Illusory Shapes via Phase Transition

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 (VIC) model proposed by Jung and Shen [{\em J. Visual Comm. Image Repres.}, {\bf 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 $\Gamma$-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.