Abstract
In many applications of geometric processing, the border of a continuous shape and of its digitization (i.e., its pixelated representation) should be matched. Assuming that the continuous-shape boundary is locally turn bounded, we prove that there exists a mapping between the boundary of the digitization and the one of the continuous shape such that these boundaries are traveled together in a cyclic order manner. Then, we use this mapping to prove the multigrid convergence of perimeter estimators that are based on polygons inscribed in the digitization. Furthermore, convergence speed is given for this class of estimators. If, moreover, the continuous curves also have a Lipschitz turn, an explicit error bound is calculated.
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Alexandrov, A.D., Reshetnyak, Y.G.: General Theory of Irregular Curves. Kluwer Academic Pulishers, New York (1989)
Asano, T., Kawamura, Y., Klette, R., Obokata, K.: Minimum-length polygons in approximation sausages. In: Arcelli, C., Cordella, L.P., Baja, G.S. (eds.) Visual Form 2001. Lecture Notes in Computer Science, vol. 2059, pp. 103–112. Springer, Berlin Heidelberg (2001). https://doi.org/10.1007/3-540-45129-3_8
Chazal, F., Cohen-Steiner, D., Lieutier, A.: A sampling theory for compact sets in Euclidean space. Discr. Comput. Geom. 41(3), 461–479 (2009). https://doi.org/10.1007/s00454-009-9144-8
Cœurjolly, 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., Lachaud, J.O., Levallois, J.: Integral based curvature estimators in digital geometry. In: B.M. R. Gonzalez-Diaz M.J. Jimenez (ed.) 17th International Conference on Discrete Geometry for Computer Imagery (DGCI 2013), Lecture Notes in Computer Science, pp. 215–227. Springer Verlag (2013). http://liris.cnrs.fr/publis/?id=5866
Coeurjolly, D., Lachaud, J.O., Roussillon, T.: Multigrid convergence of discrete geometric estimators. In: V. Brimkov, R. Barneva (eds.) Digital Geometry Algorithms, Theoretical Foundations and Applications of Computational Imaging, Lecture Notes in Computational Vision and Biomechanics, vol. 2, pp. 395–424. Springer-Verlag (2012)
Daurat, A., Tajine, M., Zouaoui, M.: Les estimateurs semi-locaux de périmètre. Tech. rep. (2011). https://hal.archives-ouvertes.fr/hal-00576881
de Vieilleville, F., Lachaud, J.O.: Experimental comparison of continuous and discrete tangent estimators along digital curves. In: Brimkov, V.E., Barneva, R.P., Hauptman, H.A. (eds.) Combinatorial Image Analysis, pp. 26–37. Springer, Berlin Heidelberg, Berlin, Heidelberg (2008)
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.) Image Analysis, pp. 988–997. Springer, Berlin Heidelberg, Berlin, Heidelberg (2005)
de Vieilleville, F., Lachaud, J.O., Feschet, F.: Maximal digital straight segments and convergence of discrete geometric estimators. J. Math. Image Vis. 27(2), 471–502 (2007)
Federer, H.: Curvature measures. Transactions of the American Mathematical Society 93(3), 418–491 (1959). http://www.jstor.org/stable/1993504
Klette, R., Rosenfeld, A.: Geometric Methods for Digital Picture Analysis. Elsevier, Amsterdam (2004)
Klette, R., Žunić, J.: Multigrid convergence of calculated features in image analysis. J. Math. Imag. Vis. 13(3), 173–191 (2000)
Klette, R., Yang, N.: Measurements of arc length’s by shortest polygonal jordan curves. http://citr.auckland.ac.nz/techreports/1998/CITR-TR-26.pdf (1998)
Klette, R., Yip, B.: Evaluation of curve length measurements. Proc. 1st Int. Conf. Pattern Recogn. 1, 1610 (2000)
