NonLocal via Local–NonLinear via Linear: A New Part-coding Distance Field via Screened Poisson Equation


Interesting phenomena in shape perception is nonlocal and nonlinear. Thus, it is crucial that a shape perception system exhibits a nonlocal and nonlinear behaviour. From the computational point of view, however, neither nonlinearity nor nonlocality is desired. We propose a repeated use of Screened Poisson PDE (leading to a sparse linear system) to compute a part coding and extracting distance field, a mapping from the shape domain \(\varOmega \subset R^n\) to the real line. Despite local and linear computations, the field exhibits highly nonlinear and nonlocal behaviour, leading to efficient and robust coding of both the local and the global structures. The proposed computation scheme is applicable to shapes in arbitrary dimensions as well as shapes implied by fragmented partial contours. The local behaviour is independent of the image context in which the shape resides.

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  1. 1.

    Aslan, C., Erdem, A., Erdem, E., Tari, S.: Disconnected skeleton: shape at its absolute scale. IEEE Trans. Pattern Anal. Mach. Intell. 30(12), 2188–2203 (2008)

    Article  Google Scholar 

  2. 2.

    Aslan, C., Tari, S.: An axis-based representation for recognition. ICCV 2, 1339–1346 (2005)

    Google Scholar 

  3. 3.

    Botsch, M., Bommes, D., Kobbelt, L.: Efficient linear system solvers for mesh processing. In: Martin, R., Bez, H., Sabin, M. (eds.) Mathematics of Surfaces XI, vol. 3604, pp. 62–83. Springer, Berlin (2005)

    Chapter  Google Scholar 

  4. 4.

    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: CVPR, vol. 2, pp. 60–65. IEEE (2005)

  5. 5.

    Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S.: Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Trans. Math. Softw. 35(3), 22:1–22:14 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cohen, E.H., Singh, M.: Geometric determinants of shape segmentation: tests using segment identification. Vision Res. 47(22), 2825–2840 (2007)

    Article  Google Scholar 

  7. 7.

    Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. UCLA CAM-report 06-57 (2006)

  8. 8.

    Giorgi, D., Biasotti, S., Paraboschi, L.: SHREC: shape retrieval contest: Watertight models track (2007)

  9. 9.

    Hassouna, M., Farag, A.: Variational curve skeletons using gradient vector flow. IEEE Trans. Pattern Anal. Mach. Intell. 31(12), 2257–2274 (2009)

    Article  Google Scholar 

  10. 10.

    Hoffman, D., Richards, W.A.: Parts of recognition. Cognition 18(1–3), 65–96 (1984)

    Article  Google Scholar 

  11. 11.

    Jung, M., Vese, L.: Nonlocal variational image deblurring models in the presence of Gaussian or impulse noise. In: SSVM, pp. 401–412. Springer (2009)

  12. 12.

    Kanizsa, G.: Organization in Vision: Essays on Gestalt Perception. Praeger, New York (1979)

    Google Scholar 

  13. 13.

    Maragos, P., Butt, M.A.: Curve evolution, differential morphology and distance transforms as applied to multiscale and Eikonal problems. Fundamentae Informatica 41(1), 91–129 (2000)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Marr, D., Nishiara, H.K.: Representation and recognition of spatial organization of three dimensional shapes. Proc. R. Soc. Lond. B Biol. Sci. 200, 269–294 (1978)

    Article  Google Scholar 

  15. 15.

    Navon, D.: Forest before trees: the precedence of global features in visual perception. Cogn. Psychol. 9(3), 355–383 (1977)

    Article  Google Scholar 

  16. 16.

    Pasupathy, A., Connor, C.E.: Population coding of shape in area V4. Nat Neurosci. 5(2), 1332–1338 (2002)

    Article  Google Scholar 

  17. 17.

    Peng, T., Jermyn, I.H., Prinet, V., Zerubia, J.: Extended phase field higher-order active contour models for networks. Int. J. Comput. Vis. 88(1), 111–128 (2010)

    Article  Google Scholar 

  18. 18.

    Rosenfeld, A., Pfaltz, J.: Distance functions on digital pictures. Pattern Recognit. 1(1), 33–61 (1968)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: CVPR, pp. 136–142. IEEE (1996)

  20. 20.

    Tari, S.: Hierarchical shape decomposition via level sets. In: ISMM, pp. 215–225. Springer (2009)

  21. 21.

    Tari, S.: Extracting parts of 2d shapes using local and global interactions simultaneosuly. In: Chen, C. (ed.) Handbook of Pattern Recognition and Computer Vision. World Scientific, Singapore (2010)

    Google Scholar 

  22. 22.

    Tari, S., Genctav, M.: From a non-local Ambrosio-Tortorelli phase field to a randomized part hierarchy tree. J. Math. Imaging Vis. 49(1), 69–86 (2014)

    Article  MATH  Google Scholar 

  23. 23.

    Tari, S., Shah, J., Pien, H.: A computationally efficient shape analysis via level sets. In: IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (1996)

  24. 24.

    Weickert, J.: Anisotropic diffusion in image processing. Teubner-Verlag, Stuttgart (1998)

    MATH  Google Scholar 

Download references


This work is funded by the Turkish National Science Foundation TUBITAK under Grant No. 112E208. We thank three anonymous reviewers for their constructive feedback.

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Correspondence to Sibel Tari.

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Genctav, M., Genctav, A. & Tari, S. NonLocal via Local–NonLinear via Linear: A New Part-coding Distance Field via Screened Poisson Equation. J Math Imaging Vis 55, 242–252 (2016).

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  • PDE-based distance transforms
  • Coding Shape
  • Screened Poisson equation
  • Feature-aware distance fields