Abstract
We provide a new method to estimate the derivatives of a digital function by linear programming or other geometrical algorithms. Knowing the digitization of a real continuous function f with a resolution h, this approach provides an approximation of the k th derivative f (k)(x) with a maximal error in \(O(h^{\frac{1}{1+k}})\) where the constant depends on an upper bound of the absolute value of the (k + 1)th derivative of f in a neighborhood of x. This convergence rate \(\frac{1}{k+1}\) should be compared to the two other methods already providing such uniform convergence results, namely \(\frac{1}{3}\) from Lachaud et. al (only for the first order derivative) and \((\frac{2}{3})^k\) from Malgouyres et al..
Chapter PDF
Similar content being viewed by others
References
Esbelin, H.-A., Malgouyres, R.: Convergence of binomial-based derivative estimation for C 2 noisy discretized curves. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 57–66. Springer, Heidelberg (2009)
Gerard, Y., Debled-Rennesson, I., Zimmermann, P.: An elementary digital plane recognition algorithm. Discrete Applied Mathematics 151, 169–183 (2005)
Gilbert, E.G., Johnson, D.W., Keerthi, S.S.: A fast procedure for computing the distance between complex objects in three-dimensional space. IEEE Journal of Robotics and Automation 4, 193–203 (1988)
Lachaud, J.O., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image and Vision Computing 25(10), 1572–1587 (2007)
Malgouyres, R., Brunet, F., Fourey, S.: Binomial convolutions and derivatives estimation from noisy discretizations. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 370–379. Springer, Heidelberg (2008)
Megiddo, N.: Linear programming in linear time when the dimension is fixed. Journal of the ACM 31(1), 114–127 (1984)
Vialard, A.: Geometrical parameters extraction from discrete paths. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 24–35. Springer, Heidelberg (1996)
de Vieilleville, F., Lachaud, J.O., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. In: Kalviainen, H., Parkkinen, J., Kaarna, A. (eds.) SCIA 2005. LNCS, vol. 3540, pp. 988–997. Springer, Heidelberg (2005)
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
Provot, L., Gérard, Y. (2011). Estimation of the Derivatives of a Digital Function with a Convergent Bounded Error. 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_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-19867-0_24
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)