den Berg, J.P., Halperin, D.: The visibility-voronoi complex and its applications, Computational Geometry 36(1), 66–87 (2007). special Issue on the 21st European Workshop on Computational Geometry. https://doi.org/10.1016/j.comgeo.2005.11.007. https://www.sciencedirect.com/science/article/pii/S0925772106000496
Wang, X., et al.: Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes, Comput.-Aided Des. 58, 51–61 (2015). solid and Physical Modeling 2014. https://doi.org/10.1016/j.cad.2014.08.023. https://www.sciencedirect.com/science/article/pii/S0010448514001924
Kim, D.S.: A single beta-complex solves all geometry problems in a molecule, pp. 254–260 (2009). https://doi.org/10.1109/ISVD.2009.41
Pal, S., Dutta, R., Bhowmick, P.: Circular arc segmentation by curvature estimation and geometric validation. Int. J. Image Graph. 12(04), 1250024 (2012). https://doi.org/10.1142/S0219467812500246
She, B., Zhu, X., ye, X., Su, K., Lee, J.: Weighted network Voronoi diagrams for local spatial analysis. Comput. Environ. Urban Syst. 52, 70–80 (2015). https://doi.org/10.1016/j.compenvurbsys.2015.03.005
Dhar, S., Pal, S.: Surface reconstruction: roles in the field of computer vision and computer graphics. Int. J. Image Graph. 22(01), 2250008 (2022). https://doi.org/10.1142/S0219467822500085
Lindow, N., Baum, D., Hege, H.-C.: Voronoi-based extraction and visualization of molecular paths. IEEE Trans. Visual Comput. Graphics 17, 2025–34 (2011). https://doi.org/10.1109/TVCG.2011.259
Brown, K.Q.: Voronoi diagrams from convex hulls. Inf. Proc. Lett. 9(5), 223–228 (1979). https://doi.org/10.1016/0020-0190(79)90074-7
Shivanasab, P., Ali Abbaspour, R.: An incremental algorithm for simultaneous construction of 2D Voronoi diagram and Delaunay triangulation based on a face-based data structure. Adv. Eng. Softw. 169, 103129 (2022). https://www.sciencedirect.com/science/article/pii/S0965997822000400
Jida, S., Ouanan, M., Aksasse, B.: Color image segmentation using Voronoi diagram and 2D histogram. Int. J. Tomogr. Simul. 30, 14–20 (2017)
Šeda, M., Pich, V.: Robot motion planning using generalised Voronoi diagrams. target 1 (2008) q2
Biswas, R., Bhowmick, P.: Construction of persistent Voronoi diagram on 3D digital plane, pp. 93–104 (2017). https://doi.org/10.1007/978-3-319-59108-7_8
Zhao, S., Evans, T.M., Zhou, X.: Three-dimensional Voronoi analysis of monodisperse ellipsoids during triaxial shear. Powder Technol. 323, 323–336 (2018). https://doi.org/10.1016/j.powtec.2017.10.023. https://www.sciencedirect.com/science/article/pii/S0032591017308197
Surajkanta, Y., Pal, S.: Recognition of spherical segments using number theoretic properties of Isothetic covers. Multimedia Tools Appl., 1–24 (2022). https://doi.org/10.1007/s11042-022-14182-3
Fortune, S.: A sweepline algorithm for Voronoi diagrams. In: Proceedings of the Second Annual Symposium on Computational Geometry, SCG ’86. Association for Computing Machinery, New York, pp. 313-322 (1986). https://doi.org/10.1145/10515.10549
Ferreira, N., Poco, J., Vo, H.T., Freire, J., Silva, C.T.: Visual exploration of big Spatio-temporal urban data: a study of New York city taxi trips. IEEE Trans. Visual Comput. Graphics 19(12), 2149–2158 (2013). https://doi.org/10.1109/TVCG.2013.226
de Berg, M., Cheong, O., van Kreveld, M.J., Overmars, M.H.: Computational Geometry: Algorithms and Applications, 3rd ed. Springer Cham (2008). https://www.worldcat.org/oclc/227584184
Saha, D., Das, N., Pal, S.: A digital-geometric approach for computing area coverage in wireless sensor networks. In: Natarajan, R. (ed.) Distributed Computing and Internet Technology, pp. 134–145. Springer International Publishing, Cham (2014). https://doi.org/10.1007/978-3-319-04483-5_15
