Shape of illusory figures

We have been developing a stochastic model for figure-ground separation. The model selects/constructs the foreground with preference for figures with `more convex' shapes. When these models are applied to illusory figures they yield perceptually accurate selection of figure and background. The approach is based on an `entropy' measure of a region diffusion Markov model from a set of local figure/ground hypothesis. The contour boundaries are implicitly represented, via the thresholding of the diffusion result. What optimal properties do the illusory contours satisfies ? We show that the entropy criteria selects contours such as to minimize a Taylor series of the even derivatives with respect to the length of the contour. The coefficients are positive and they get exponentially smaller as the derivatives increase. The zeroth order term suggest that small length contours are preferred, the second order terms suggests that curvature-like term is minimized (with less strength compared to the zero order one), and higher order derivatives give additional contour smoothness constraints.