Establishing an accurate mathematical model and a design-effective algorithm is a dilemma in image restoration. The non-smooth energy functional model effectively describes image features but is difficult to use in a design-efficient computational algorithm. In this study

a new non-smooth energy functional regularization model that consists of fitting and regularization terms is developed. An efficient alternative iterative algorithm is deduced. First

for an image made blurry by system and salt-and-pepper noise in a tight frame domain

the fitting term is described by the L norm;the regularization term is established by the semi-norm of a weight-bound variation function. Second

the regularization model of image restoration is converted into an augmentation Lagrange model by introducing an auxiliary variable. Third

the transformed model is decomposed into two sub-problems by employing the variable splitting technique. Finally

by employing Fenchel transform and the fixed-point iterative principle

the sub-problem is transformed into dual and relaxed iterative sub-problems. The convergence property of the sub-problems is proven. An alternate iterative algorithm is proposed for the non-smooth property of the image restoration model. Compared with traditional algorithms

the proposed algorithm can effectively restore blurry images made so by system and salt-and-pepper noise and can increase the peak signal-to-noise ratio to approximately 0.5 dB to 1 dB. Results show that the proposed algorithm can effectively protect image edges and can achieve a high peak signal-to-noise ratio and structural similarity index measure. The proposed algorithm also has high convergence speed and can restore images rendered blurry by salt-and -pepper noise.