Abstract
In this paper we prove that the function giving the frequency of a class of patterns of digital planes with respect to the slopes of the plane is continuous and piecewise affine, moreover the regions of affinity are precised. It allows to prove some combinatorial properties of a class of patterns called (m,n)-cubes. This study has also some consequences on local estimators of area: we prove that the local estimators restricted to regions of plane never converge to the exact area when the resolution tends to zero for almost all slope of plane. Actually all the results of this paper can be generalized for the regions of hyperplanes for any dimension d ≥ 3.
The proofs of some results used in this article are contained in the extended version of this paper [1].
Chapter PDF
Similar content being viewed by others
References
Daurat, A., Tajine, M., Zouaoui, M.: About the frequencies of some patterns in digital planes. Application to area estimators - extended version (preprint, 2007), http://hal.archives-ouvertes.fr/hal-00174960
Forchhammer, S.: Digital plane and grid point segments. Comput. Vis. Graph. Image Process 47, 373–384 (1989)
Françon, J., Schramm, J.M., Tajine, M.: Recognizing arithmetic straight lines and planes. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 141–150. Springer, Heidelberg (1996)
Schramm, J.M.: Coplanar tricubes. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 87–98. Springer, Heidelberg (1997)
Reveillès, J.P.: Combinatorial pieces in digital lines and planes. In: Proc. SPIE Vision Geometry IV, vol. 2573, pp. 23–34 (1995)
Gérard, Y.: Contribution à la Géométrie Discrète. PhD thesis, Université d’Auvergne, Clermont-Ferrand (1999)
Gérard, Y.: Local Configurations of Digital Hyperplanes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 65–95. Springer, Heidelberg (1999)
Vittone, J., Chassery, J.M. (n,m)-Cubes and Farey Nets for Naive Planes Understanding. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 76–90. Springer, Heidelberg (1999)
Vittone, J.: Caractérisation et reconnaissance de droites et de plans en géométrie discrète. PhD thesis, Université Joseph Fourier, Grenoble (1999)
Vuillon, L.: Combinatoire des motifs d’une suite sturmienne bidimensionnelle. Theoret. Comput. Sci. 209, 261–285 (1998)
Vuillon, L.: Local configurations in a discrete plane. Bull. Belg. Math. Soc. Simon Stevin 6(4), 625–636 (1999)
Brimkov, V.E., Andres, E., Barneva, R.P.: Object discretizations in higher dimensions. Pattern Recogn. Lett. 23(6), 623–636 (2002)
Veelaert, P.: Digital planarity of rectangular surface segments. IEEE Trans. Pattern Anal. Mach. Intell. 16(6), 647–652 (1994)
Brimkov, V., Coeurjolly, D., Klette, R.: Digital planarity - a review. Discrete Appl. Math. 155(4), 468–495 (2007)
Lindblad, J.: Surface area estimation of digitized 3d objects using weighted local configurations. Image Vis. Comput. 23(2), 111–122 (2005)
Kenmochi, Y., Klette, R.: Surface area estimation for digitized regular solids. In: Proc. SPIE, Vision Geometry IX, vol. 4117, pp. 100–111 (2000)
Tajine, M., Daurat, A.: On Local Definitions of Length of Digital Curves. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 114–123. Springer, Heidelberg (2003)
Coeurjolly, D., Sivignon, I., Dupont, F., Feschet, F., Chassery, J.M.: On digital plane preimage structure. Discrete Appl. Math. 151(1-3), 78–92 (2005)
Berthé, V., Fiorio, C., Jamet, D., Philippe, F.: On some applications of generalized functionality for arithmetic discrete planes. Image Vis. Comput. 25(10), 1671–1684 (2007)
Berenstein, C., Lavine, D.: On the number of digital straight line segments. IEEE Trans. On Pattern Analysis and Machine Intelligence 10(6), 880–887 (1988)
Mignosi, F.: On the number of factors of Sturmian words. Theoret. Comput. Sci. 82(1), 71–84 (1991)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Daurat, A., Tajine, M., Zouaoui, M. (2008). About the Frequencies of Some Patterns in Digital Planes Application to Area Estimators. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds) Discrete Geometry for Computer Imagery. DGCI 2008. Lecture Notes in Computer Science, vol 4992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79126-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-79126-3_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79125-6
Online ISBN: 978-3-540-79126-3
eBook Packages: Computer ScienceComputer Science (R0)