Abstract
In recent years, the theory behind distance functions defined by neighbourhood sequences has been developed in the digital geometry community. A neighbourhood sequence is a sequence of integers, where each element defines a neighbourhood. In this paper, we establish the equivalence between the representation of convex digital disks as an intersection of half-planes (\(\mathcal{H}\)-representation) and the expression of the distance as a maximum of non-decreasing functions.
Both forms can be deduced one from the other by taking advantage of the Lambek-Moser inverse of integer sequences.
Examples with finite sequences, cumulative sequences of periodic sequences and (almost) Beatty sequences are given. In each case, closed-form expressions are given for the distance function and \(\mathcal{H}\)-representation of disks. The results can be used to compute the pair-wise distance between points in constant time and to find optimal parameters for neighbourhood sequences.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Beatty, S.: Problem 3173. The American Mathematical Monthly 33(3), 159 (1926)
Borgefors, G.: Distance transformations in arbitrary dimensions. Computer Vision, Graphics, and Image Processing 27(3), 321–345 (1984)
Hajdu, A., Hajdu, L.: Approximating the Euclidean distance using non-periodic neighbourhood sequences. Discrete Mathematics 283(1-3), 101–111 (2004)
Lambek, J., Moser, L.: Inverse and complementary sequences of natural numbers. The American Mathematical Monthly 61(7), 454–458 (1954)
Montanari, U.: A method for obtaining skeletons using a quasi-Euclidean distance. Journal of the ACM 15(4), 600–624 (1968)
Normand, N., Evenou, 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., Arlicot, A.: Path-Based Distance with Varying Weights and Neighborhood Sequences. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 199–210. Springer, Heidelberg (2011)
Normand, N., Strand, R., Evenou, P., Arlicot, A.: Minimal-delay distance transform for neighborhood-sequence distances in 2D and 3D. Computer Vision and Image Understanding (2013)
Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. Journal of the ACM 13(4), 471–494 (1966)
Rosenfeld, A., Pfaltz, J.L.: Distances functions on digital pictures. Pattern Recognition 1(1), 33–61 (1968)
Strand, R.: Weighted distances based on neighbourhood sequences. Pattern Recognition Letters 28(15), 2029–2036 (2007)
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
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Normand, N., Strand, R., Evenou, P. (2013). Digital Distances and Integer Sequences. In: Gonzalez-Diaz, R., Jimenez, MJ., Medrano, B. (eds) Discrete Geometry for Computer Imagery. DGCI 2013. Lecture Notes in Computer Science, vol 7749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37067-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-37067-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-37066-3
Online ISBN: 978-3-642-37067-0
eBook Packages: Computer ScienceComputer Science (R0)