A combined higher order non-convex total variation with overlapping group sparsity for Poisson noise removal

Poisson noise removal is a fundamental image restoration task in imaging science due to the Poisson statistics of the noise. The total variation (TV) image restoration has been promising for Poisson noise removal. However, TV-based denoising methods suffer from the staircase artifacts which makes the restored image blocky. Apart from that, the ℓ₁-norm penalization in TV restoration tends to over-penalize signal entries. To address these shortcomings, in this paper, we propose a combined regularization method that uses two regularization functions. Specifically, a combination of a non-convex ℓ_p-norm, 0 < p< 1 higher order TV, and an overlapping group sparse TV (OGSTV) is proposed as a regularizer. The combination of a higher order non-convex TV and an overlapping group sparse (OGS) regularization serves as a means to preserve natural-looking images with sharp edges and eliminate the staircase artifacts. Meanwhile, to effectively denoise Poisson noise, a Kullback–Leibler (KL) divergence data fidelity is used for the data fidelity which better captures the Poisson noise statistic. To solve the resulting non-convex minimization problem of the proposed method, an alternating direction method of multipliers (ADMM)-based iterative re-weighted ℓ₁ (IRℓ₁) based algorithm is formulated. Comparative analysis against KL-TV, KL-TGV and, KL-OGS TV for restoring blurred images contaminated with Poisson noise attests to the good performance of the proposed method in terms of peak signal-to-noise ratio (PSNR) and structure similarity index measure (SSIM).