Abstract
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.
Similar content being viewed by others
References
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)
Aslan, C., Tari, S.: An axis-based representation for recognition. ICCV 2, 1339–1346 (2005)
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)
Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: CVPR, vol. 2, pp. 60–65. IEEE (2005)
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)
Cohen, E.H., Singh, M.: Geometric determinants of shape segmentation: tests using segment identification. Vision Res. 47(22), 2825–2840 (2007)
Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. UCLA CAM-report 06-57 (2006)
Giorgi, D., Biasotti, S., Paraboschi, L.: SHREC: shape retrieval contest: Watertight models track (2007)
Hassouna, M., Farag, A.: Variational curve skeletons using gradient vector flow. IEEE Trans. Pattern Anal. Mach. Intell. 31(12), 2257–2274 (2009)
Hoffman, D., Richards, W.A.: Parts of recognition. Cognition 18(1–3), 65–96 (1984)
Jung, M., Vese, L.: Nonlocal variational image deblurring models in the presence of Gaussian or impulse noise. In: SSVM, pp. 401–412. Springer (2009)
Kanizsa, G.: Organization in Vision: Essays on Gestalt Perception. Praeger, New York (1979)
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)
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)
Navon, D.: Forest before trees: the precedence of global features in visual perception. Cogn. Psychol. 9(3), 355–383 (1977)
Pasupathy, A., Connor, C.E.: Population coding of shape in area V4. Nat Neurosci. 5(2), 1332–1338 (2002)
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)
Rosenfeld, A., Pfaltz, J.: Distance functions on digital pictures. Pattern Recognit. 1(1), 33–61 (1968)
Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: CVPR, pp. 136–142. IEEE (1996)
Tari, S.: Hierarchical shape decomposition via level sets. In: ISMM, pp. 215–225. Springer (2009)
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)
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)
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)
Weickert, J.: Anisotropic diffusion in image processing. Teubner-Verlag, Stuttgart (1998)
Acknowledgments
This work is funded by the Turkish National Science Foundation TUBITAK under Grant No. 112E208. We thank three anonymous reviewers for their constructive feedback.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10851-015-0614-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-015-0614-8