Fractional-order total variation combined with sparsifying transforms for compressive sensing sparse image reconstruction

• A FrTV combined with sparsifying transforms for CS sparse image reconstruction model is proposed.
• A method for estimating the regularization parameter is proposed.
• A gradient projection algorithm is developed to solve this reconstruction model.
• The algorithm has a higher PSNR and better detail preservation.

The total variation (TV) model has been considered to be one of the most successful and representative model for compressive sensing (CS) sparse image reconstruction due to its advantage of preserving image edges. However, TV regularized term often favors piecewise constant solution and therefore it fails to preserve the details and textures. To overcome this defect and reconstruct the fine details, this paper proposes a two-dimensional CS sparse image reconstruction model by introducing the fractional-order TV (FrTV) regularization constraint into CS optimization problem. Furthermore, in order to achieve sparser representation flexibly, a combination of discrete wavelet transform and curvelet transform ℓ1ℓ1-norm regularization is also incorporated into the cost function and a method for estimating the regularization parameter that trades off the two terms in the cost function is proposed. By using a smooth approximation of the ℓ1ℓ1-norm, a gradient projection algorithm is derived to solve the combined FrTV and sparsifying transforms constrained minimization problem effectively. Compared with several state-of-the-art reconstruction algorithms, the proposed algorithm is more efficient and robust, not only yielding higher peak-signal-to-noise ratio, but also reconstructing the fine details and textures more efficiently.