Variational methods with regularization techniques have become an important kind of methods image restoration. The convex total variation (TV) regularization, although achieved great successes,suffers from a contrast reduction effect. Recently nonconvex regularization techniques become popular. In this talk, I will mainly present two parts. The first one is a motivation of using nonconvex regularizations and a general truncated regularizationframework. The second is a lower bound theory for nonconvex regularized models, which shows the good edge recovery property.