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\(\alpha \)-Pixels for Hierarchical Analysis of Digital Objects

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Scale Space and Variational Methods in Computer Vision (SSVM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14009))

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Abstract

In this paper, we apply a scale space analysis method to digital geometry. We deal with the reconstruction of Euclidean polylines controlling the size of pixels in discrete space. Numerical examples show that the size of pixels of digital geometry processes the scale for the hierarchical reconstruction of Euclidean objects from digital objects.

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References

  1. Rosenfeld, A., Klette, R.: Digital Geometry: Geometric Methods for Digital Image Analysis (The Morgan Kaufmann Series in Computer Graphics). Morgan Kaufmann, San Diego (2004)

    MATH  Google Scholar 

  2. Bruckstein, A.M.: Digital geometry in image-based metrology. In: Brimkov, V., Barneva, R. (eds.) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol. 2. Springer, Cham (2012). https://doi.org/10.1007/978-94-007-4174-4_1

    Chapter  MATH  Google Scholar 

  3. Imiya, A., Sato, K.: Shape from silhouettes in discrete space. In: Brimkov, V., Barneva, R. (eds.) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol. 2. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-007-4174-4_11

    Chapter  Google Scholar 

  4. Coeurjolly, D., Zerarga, L.: Supercover model, digital straight line recognition and curve reconstruction on the irregular isothetic grids. Comput. Graph. 30, 46–53 (2006). https://doi.org/10.1016/j.cag.2005.10.009

    Article  Google Scholar 

  5. Amenta, N., Choi, S., Dey, T.K., Leekha, N.: A simple algorithm for homeomorphic surface reconstruction. In: SCG 2000: Proceedings of the Sixteenth Annual Symposium on Computational Geometry, pp. 213–222 (2000). https://doi.org/10.1145/336154.336207

  6. Karasick, M., Lieber, D., Nackman, L.: Efficient Delaunay triangulation using rational arithmetic. ACM Trans. Graph. 10, 71–91 (1991)

    Article  Google Scholar 

  7. Alefeld, G., Mayer, G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000). https://doi.org/10.1016/S0377-0427(00)00342-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Ratschek, H.J., Rokne, J.: Geometric Computations with Interval and New Robust Methods Applications in Computer Graphics, GIS and Computational Geometry. Woodhead Publishing Limited (2003)

    Google Scholar 

  9. Vacavant, A., Roussillon, T., Kerautret, B., Lachaud, J.-O.: A combined multi-scale/irregular algorithm for the vectorization of noisy digital contours. CVIU 117, 438–450 (2013). https://doi.org/10.1016/j.cviu.2012.07.006

    Article  Google Scholar 

  10. Vacavant, A.: Fast distance transformation on two-dimensional irregular grids. Pattern Recogn. 43, 3348–3358 (2010). https://doi.org/10.1016/j.patcog.2010.04.018

    Article  MATH  Google Scholar 

  11. Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986). https://doi.org/10.4310/jdg/1214439902

    Article  MathSciNet  MATH  Google Scholar 

  12. Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987). https://doi.org/10.4310/jdg/1214441371

    Article  MathSciNet  MATH  Google Scholar 

  13. Bruckstein, A.M., Sapiro, G., Shaked, D.: Evolutions of planar polygons. IJPRAI 9, 991–1014 (1995). https://doi.org/10.1142/S0218001495000407

    Article  Google Scholar 

  14. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press (1983)

    Google Scholar 

  15. Barrodale, I., Roberts, F.D.K.: An improved algorithm for discrete \(l_1\) linear approximation. SIAM J. Numer. Anal. 10, 839–848 (1973). https://www.jstor.org/stable/2156318

  16. Wstson, G.A.: Approximation in normed linear spaces. J. Comput. Appl. Math. 121, 1–36 (2000). https://doi.org/10.1016/S0377-0427(00)00333-2

    Article  MathSciNet  Google Scholar 

  17. Matouǎek, J., Gärtner, B.: Understanding and using linear programming. Springer (2007). https://doi.org/10.1007/978-3-540-30717-4

    Article  Google Scholar 

  18. Imai, H., Kato, K., Yamamoto, P.: A linear-time algorithm for linear \(L_1\) approximation of points. Algorithmica 4, 77–96 (1989). https://doi.org/10.1007/BF01553880

    Article  MathSciNet  MATH  Google Scholar 

  19. Aronov, B., Asano, T., Katoh, N., Mehlhorn, K., Tokuyama, T.: Polyline fitting of planar points under min-sum criteria. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 77–88. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30551-4_9

    Chapter  Google Scholar 

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Acknowledgement

This research was supported by the Grants-in-Aid for Scientific Research funded by the Japan Society for the Promotion of Science, under 20K11881.

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Authors and Affiliations

  1. School of Science and Engineering, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan

    Kouki Tosaka

  2. Institute of Management and Information Technologies, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan

    Atsushi Imiya

Authors
  1. Kouki Tosaka
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  2. Atsushi Imiya
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Corresponding author

Correspondence to Atsushi Imiya .

Editor information

Editors and Affiliations

  1. CNRS, Université Côte d'Azur, Sophia-Antipolis, France

    Luca Calatroni

  2. University of Insubria, Como, Italy

    Marco Donatelli

  3. University of Bologna, Bologna, Italy

    Serena Morigi

  4. University of Modena and Reggio Emilia, Modena, Italy

    Marco Prato

  5. University of Genova, Genova, Italy

    Matteo Santacesaria

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Tosaka, K., Imiya, A. (2023). \(\alpha \)-Pixels for Hierarchical Analysis of Digital Objects. In: Calatroni, L., Donatelli, M., Morigi, S., Prato, M., Santacesaria, M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, vol 14009. Springer, Cham. https://doi.org/10.1007/978-3-031-31975-4_51

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  • DOI: https://doi.org/10.1007/978-3-031-31975-4_51

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-31974-7

  • Online ISBN: 978-3-031-31975-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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