  In this manuscript, we research on the behaviors of surrogates for the rank
function on different image processing problems and their optimization
algorithms. We first propose a novel nonconvex rank surrogate on the general
rank minimization problem and apply this to the corrupted image completion
problem. Then, we propose that nonconvex rank surrogates can be introduced into
two well-known sparse and low-rank models: Robust Principal Component Analysis
(RPCA) and Low-Rank Representation (LRR). For optimization, we use alternating
direction method of multipliers (ADMM) and propose a trick, which is called the
dual momentum. We add the difference of the dual variable between the current
and the last iteration with a weight. This trick can avoid the local minimum
problem and make the algorithm converge to a solution with smaller recovery
error in the nonconvex optimization problem. Also, it can boost the convergence
when the variable updates too slowly. We also give a severe proof and verify
that the proposed algorithms are convergent. Then, several experiments are
conducted, including image completion, denoising, and spectral clustering with
outlier detection. These experiments show that the proposed methods are
effective in image and signal processing applications, and have the best
performance compared with state-of-the-art methods.
