Restoration of Poisson noise corrupted digital images with nonlinear PDE based filters along with the choice of regularization parameter estimation

• Proposed three nonlinear PDE based filters adapted to Poisson noise statistics for image restoration.
• Methods for the regularization parameter estimation have been examined and introduced.
• The examined regularization parameter estimation methods are GCV, discrepancy principles and UPRE.
• Performance analysis and comparison of the proposed methods are presented.
• The modified anisotropic diffusion based method along with GCV parameter estimation works better.

In this paper, the reconstruction of three nonlinear partial differential equation (PDE) based filters adapted to Poisson noise statistics have been proposed in a variational framework for restoration and enhancement of digital images corrupted with Poisson noise. The proposed and examined PDE based filters include total variation adapted to Poisson noise in L-1 framework; anisotropic diffusion; and complex diffusion based methods adapted to Poisson noise in L-2 framework. The resulting filters contain two terms namely data fidelity and regularization or smoothing function. The data fidelity term is Poisson likelihood term and the regularization functions are PDE based filters. Other choices for the regularization functions have also been presented. The two terms in the proposed filters are coupled with a regularization parameter lambda which makes a proper balance between the two terms during the filtering process. The choice of method for estimation of regularization parameter lambda plays an important role. In this study, the various regularization parameter estimation methods for Poisson noise have also been presented and their suitability has been examined. The resulting optimization problems are further investigated for efficient implementation for large scale problems. For estimating the regularization parameter, three choices are considered for Poisson noise case which are discrepancy principles, generalized cross validations (GCV), and unbiased predictive risk estimate (UPRE). GCV and UPRE functions are further other optimization problems in addition to main image reconstruction problem. For minimizing the GCV and UPRE functions, the methods of Conjugate Gradients (CG) is used. For digital implementations, all schemes have been discretized using finite difference scheme. The comparative analysis of the proposed methods are presented in terms of relative norm error, improvement in SNR, MSE, PSNR, CP and MSSIM for an adaptive value of regularization parameter calculated by every methods in consideration. Finally, from the obtained results it is observed that the anisotropic diffusion based method adapted to Poisson noise gives better results in comparison to other methods in consideration along with choice of GCV for regularization parameter selection.