Allen, S.R., Barba, L., Iacono, J., Langerman, S.: Incremental Voronoi diagrams. CoRR abs/1603.08485 (2016). http://arxiv.org/abs/1603.08485
Sherbrooke, E., Patrikalakis, N., Brisson, E.: Computation of the medial axis transform of 3-D Polyhedra (1995). https://doi.org/10.1145/218013.218059
Albers, G., Guibas, L.J., Mitchell, J.S.B., Roos, T.: Voronoi diagrams of moving points. Int. J. Comput. Geom. Appl. 08(03), 365–379 (1998). https://doi.org/10.1142/S0218195998000187
Meijster, A., Roerdink, J., Hesselink, W.: A general algorithm for computing distance transforms in linear time. In: Goutsias, J., Vincent, L., Bloomberg, D.S. (eds.) Mathematical Morphology and its Applications to Image and Signal Processing. Computational Imaging and Vision, vol. 18, pp. 331–340. Springer, Boston (2002). https://doi.org/10.1007/0-306-47025-X_36
Arora, S., Raghavan, P., Rao, S.: Approximation schemes for Euclidean k-medians and related problems. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC ’98, pp. 106-113. Association for Computing Machinery, New York (1998). https://doi.org/10.1145/276698.276718
Borradaile, G., Klein, P.N., Mathieu, C.: A polynomial-time approximation scheme for euclidean steiner forest (2015)
Sugihara, K., Iri, M.: A robust topology-oriented incremental algorithm for Voronoi diagrams. Int. J. Comput. Geom. Appl. 04(02), 179–228 (1994). https://doi.org/10.1142/S0218195994000124
Hoff, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’99. ACM Press/Addison-Wesley Publishing Co., USA, pp. 277–286 (1999). https://doi.org/10.1145/311535.311567
Rong, G., Tan, T.-S.: Jump flooding in GPU with applications to Voronoi diagram and distance transform, vol. 2006, pp. 109–116 (2006). https://doi.org/10.1145/1111411.1111431
Masood, T.B., Malladi, H.K., Natarajan, V.: Facet-JFA: faster computation of discrete Voronoi diagrams. In: Proceedings of the 2014 Indian Conference on Computer Vision Graphics and Image Processing, ICVGIP ’14. Association for Computing Machinery, New York (2014). https://doi.org/10.1145/2683483.2683503
Bhowmick, P., Bhattacharya, B.B.: Number-theoretic interpretation and construction of a digital circle. Discr. Appl. Math. 156(12), 2381–2399 (2008). https://doi.org/10.1016/j.dam.2007.10.022. https://www.sciencedirect.com/science/article/pii/S0166218X07004817
Bera, S., Bhowmick, P., Bhattacharya, B.: On the characterization of absentee-voxels in a spherical surface and volume of revolution in \({\mathbb{z}}^3\) z 3. J. Math. Imaging Vision 56 (2016). https://doi.org/10.1007/s10851-016-0654-8
Andres, E., Richaume, L., Largeteau-Skapin, G.: Digital surface of revolution with hand-drawn generatrix. J. Math. Imaging Vis. (2017). https://doi.org/10.1007/s10851-017-0708-6
Andres, E.: Discrete circles, rings and spheres. Comput. Graph. 18(5), 695–706 (1994). https://doi.org/10.1016/0097-8493(94)90164-3
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Ethics declarations
Conflict of Interest
The authors declare that there is no conflict of interest.
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Dhar, S., Pal, S. (2024). A Fast and Efficient Algorithm for Construction of Discrete Voronoi Diagram. In: Kaur, H., Jakhetiya, V., Goyal, P., Khanna, P., Raman, B., Kumar, S. (eds) Computer Vision and Image Processing. CVIP 2023. Communications in Computer and Information Science, vol 2011. Springer, Cham. https://doi.org/10.1007/978-3-031-58535-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-031-58535-7_25
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-58534-0
Online ISBN: 978-3-031-58535-7
eBook Packages: Computer ScienceComputer Science (R0)