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Reconstructing Objects from Noisy Images at Low Resolution

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 11510)


We study the problem of reconstructing small objects from their low-resolution images, by modelling them as r-regular objects. Previous work shows how the boundary constraints imposed by r-regularity allows bounds on estimation error for noise-free images. In order to utilize this for noisy images, this paper presents a graph-based framework for reconstructing noise-free images from noisy ones. We provide an optimal, but potentially computationally demanding algorithm, as well as a greedy heuristic for reconstructing noise-free images of r-regular objects from images with noise.


  • Object reconstruction
  • r-regularity

This research was supported by Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation. The authors thank François Lauze and Pawel Winter for valuable discussions.

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  1. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)

    CrossRef  Google Scholar 

  2. Chan, T., Vese, L.: Active contours without edges. IEEE TIP 10(2), 266–277 (2001)

    MATH  Google Scholar 

  3. Cootes, T.F., Taylor, C.J.: Active shape models - “smart snakes”. In: Hogg, D., Boyle, R. (eds.) BMVC92. Springer, London (1992).

    CrossRef  Google Scholar 

  4. Duarte, P., Torres, M.J.: Smoothness of boundaries of regular sets. J. Math. Imaging Vis. 48(1), 106–113 (2014)

    CrossRef  MathSciNet  Google Scholar 

  5. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. TPAMI 6(6), 721–741 (1984)

    CrossRef  Google Scholar 

  6. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. IJCV 1(4), 321–331 (1988)

    CrossRef  Google Scholar 

  7. Latecki, L., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8, 131–159 (1998)

    CrossRef  MathSciNet  Google Scholar 

  8. Litjens, G., et al.: A survey on deep learning in medical image analysis. MedIA 42, 60–88 (2017)

    Google Scholar 

  9. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–684 (1989)

    CrossRef  MathSciNet  Google Scholar 

  10. Pavlidis, T.: Algorithms for Graphics and Image Processing. Digital System Design Series. Springer, Heidelberg (1982).

    CrossRef  MATH  Google Scholar 

  11. Ronse, C., Tajine, M.: Discretization in Hausdorff space. J. Math. Imaging Vis. 12(3), 219–242 (2000)

    CrossRef  MathSciNet  Google Scholar 

  12. SDSS, S.D.S.S.

  13. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press Inc., Orlando (1983)

    Google Scholar 

  14. Stelldinger, P., Köthe, U.: Shape preservation during digitization: tight bounds based on the morphing distance. In: Michaelis, B., Krell, G. (eds.) DAGM 2003. LNCS, vol. 2781, pp. 108–115. Springer, Heidelberg (2003).

    CrossRef  Google Scholar 

  15. Stelldinger, P., Köthe, U.: Towards a general sampling theory for shape preservation. Image Vision Comput. 23(2), 237–248 (2005)

    CrossRef  Google Scholar 

  16. Svane, H., du Plessis, A.: Reconstruction of r-regular objects from trinary images, to be published in Ph.D thesis of H. Svane (2019). Until then available on Arxiv

    Google Scholar 

  17. V12.8.0, I.I.C.O.S.

  18. Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A.: A geometric snake model for segmentation of medical imagery. IEEE TMI 16(2), 199–209 (1997)

    Google Scholar 

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Correspondence to Helene Svane .

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Svane, H., Feragen, A. (2019). Reconstructing Objects from Noisy Images at Low Resolution. In: Conte, D., Ramel, JY., Foggia, P. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2019. Lecture Notes in Computer Science(), vol 11510. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-20080-0

  • Online ISBN: 978-3-030-20081-7

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