Recently

the Hessian Nuclear norm regularization method has been a preferable higher order regularization scheme for image restoration

but with the Hessian Nuclear norm regularization term been highly non-linear and non-differentiable

image deblurring and denoising processes are highly coupled so that their minimization algorithms are with highly computational complexity.In this paper

we employ variable splitting to design a fast alternating iterative minimization algorithm with closed-form solutions for image restoration

in which we separate image restoration into image deblurring and denoising.Furthermore

we show the convergence of our proposed algorithm.Finally

experimental results demonstrate the effectiveness of the proposed method which consists in not only giving the improved performance in terms of peak signal to noise ratio (PSNR)

but also exhibiting a much faster convergence rate than the previous majorization-minimization (MM) method for Hessian Nuclear norm regularization based image restoration.