Abstract
In this paper, we present a method that we call on-surface convolution which extends the classical notion of a 2D digital filter to the case of digital surfaces (following the cuberille model). We also define an averaging mask with local support which, when applied with the iterated convolution operator, behaves like an averaging with large support. The interesting property of the latter averaging is the way the resulting weights are distributed: they tend to decrease following a “continuous” geodesic distance within the surface. We eventually use the iterated averaging followed by convolutions with differentiation masks to estimate partial derivatives and then normal vectors over a surface. We provide an heuristics based on [14] for an optimal mask size and show results.
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
Artzy, E., Frieder, G., Herman, G.T.: The theory, design, implementation and evaluation of a three dimensional surface detection algorithm. Computer graphics and image processing 15(1), 1–23 (1981)
Chen, L.-S., Herman, G.T., Reynolds, R.A., Udupa, J.K.: Surface shading in the cuberille environment. IEEE Journal of computer graphics and Application 5(12), 33–43 (1985)
Coeurjolly, D., Flin, F., Teytaud, O., Tougne, L.: Multigrid convergence and surface area estimation. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 119–124. Springer, Heidelberg (2003)
Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 252–257 (2004)
Debled-Rennesson, I., Reveillès, J.-P.: A linear algorithm for segmentation of digital curves. International Journal of Pattern Recognition and Artificial Intelligence 9(4), 635–662 (1995)
Dorst, L., Smeulders, A.W.M.: Length estimators for digitized contours. Computer Vision, Graphics, and Image Processing 40(3), 311–333 (1987)
Herman, G.T.: Discrete multidimensional Jordan surfaces. CVGIP: Graphical Models and Image Processing 54(6), 507–515 (1992)
Herman, G.T., Liu, H.K.: Three-dimensional display of human organs from computed tomograms. Computer Graphics and Image Processing 9, 1–29 (1979)
Klette, R., Sun, H.J.: A global surface area estimation algorithm for digital regular solids. Technical report, Computer science department of the University of Auckland (September 2000)
Lachaud, J.-O., Vialard, A.: Geometric measures on arbitrary dimensional digital surfaces. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 434–443. Springer, Heidelberg (2003)
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)
Lenoir, A.: Des outils pour les surfaces discrètes. PhD thesis, Université de Caen (in French) (1999)
Lenoir, A., Malgouyres, R., Revenu, M.: Fast computation of the normal vector field of the surface of a 3-d discrete object. In: Miguet, S., Ubéda, S., Montanvert, A. (eds.) DGCI 1996. LNCS, vol. 1176, pp. 101–112. Springer, Heidelberg (1996)
Malgouyres, R., Brunet, F., Fourey, S.: Binomial convolutions and derivatives estimation from noisy discretizations. In: Coeurjolly, D., et al. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 369–377. Springer, Heidelberg (2008)
Papier, L.: Polyédrisation et visualisation d’objets discrets tridimensionnels. PhD thesis, Université Louis Pasteur, Strasbourg, France (in French) (1999)
Papier, L., Françon, J.: Évalutation de la normale au bord d’un objet discret 3D. Revue de CFAO et d’informatique graphique 13, 205–226 (1998)
Sivignon, I., Dupont, F., Chassery, J.-M.: Discrete surfaces segmentation into discrete planes. In: Klette, R., Žunić, J. (eds.) IWCIA 2004. LNCS, vol. 3322, pp. 458–473. Springer, Heidelberg (2004)
Stoker, J.J.: Differential geometry. Wiley, Chichester (1989)
Tellier, P., Debled-Rennesson, I.: 3d discrete normal vectors. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 447–458. Springer, Heidelberg (1999)
Windreich, G., Kiryati, N., Lohmann, G.: Voxel-based surface area estimation: from theory to practice. Pattern Recognition 36(11), 2531–2541 (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fourey, S., Malgouyres, R. (2008). Normals and Curvature Estimation for Digital Surfaces Based on Convolutions. 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_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-79126-3_26
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)
