Image restoration based on fractional-order model with decomposition: texture and cartoon

Inspired by the work of Daubechies and Teschke (Appl Comput Harmon Anal 19(1):1–16. https://doi.org/10.1016/j.acha.2004.12.004 , 2005), we propose an image deblurring and denoising method based on fractional-order model with simultaneous decomposition. We use fractional-order derivative as the regularization term of cartoon part to avoid blocky effect. We replace the BV regularization term by $$B^\beta _q(L_p(\varOmega ))$$ term, and $$B^{-1}_1(L_1(\varOmega ))$$ term for the regularization of texture part. To promote sparsity, we add a nonconvex regularization term which is the weighted difference of $$l_1$$ -norm and $$l_2$$ -norm based on wavelet frame to the regularization term. The model can be solved by alternating direction method of multipliers (ADMM). The comparative experimental results show that the capability of preserving the edges and textural details of our algorithms. Our fractional-order algorithms are superior to that of traditional integer-order algorithms especially for images with texture.