Coupled image restoration model with non‐convex non‐smooth ℓ _p wavelet frame and total variation regularisation

Normal curvature‐induced variational model for image restoration

“…One of the most famous regularisation terms is the total variation (TV) regulariser, which was first proposed for image denoising [6] and turned out to be very effective for removing noise while preserving image edges. Owing to its good ability to preserve edges, TV has been widely applied to many image processing inverse problems [7][8][9][10][11][12][13][14]. Since edges in an image lead to outliers in the regularisation term, then Fu et al [15] propose the l 1 norm-based regularisation for image restoration, which is closely related to the TV regularisation method.…”

“…Next, we will give the discrete formulation of the proposed model (10). In the discrete case, we stack an image of size r × c by column into a vector of size N = r × c. More specifically, we assume the periodic boundary condition for images and use the forward finite differences to approximate the second-order derivatives [31].…”

A nonconvex $$\hbox{TV}_q-l_1$$ regularization model and the ADMM based algorithm

“…To further encourage the sparsity of the solutions, some nonconvex regularizers are proposed since nonconvex functions are much closer to the -norm than convex counterparts 20 , 21 . Since the seminal work of Geman and Geman in 22 , various nonconvex regularization models have been proposed, such as 23 – 27 . Although nonconvex optimization problems cannot guarantee the existence and uniqueness of the solution, and will lead to complex numerical calculation, a variety of applications (e.g., 28 – 31 ) have shown that nonconvex regularization models outperform the convex counterparts, and yield the restorations of high quality with sharp and neat edges.…”