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


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.


Illusory shapes Phase transition  Null hypothesis  Convergence 



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.


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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

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