Journal of Mathematical Imaging and Vision

, Volume 55, Issue 2, pp 242–252 | Cite as

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.


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



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


  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)CrossRefGoogle 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)CrossRefGoogle 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)Google Scholar
  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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohen, E.H., Singh, M.: Geometric determinants of shape segmentation: tests using segment identification. Vision Res. 47(22), 2825–2840 (2007)CrossRefGoogle Scholar
  7. 7.
    Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. UCLA CAM-report 06-57 (2006)Google Scholar
  8. 8.
    Giorgi, D., Biasotti, S., Paraboschi, L.: SHREC: shape retrieval contest: Watertight models track (2007)Google Scholar
  9. 9.
    Hassouna, M., Farag, A.: Variational curve skeletons using gradient vector flow. IEEE Trans. Pattern Anal. Mach. Intell. 31(12), 2257–2274 (2009)CrossRefGoogle Scholar
  10. 10.
    Hoffman, D., Richards, W.A.: Parts of recognition. Cognition 18(1–3), 65–96 (1984)CrossRefGoogle 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)Google Scholar
  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)MathSciNetMATHGoogle 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)CrossRefGoogle Scholar
  15. 15.
    Navon, D.: Forest before trees: the precedence of global features in visual perception. Cogn. Psychol. 9(3), 355–383 (1977)CrossRefGoogle Scholar
  16. 16.
    Pasupathy, A., Connor, C.E.: Population coding of shape in area V4. Nat Neurosci. 5(2), 1332–1338 (2002)CrossRefGoogle 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)CrossRefGoogle Scholar
  18. 18.
    Rosenfeld, A., Pfaltz, J.: Distance functions on digital pictures. Pattern Recognit. 1(1), 33–61 (1968)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: CVPR, pp. 136–142. IEEE (1996)Google Scholar
  20. 20.
    Tari, S.: Hierarchical shape decomposition via level sets. In: ISMM, pp. 215–225. Springer (2009)Google Scholar
  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)CrossRefMATHGoogle 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)Google Scholar
  24. 24.
    Weickert, J.: Anisotropic diffusion in image processing. Teubner-Verlag, Stuttgart (1998)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Middle East Technical UniversityAnkaraTurkey

Personalised recommendations