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Analysis of Noisy Digital Contours with Adaptive Tangential Cover

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Abstract

The notion of tangential cover, based on maximal segments, is a well-known tool to study the geometrical characteristics of a discrete curve. However, it is not robust to noise, while extracted contours from digital images typically contain noise, and this makes the geometric analysis tasks on such contours difficult. To tackle this issue, we investigate in this paper a discrete structure, named adaptive tangential cover (ATC), which is based on the notion of tangential cover and on a local noise estimator. More specifically, the ATC is composed of maximal segments with different widths deduced from the local noise values estimated at each point of the contour. Furthermore, a parameter-free algorithm is also presented to compute ATC. This study leads to the proposal of several applications of ATC on noisy digital contours: dominant point detection, contour length estimator, tangent/normal estimator, detection of convex and concave parts. An extension of ATC to 3D curves is also proposed in this paper. The experimental results demonstrate the efficiency of this new notion.

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Notes

  1. \([\![ a,b ]\!]\) indicates all integers between a and b.

  2. valid in the sense of there are no two points of 3D digital curve having the same projection onto a 2D plane.

References

  1. Reveillès, J.-P.: Géométrie discrète, calculs en nombre entiers et algorithmique, thèse d’état. Université Louis Pasteur, Strasbourg (1991)

  2. Feschet, F., Tougne, L.: Optimal time computation of the tangent of a discrete curve: application to the curvature. In: DGCI, vol. 1568 of LNCS, pp. 31–40 (1999)

  3. Feschet, F.: Canonical representations of discrete curves. Pattern Anal. Appl. 8(1–2), 84–94 (2005)

    Article  MathSciNet  Google Scholar 

  4. Lachaud, J.: Digital shape analysis with maximal segments. In: Applications of Discrete Geometry and Mathematical Morphology—First International Workshop, WADGMM 2010, Istanbul, Turkey, pp. 14–27 (2010)

  5. Kerautret, B., Lachaud, J.-O.: Meaningful scales detection along digital contours for unsupervised local noise estimation. IEEE Trans. Pattern Anal. Mach. Intell. 34(12), 2379–2392 (2012)

    Article  Google Scholar 

  6. Kerautret, B., Lachaud, J.-O.: Meaningful scales detection: an unsupervised noise detection algorithm for digital contours. Image Process. Line 4, 98–115 (2014)

    Article  Google Scholar 

  7. Malgouyres, R., Brunet, F., Fourey, S.: Binomial convolutions and derivatives estimations from noisy discretizations. In: Proceedings of the International Conference on DGCI, vol. 4992 of LNCS, Springer, pp. 370–379 (2008)

  8. Cuel, L., Lachaud, J., Mérigot, Q., Thibert, B.: Robust geometry estimation using the generalized voronoi covariance measure. SIAM J. Imaging Sci. 8(2), 1293–1314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Debled-Rennesson, I., Feschet, F., Rouyer-Degli, J.: Optimal blurred segments decomposition of noisy shapes in linear time. Comput. Graph. 30(1), 30–36 (2006)

    Article  MATH  Google Scholar 

  10. Nguyen, T.P., Debled-Rennesson, I.: On the local properties of digital curves. Int. J. Shape Model. 14(2), 105–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nguyen, T.P., Debled-Rennesson, I.: A discrete geometry approach for dominant point detection. Pattern Recognit. 44(1), 32–44 (2011)

    Article  MATH  Google Scholar 

  12. Ngo, P., Nasser, H., Debled-Rennesson, I.: Efficient dominant point detection based on discrete curve structure. In: International Workshop on Combinatorial Image Analysis IWCIA), Kolkata, India, November, vol. 9448 of LNCS (2015)

  13. Nguyen, T.P., Kerautret, B., Debled-Rennesson, I., Lachaud, J.: Unsupervised, fast and precise recognition of digital arcs in noisy images. In: Computer Vision and Graphics—International Conference, ICCVG 2010, Warsaw, Poland, September, vol. 6374 of LNCS (2010)

