Advertisement

Journal of Mathematical Imaging and Vision

, Volume 53, Issue 3, pp 303–313 | Cite as

Illusory Shapes via First-Order Phase Transition and Approximation

  • Yoon Mo Jung
  • Jianhong Jackie ShenEmail author
Article
  • 186 Downloads

Abstract

We propose a new variational illusory shape (VIS) model via phase fields and phase transitions. It is inspired by the first-order variational illusory contour model proposed by Jung and Shen (J Visual Commun Image Represent 19:42–55, 2008). Under the new VIS model, illusory shapes are represented by phase values close to 1 while the rest by values close to 0. The 0–1 transition is achieved by an elliptic energy with a double-well potential, as in the theory of \(\varGamma \)-convergence. The VIS model is non-convex, with the zero field as its trivial global optimum. To seek visually meaningful local optima that can induce illusory shapes, an iterative algorithm is designed and its convergence behavior is closely studied. Several generic numerical examples confirm the versatility of the model and the algorithm.

Keywords

Illusory shapes Phase transition  Null hypothesis  Convergence 

Notes

Acknowledgments

Jung has been supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) (2012R1A1A1015492, 2014R1A1A2054763). Shen has been supported by the National Science Foundation (NSF) of USA.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \({\varGamma }\)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. 6–B, 105–123 (1992)MathSciNetGoogle Scholar
  4. 4.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2001)Google Scholar
  5. 5.
    Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34, 1948–1979 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, vol. 120, 2nd edn. Academic Press Inc, New York (1986)zbMATHGoogle Scholar
  7. 7.
    Braides, A.: \({\varGamma }\)-Convergence for Beginners. Oxford University Press, Oxford (2002)Google Scholar
  8. 8.
    Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. SIAM Publisher, Philadelphia (2005)CrossRefGoogle Scholar
  9. 9.
    Chan, T.F., Zhu, W.: Capture illusory contours: a level set based approach. UCLA CAM Report 03-65 (2003)Google Scholar
  10. 10.
    Chan, T.F., Kang, S.-H., Shen, J.: Euler’s elastica and curvature based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2002)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Chandler, D.: Introduction to Modern Statistical Mechanics. Oxford University Press, New York (1987)Google Scholar
  12. 12.
    Dal Maso, G.: An Introduction to \(\varGamma \)-Convergence. Birkhauser, Boston (1992)Google Scholar
  13. 13.
    Folland, G.B.: Real Analysis—Modern Techniques and Their Applications, 2nd edn. Wiley, Hoboken (1999)zbMATHGoogle Scholar
  14. 14.
    Freedman, D., Pisani, R., Purves, R.: Statistics. W. W. Norton and Co, New York (2007)Google Scholar
  15. 15.
    Fukushima, K.: Neural network model for completing occluded contours. Neural Netw. 23, 528–540 (2010)CrossRefGoogle Scholar
  16. 16.
    Grosof, D.H., Shapley, R.M., Hawken, M.J.: Macaque V1 neurons can signal illusory contours. Nature 365, 550–552 (1993)CrossRefGoogle Scholar
  17. 17.
    Hales, R.: Jordan’s proof of the Jordan curve theorem. Stud. Logic Gramm. Rhetor. 10(23), 45–60 (2007)Google Scholar
  18. 18.
    Han, F., Zhu, S.C.: Bottom-up/top-down image parsing with attribute graph grammars. IEEE Trans. Pattern Anal. Mach. Intell. 31(1), 59–73 (2009)Google Scholar
  19. 19.
    Jung, Y.M., Shen, J.: First-order modeling and stability analysis of illusory contours. J. Visual Commun. Image Represent. 19, 42–55 (2008)CrossRefGoogle Scholar
  20. 20.
