A New Foreground and Background Image Segmentation Method Based on a Convex Shape Prior and a Nonconvex Regularizer

Foreground and background image segmentation plays an important role in the field of image processing. Convex shape priors are introduced in some image segmentation methods to estimate those implicit boundaries of image objects when these convex objects are partly obscured by unrelated surroundings that appeared in images. Classical convex shape priors based image segmentation methods use binary label functions to distinguish different image regions, which are sensitive to model parameter choices and often show poor performance in segmentation results. To solve these problems, in this paper fuzzy membership functions, which can be seen as relaxed forms of typical binary label functions, are chosen to represent different image parts, and a new variational model with respect to fuzzy membership functions is proposed in which a convex shape prior on fuzzy membership functions is designed to guarantee estimated image foregrounds having convex shapes in geometry, while to avoid the indicator function from being over-smoothed, a minimax concave regularization term is introduced to measure the smoothness of fuzzy membership functions. In order to construct an efficient algorithm to obtain desired solutions, the original model we proposed is transformed into an equivalent constrained minimization problem using the variable splitting technology, and then numerical solutions for the equivalent constrained model can be solved by an iteration algorithm which integrates the alternating direction method of multipliers, the fast Fourier transform, and the soft thresholding method. Numerical results demonstrate our proposed image segmentation method can keep estimated foreground regions convex in geometry. Compared with classical image segmentation methods, our proposed method shows more robust in model parameter choices and can obtain better convex shape in image segmentation results.