Abstract
In many applications, separable algorithms have demonstrated their efficiency to perform high performance and parallel volumetric computations, such as distance transformation or medial axis extraction. In the literature, several authors have discussed about conditions on the metric to be considered in a separable approach. In this article, we present generic separable algorithms to efficiently compute Voronoi maps and distance transformations for a large class of metrics. Focusing on path based norms (chamfer masks, neighborhood sequences, ...), we detail a subquadratic algorithm to compute such volumetric transformation in dimension 2. More precisely, we describe a O(log2 m·N 2) algorithm for shapes in a N×N domain with chamfer norm of size m.
This work has been mainly funded by ANR-11-BS02-009 and ANR-11-IDEX-0007-02 PALSE/2013/21 research grants.
Chapter PDF
References
DGtal: Digital geometry tools and algorithms library, http://dgtal.org
Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics, and Image Processing 34(3), 344–371 (1986)
Breu, H., Gil, J., Kirkpatrick, D., Werman, M.: Linear time euclidean distance transform algorithms. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(5), 529–533 (1995)
Coeurjolly, D.: Volumetric Analysis of Digital Objects Using Distance Transformation: Performance Issues and Extensions. In: Köthe, U., Montanvert, A., Soille, P. (eds.) WADGMM 2010. LNCS, vol. 7346, pp. 82–92. Springer, Heidelberg (2012)
Danielsson, P.E.: Euclidean distance mapping. Computer Graphics and Image Processing 14, 227–248 (1980)
Fouard, C., Malandain, G.: 3-D chamfer distances and norms in anisotropic grids. Image and Vision Computing 23, 143–158 (2005)
Hirata, T.: A unified linear-time algorithm for computing distance maps. Information Processing Letters 58(3), 129–133 (1996)
Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Series in Computer Graphics and Geometric Modeling. Morgan Kaufmann (2004)
Maurer, C., Qi, R., Raghavan, V.: A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions. IEEE Trans. Pattern Analysis and Machine Intelligence 25, 265–270 (2003)
Meijster, A., Roerdink, J.B.T.M., Hesselink, W.H.: A general algorithm for computing distance transforms in linear time. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 331–340. Kluwer (2000)
Mukherjee, J., Das, P.P., Kumar, M.A., Chatterji, B.N.: On approximating euclidean metrics by digital distances in 2D and 3D. Pattern Recognition Letters 21(6-7), 573–582 (2000)
Normand, N., Évenou, P.: Medial axis lookup table and test neighborhood computation for 3D chamfer norms. Pattern Recognition 42(10), 2288–2296 (2009)
Normand, N., Strand, R., Evenou, P.: Digital distances and integer sequences. In: Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B. (eds.) DGCI 2013. LNCS, vol. 7749, pp. 169–179. Springer, Heidelberg (2013)
Ragnemalm, I.: Contour processing distance transforms, pp. 204–211. World Scientific (1990)
Rosenfeld, A., Pfaltz, J.: Distance functions on digital pictures. Pattern Recognition 1, 33–61 (1968)
Rosenfeld, A., Pfaltz, J.: Sequential operations in digital picture processing. Journal of the ACM (JACM) 13, 471–494 (1966), http://portal.acm.org/citation.cfm?id=321357
Strand, R.: Distance Functions and Image Processing on Point-Lattices With Focus on the 3D Face- and Body-centered Cubic Grids. Phd thesis, Uppsala Universitet (2008)
Thiel, E.: Géométrie des distances de chanfrein. Ph.D. thesis, Aix-Marseille 2 (2001)
Verwer, B.J.H., Verbeek, P.W., Dekker, S.T.: An efficient uniform cost algorithm applied to distance transforms. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(4), 425–429 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Coeurjolly, D. (2014). 2D Subquadratic Separable Distance Transformation for Path-Based Norms. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds) Discrete Geometry for Computer Imagery. DGCI 2014. Lecture Notes in Computer Science, vol 8668. Springer, Cham. https://doi.org/10.1007/978-3-319-09955-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-09955-2_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09954-5
Online ISBN: 978-3-319-09955-2
eBook Packages: Computer ScienceComputer Science (R0)