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Illusory Shapes via First-Order Phase Transition and Approximation

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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.

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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.

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Authors and Affiliations

  1. Department of Computational Science and Engineering, Yonsei University, Seoul, Korea

    Yoon Mo Jung

  2. Department of Industrial and Systems Engineering, University of Illinois, Urbana, IL, 61801, USA

    Jianhong Jackie Shen

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  1. Yoon Mo Jung
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  2. Jianhong Jackie Shen
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Correspondence to Jianhong Jackie Shen.

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Dedicated to Gil Strang on the Occasion of His 80th Birthday.

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Jung, Y.M., Shen, J.J. Illusory Shapes via First-Order Phase Transition and Approximation. J Math Imaging Vis 53, 303–313 (2015). https://doi.org/10.1007/s10851-015-0580-1

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