    Kang, S.-H., Zhu, W., Shen, J.: Illusory shapes via corner fusion. SIAM J. Imaging Sci. 7(4), 1907–1936 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Knill, D.C., Richards, W.: Perception as Bayesian Inference. Cambridge University Press, New York (1996)zbMATHCrossRefGoogle Scholar
  23. 23.
    Lee, T.S., Mumford, D.: Hierarchical Bayesian inference in the visual cortex. J. Opt. Soc. Am. A 20(7), 1434–1448 (2003)CrossRefGoogle Scholar
  24. 24.
    Lee, T.S., Nguyen, M.: Dynamics of subjective contour formation in the early visual cortex. Proc. Natl. Acad. Sci. USA 98, 1907–1911 (2001)CrossRefGoogle Scholar
  25. 25.
    Léveillé, J., Versace, M., Grossberg, S.: Running as fast as it can: how spiking dynamics form object groups in the laminar circuits of visual cortex. J. Comput. Neurosci. 28, 323–346 (2010)CrossRefGoogle Scholar
  26. 26.
    March, R.: Visual reconstruction with discontinuities using variational methods. Image Vis. Comput. 10, 30–38 (1992)CrossRefGoogle Scholar
  27. 27.
    March, R., Dozio, M.: A variational method for the recovery of smooth boundaries. Image Vis. Comput. 15, 705–712 (1997)CrossRefGoogle Scholar
  28. 28.
    Mumford, D.: Elastica and computer vision. In: Bajaj, C.L. (ed.) Algebraic Geometry and its Applications, pp. 491–506. Springer, New York (1994)CrossRefGoogle Scholar
  29. 29.
    Murray, M.M., Herrmann, C.S.: Illusory contours: a window onto the neurophysiology of constructing perception. Trends Cognit. Sci. 17(9), 471–481 (2013)CrossRefGoogle Scholar
  30. 30.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(12), 12–49 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Sarti, A., Malladi, R., Sethian, J.A.: Subjective surfaces: a geometric model for boundary completion. Int. J. Comput. Vis. 46(3), 201–221 (2002)zbMATHCrossRefGoogle Scholar
  32. 32.
    Sauer, T.: Numerical Analysis. Pearson, Boston (2011)Google Scholar
  33. 33.
    Shen, J.: \(\varGamma \)-convergence approximation to piecewise constant Mumford-Shah segmentation. Lect. Notes Comput. Sci. 3708, 499–506 (2005)CrossRefGoogle Scholar
  34. 34.
    Shen, J.: A stochastic-variational model for soft Mumford-Shah segmentation. Int. J. Biomed. Imaging 2006(92329), 1–14 (2006)CrossRefGoogle Scholar
  35. 35.
    Stanley, D.A., Rubin, N.: fMRI activation in response to illusory contours and salient regions in the human lateral occipital complex. Neuron 37, 323–331 (2003)CrossRefGoogle Scholar
  36. 36.
    Vese, L.A.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim. 44(2), 131–161 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    von der Heydt, R., Peterhans, E.: Mechanism of contour perception in monkey visual cortex. I. Lines of pattern discontinuity. J. Neurosci. 9, 1731–1748 (1989)Google Scholar
  38. 38.
    von der Heydt, R., Peterhans, E., Baumgartner, G.: Illusory contours and cortical neuron responses. Science 224, 1260–1262 (1984)CrossRefGoogle Scholar
  39. 39.
    Wu, T.F., Zhu, S.C.: A numeric study of the bottom-up and top-down inference processes in and-or graphs. Int. J. Comput. Vis. 93(2), 226–252 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Yoshino, A., Kawamoto, M., Yhoshida, T., Kobayashi, N., Shigemura, J., Takahashi, Y., Nomura, S.: Activation time course of responses to illusory contours and salient region: a high-density electrical mapping comparison. Brain Res. 1071, 137–144 (2006)CrossRefGoogle Scholar
  41. 41.
    Zhu, W., Chan, T.: Illusory contours using shape information. UCLA CAM Tech. Report 03-09 (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea
  2. 2.Department of Industrial and Systems EngineeringUniversity of IllinoisUrbanaUSA

Personalised recommendations