Lachaud, J., Thibert, B.: Properties of gauss digitized shapes and digital surface integration. J. Math. Imaging Vis. 54(2), 162–180 (2016). https://doi.org/10.1007/s10851-015-0595-7
Lachaud, J.O.: Espaces non-euclidiens et analyse d’image : modèles déformables riemanniens et discrets, topologie et géométrie discrète. Université Sciences et Technologies, Bordeaux I, Habilitation à diriger des recherches en informatique (2006)
Latecki, L., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Comput. Vis. Image Understand. 61(1), 70–83 (1995). https://doi.org/10.1006/cviu.1995.1006
Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8, 131–159 (1998)
Le Quentrec, É., Mazo, L., Baudrier, É., Tajine, M.: Local turn-boundedness: A curvature control for a good digitization. In: Couprie, M., Cousty, J., Kenmochi, Y., Mustafa, N. (eds.) Discrete Geometry for Computer Imagery, pp. 51–61. Springer International Publishing, Cham (2019)
Le Quentrec, E., Mazo, L., Baudrier, E., Tajine, M.: Local turn-boundedness, a curvature control for continuous curves with application to digitization. J. Math. Imaging Vis. 62, 673–692 (2020). https://doi.org/10.1007/s10851-020-00952-x. http://icube-publis.unistra.fr/2-LMBT20
Le Quentrec, E., Mazo, L., Baudrier, É., Tajine, M.: LTB curves with Lipschitz turn are par-regular. Research Report, Laboratoire ICube, université de Strasbourg (2021). https://hal.archives-ouvertes.fr/hal-03480735v1
Mazo, L., Baudrier, É.: Non-local estimators: a new class of multigrid convergent length estimators. Theor. Comput. Sci. 645, 128–146 (2016). https://doi.org/10.1016/j.tcs.2016.07.007. http://icube-publis.unistra.fr/2-MB16
Mazo, L., Baudrier, E.: Non-local length estimators and concave functions. Theor. Comput. Sci. 690, 73–90 (2017). https://doi.org/10.1016/j.tcs.2017.06.005. http://icube-publis.unistra.fr/2-MB17
Meine, H., Köthe, U., Stelldinger, P.: A topological sampling theorem for robust boundary reconstruction and image segmentation. Discr. Appl. Math. 157(3), 524–541 (2009). https://doi.org/10.1016/j.dam.2008.05.031. http://www.sciencedirect.com/science/article/pii/S0166218X08002643. International Conference on Discrete Geometry for Computer Imagery
Milnor, J.W.: On the total curvature of knots. Ann. Math. Second Ser. 52, 248–257 (1950)
Ngo, P., Passat, N., Kenmochi, Y., Debled-Rennesson, I.: Geometric preservation of 2D digital objects under rigid motions. J. Math. Imag. Vis. 61, 204–223 (2019). https://doi.org/10.1007/s10851-018-0842-9. https://hal.univ-reims.fr/hal-01695370
Pavlidis, T.: Algorithms for Graphics and Image Processing. Springer-Verlag, Berlin-Heidelberg (1982)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press Inc, USA (1983)
Sloboda, F., Zatko, B., Stoer, J.: On approximation of planar one-dimensional continua. Adv. Digital Comput. Geom. pp. 113–160 (1998)
Stelldinger, P., Terzic, K.: Digitization of non-regular shapes in arbitrary dimensions. Image Vis. Comput. 26(10), 1338–1346 (2008). https://doi.org/10.1016/j.imavis.2007.07.013. http://www.sciencedirect.com/science/article/pii/S0262885607001370
Tajine, M., Daurat, A.: Patterns for multigrid equidistributed functions: Application to general parabolas and length estimation. Theor. Comput. Sci. 412(36), 4824–4840 (2011)
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Le Quentrec, É., Mazo, L., Baudrier, É. et al. Monotonic Sampling of a Continuous Closed Curve with Respect to Its Gauss Digitization: Application to Length Estimation. J Math Imaging Vis 64, 869–891 (2022). https://doi.org/10.1007/s10851-022-01098-8
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DOI: https://doi.org/10.1007/s10851-022-01098-8