We propose a novel method for constructing illusory foreground and background shapes from convex corners. We first introduce a new class of visual cues called corner bases, which expands zero-dimensional corner points to two-dimensional microstructures. These corner bases are then fused together by the functionalized elastica energy imposed upon an admissible phase field. The optimal phase field segments the visual field into disjoint connected components, which are further fused via a simple connectivity principle to construct both foreground illusory shapes and background occluded shapes. Robust and efficient numerical schemes are developed, and several generic examples are presented. Cognitive implications are highlighted.

We propose a new variational model for image denoising, which employs the $L^{1}$-norm of the mean curvature of the image surface $(x,f(x))$ of a given image $f:\Omega\rightarrow\mathbb{R}$. Besides eliminating noise and preserving edges of objects efficiently, our model can keep corners of objects and greyscale intensity contrasts of images and also remove the staircase effect. In this paper, we analytically study the proposed model and justify why our model can preserve object corners and image contrasts. We apply the proposed model to the denoising of curves and plane images, and also compare the results with those obtained by using the classical Rudin–Osher–Fatemi model [Phys. D, 60 (1992), pp. 259–268].

We propose a level set based variational approach that incorporates shape priors into Chan-Vese's model for the shape prior segmentation problem. In our model, besides the level set function for segmentation, as in Cremers' work, we introduce another labelling level set function to indicate the regions on which the prior shape should be compared. Our model can segment an object, whose shape is similar to the given prior shape, from a background where there are several objects. Moreover, we provide a proof for a fast solution principle, which was mentioned by F. Gibou et al., and similar to the one proposed in [B. Song et al., (2002)], for minimizing Chan-Vese's segmentation model without length term. We extend the principle to the minimization of our prescribed functionals.