  14. Nguyen, T.P., Debled-Rennesson, I.: Decomposition of a curve into arcs and line segments based on dominant point detection. In: Image Analysis—17th Scandinavian Conference, SCIA, vol. 6688 of LNCS, pp. 794–805 (2011)

  15. Ngo, P., Nasser, H., Debled-Rennesson, I., Kerautret, B.: Adaptive tangential cover for noisy digital contours. In: Discrete Geometry for Computer Imagery—19th IAPR International Conference, DGCI 2016, Nantes, France, vol. 9647 of LNCS, pp. 439–451 (2016)

  16. Kerautret, B., Lachaud, J.-O., Said, M.: Meaningful thickness detection on polygonal curve. In: Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, SciTePress, pp. 372–379 (2012)

  17. Lachaud, J.-O., Vialard, A., de Vieilleville, F.: Fast, accurate and convergent tangent estimation on digital contours. Image Vis. Comput. 25(10), 1572–1587 (2007)

    Article  Google Scholar 

  18. Lachaud, J.-O., Vialard, A., de Vieilleville, F.: Analysis and comparative evaluation of discrete tangent estimators. In: Andrs, E., Damiand, G., Lienhardt, P. (eds.) Proceedings of the International Conference on Discrete Geometry for Computer Imagery (DGCI’2005), Poitiers, France, vol. 3429 of LNCS, Springer, pp. 140–251 (2005)

  19. Faure, A., Buzer, L., Feschet, F.: Tangential cover for thick digital curves. Pattern Recognit. 42(10), 2279–2287 (2009)

    Article  MATH  Google Scholar 

  20. Coeurjolly, D., Kerautret, B., Lachaud, J.-O.: Extraction of connected region boundary in multidimensional images. Image Process. Line 4, 30–43 (2014)

    Article  Google Scholar 

  21. Kanungo, T.: Document degradation models and a methodology for degradation model validation. Ph.D. thesis, University of Washington (1996)

  22. DGtalTools project: Tools associated with dgtal library. https://github.com/DGtal-team/DGtalTools

  23. Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators of digital curves. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 252–258 (2004)

    Article  Google Scholar 

  24. DGtal: Digital geometry tools and algorithms library. http://libdgtal.org

  25. Dörksen-Reiter, H., Debled-Rennesson, I.: Convex and concave parts of digital curves. In: Geometric Properties for Incomplete Data, Computational Imaging and Vision, vol. 31, pp. 145–160. Springer (2006)

  26. Coeurjolly, D., Debled-Rennesson, I., Teytaud, O.: Segmentation and length estimation of 3d discrete curves. In: Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000], pp. 299–317 (2000)

  27. Imagene, Generic digital Image library, http://gforge.liris.cnrs.frs/projects/imagene

  28. Adaptive tangential cover for noisy digital contours: Online demonstration. http://ipol-geometry.loria.fr/~kerautre/ipol_demo/ATC_IPOLDemo/

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Acknowledgements

The authors would like to thank the referees for useful comments.

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Authors and Affiliations

  1. Loria, Campus Scientifique, BP 239, 54506, Vandoeuvre-lès-Nancy, France

    Phuc Ngo, Isabelle Debled-Rennesson, Bertrand Kerautret & Hayat Nasser

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  1. Phuc Ngo
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  2. Isabelle Debled-Rennesson
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  3. Bertrand Kerautret
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  4. Hayat Nasser
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Correspondence to Phuc Ngo.

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Ngo, P., Debled-Rennesson, I., Kerautret, B. et al. Analysis of Noisy Digital Contours with Adaptive Tangential Cover. J Math Imaging Vis 59, 123–135 (2017). https://doi.org/10.1007/s10851-017-0723-7

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  • DOI: https://doi.org/10.1007/s10851-017-0723-7

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