Vectorial additive half-quadratic minimization for isotropic regularization

We propose the vectorial additive half-quadratic (VAHQ) algorithm to minimize the isotropic regularized cost function with the general edge-preserving potential functions (PFs) in image restoration. By introducing an auxiliary vectorial variable, the cost function is changed into an augmented one which can be alternately minimized. One minimization is solved with an explicit expression, the other is implemented by Fast Fourier Transform (FFT). VAHQ is shown to globally converge to a stationary point for nonconvex PFs providing all stationary points are isolated and to a unique minimum for convex PFs without any isolation assumption, on an extended domain of parameters. What is more, the linear convergence rate of VAHQ is proved to be less than 1, the explicit expression of the optimal parameters and the optimal bound of the convergence rate are present for convex PFs. Image restoration examples show the restoration performance of nonconvex PFs, the lower computation cost of our algorithm compared to Majorize-Minimize memory gradient (MMMG) algorithm with the help of FFT and some interesting phenomena confirming our conclusions on the convergence domain and the convergence rate.