Abstract
We propose a method that we call auto-adaptive convolution which extends the classical notion of convolution in pictures analysis to function analysis on a discrete set. We define an averaging kernel which takes into account the local geometry of a discrete shape and adapts itself to the curvature. Its defining property is to be local and to follow a normal law on discrete lines of any slope. We used it together with classical differentiation masks to estimate first and second derivatives and give a curvature estimator of discrete functions.
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References
Andres, E.: Modélisation Analytique Discrète d’Objets Géométriques. Habilitation à diriger des recherches, UFR Sciences Fondamentale et Appliquées, Université de Poitiers, France (2000)
Billingsley, P.: Convergence of probability measures, 2nd edn. Wiley Series in Probability and Statistics. John Wiley & Sons Inc., New York (1999)
Coeurjolly, D., Debled-Rennesson, I., Teytaud, O.: Segmentation and length estimation of 3D discrete curves. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 299–317. Springer, Heidelberg (2002)
Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 252–257 (2004)
Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.): DGCI 2008. LNCS, vol. 4992. Springer, Heidelberg (2008)
Debled-Rennesson, I., Reveillès, J.P.: A linear algorithm for segmentation of digital curves. IJPRAI 9(4), 635–662 (1995)
Feschet, F., Tougne, L.: Optimal time computation of the tangent of a discrete curve: Application to the curvature. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 31–40. Springer, Heidelberg (1999)
Fourey, S., Malgouyres, R.: Normals and curvature estimation for digital surfaces based on convolutions. In: Coeurjolly, D., et al. (eds.) [5], pp. 287–298
Gebal, K., Bærentzen, J.A., Aanæs, H., Larsen, R.: Shape Analysis Using the Auto Diffusion Function. Computer Graphics Forum 28(5), 1405–1413 (2009)
Konrad, P., Marc, A., Michael, K. (eds.): Symposium on Graphics Processing. Eurographics Association (2009)
Lachaud, J.O., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image Vision Comput. 25(10), 1572–1587 (2007)
Malgouyres, R., Brunet, F., Fourey, S.: Binomial convolutions and derivatives estimation from noisy discretizations. In: Coeurjolly, et al. (eds.) [5], pp. 370–379
Matas, J., Shao, Z., Kittler, J.: Estimation of curvature and tangent direction by median filtered differencing. In: Braccini, C., Vernazza, G., DeFloriani, L. (eds.) ICIAP 1995. LNCS, vol. 974, pp. 83–88. Springer, Heidelberg (1995)
Nguyen, T.P., Debled-Rennesson, I.: Curvature estimation in noisy curves. In: Kropatsch, W.G., Kampel, M., Hanbury, A. (eds.) CAIP 2007. LNCS, vol. 4673, pp. 474–481. Springer, Heidelberg (2007)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2(1), 15–36 (1993)
Reveillès, J.P.: Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique. Ph.D. Thesis, Université Louis Pasteur, Strasbourg, France (1991)
Rosenfeld, A.: Digital straight line segments. IEEE Transactions on Computers 23(12), 1264–1269 (1974)
Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Konrad, et al. (eds.) [10], pp. 1383–1392, http://www.eg.org/EG/DL/CGF/volume28/issue5/v28i5pp1383-1392.pdf
Worring, M., Smeulders, A.W.: Digital curvature estimation. CVGIP: Image Understanding 58(3), 366–382 (1993)
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Fiorio, C., Mercat, C., Rieux, F. (2010). Curvature Estimation for Discrete Curves Based on Auto-adaptive Masks of Convolution. In: Barneva, R.P., Brimkov, V.E., Hauptman, H.A., Natal Jorge, R.M., Tavares, J.M.R.S. (eds) Computational Modeling of Objects Represented in Images. CompIMAGE 2010. Lecture Notes in Computer Science, vol 6026. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12712-0_5
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DOI: https://doi.org/10.1007/978-3-642-12712-0_5
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