A Convex Image Segmentation: Extending Graph Cuts and Closed-Form Matting

Image matting and segmentation are two closely related topics that concern extracting the foreground and background of an image. While the methods based on global optimization are popular in both fields, the cost functions and the optimization methods have been developed independently due to the different interests of the fields: graph cuts optimize combinatorial functions yielding hard segments, and closedform matting minimizes quadratic functions yielding soft matte. In this paper, we note that these seemingly different costs can be represented in very similar convex forms, and suggest a generalized framework based on convex optimization, which reveals a new insight. For the optimization, a primal-dual interior point method is adopted. Under the new perspective, two novel formulations are presented showing how we can improve the state-of-the-art segmentation and matting methods. We believe that this will pave the way for more sophisticated formulations in the future.