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Poisson Image Denoising Using Minimax Concave and Reweighted $\ell_1$ Penalties: Nonblind and Blind Approaches
Reza Parvaz
TL;DR
Poisson noise and blur complicate image restoration, particularly when the PSF is unknown, and edge preservation is a key regularization concern. The paper proposes MCP and reweighted fractional-order regularization with ADMM-based approaches for nonblind and blind restoration. Experiments are reported to confirm accuracy and convergence behavior.
Problem
Poisson noise and blur degrade images, and restoration is more complicated when the PSF is unknown; regularization choice also affects restored-image quality and edge preservation.
Source: §1.1 · p. 2, §1.2 · pp. 2–3Method
The paper combines transform-domain MCP with reweighted ℓ1 fractional-order regularization and uses ADMM, extending the model to blind deconvolution.
Source: §3.1 · p. 5, §6 · p. 25Results
The authors report that numerical experiments confirm the proposed method’s performance in accuracy and convergence behavior.
Source: §6 · p. 26Takeaways & Limitations
The proposed formulation covers Poisson image restoration in both nonblind and blind settings.
Source: §5 · p. 24, §6 · p. 25
Abstract
from arXiv · showhide
Images are important tools in various sciences. Despite the development of photo-taking tools, creating clear and image without noise remains challenging in practice. In particular, Poisson noise has an effect on medical and astronomical images, and reduces their quality. Additionally, blur is another factor that has an effect on image quality. The problem of image restoration becomes very complicated when we have no information about the Point Spread Function (PSF). These types of problems are known as blind case. However, in some images, such as some astronomical images, the type of PSF can be specified, and these types of problems are known as nonblind problems. Total Variation (TV) is a widely used method for solving such inverse problems, where the selection of the penalty function is the most critical factor that affects the method's performance. In this paper, to improve edge preservation, we employ a reweighted $\ell_1$-regularization of the fractional order derivative. Furthermore, we propose a nonblind and blind image deblurring approach under Poisson noise using the Minimax Concave Penalty (MCP), which is a continuous, sparsity promoting, and nearly unbiased regularizer. This formulation leads to a nonconvex optimization model. To solve the proposed model, we introduce an efficient numerical algorithm based on the Alternating Direction Method of Multipliers (ADMM) and provide an analysis of its convergence. Finally, the effectiveness of the proposed algorithm are demonstrated through extensive experiments on various images.
1 Introduction
The introduction frames Poisson-noise image restoration as a problem of recovering clear images from blur and noise, with edge preservation motivating the choice of regularizer. It proposes combining transform-domain MCP sparsity with adaptive reweighted fractional-order regularization.
1.1 Problem Description
- Poisson noise and blur degrade images, while unknown PSFs make restoration a blind problem and known PSFs define the nonblind case.The image model applies convolution by the PSF followed by Poisson corruption.Source: §1.1 · p. 2
1.2 Literature Review and Motivation
- The regularizer strongly affects restoration quality, and preserving edges in blurred, noisy images is a central motivation.The introduction also identifies staircase artifacts as a concern with total variation and motivates derivative-based alternatives.Source: §1.2 · pp. 2–3
- Fractional-order derivatives are presented as balancing edge preservation and smoothness, while reweighted ℓ1 promotes sparsity and MCP offers a nonconvex alternative to standard norms.The cited discussion contrasts fractional derivatives with limitations of Laplacian-based regularization and describes MCP as closer to ℓ0 sparsity promotion.Source: §1.2 · p. 3
- The proposed framework combines transform-domain MCP with adaptive reweighted fractional-order regularization instead of the MCP-TGV framework.The authors describe the transform-domain term as promoting sparse representations and the fractional-order term as adaptively enhancing gradient sparsity.Source: §1.2 · p. 3
2 Preliminaries and Notation
The preliminaries establish notation and introduce fractional-order gradients and MCP tools used in the paper. They also point to a vector MCP optimization lemma.
2 Preliminaries and Notation
- The section introduces notation including the FFT, inverse FFT, gamma function, and framelet transform matrix.The supplied passage begins the description of the framelet transform but does not include its full construction.Source: §2 · p. 4
- The fractional-order Grünwald–Letnikov gradient is introduced with derivative order β and horizontal and vertical fractional derivatives.The supplied passages identify β as the derivative order but do not provide enough intact text to summarize the operator formula.Source: §2 · p. 4
- The preliminaries define scalar and vector MCP forms and state a lemma about an optimization problem involving MCP parameters γ and η.The supplied text identifies γ and η as MCP parameters but omits the formulas and lemma solution.Source: §2 · pp. 4–5
3 Proposed Model and Numerical Algorithm
The paper formulates Poisson deconvolution with MCP and reweighted ℓ1 regularization, then develops ADMM-based numerical methods for nonblind and blind restoration. The blind approach estimates both the image and PSF.
3.1 Proposed Model
- The proposed model combines MCP and reweighted ℓ1 regularization to promote transform-domain sparsity while protecting edge structures.A nonnegativity indicator enforces x ≥ 0.Source: §3.1 · p. 5
3.2 Numerical Algorithm
- The nonblind model is solved with ADMM by introducing auxiliary variables and splitting the constraints into subproblems.The constraints link the variables to the blur operator, framelet transform, fractional gradient, and image.Source: §3.2 · pp. 5–6
- With periodic boundary conditions, the x-subproblem can be computed efficiently in the Fourier domain because its operators are BCCB and diagonalizable by the discrete Fourier transform.The method also updates reweighted ℓ1 weights using image-gradient magnitudes at each iteration.Source: §3.2 · pp. 6–7
3.3 Blind Problem
- The blind approach estimates both the clear image and PSF, updating the kernel with an EM step and using cropped convolution output to address boundary effects.The algorithm is described as using fixed-point iteration, and its output includes the restored image and PSF.Source: §3.3 · pp. 8, 10, 11
4 Convergence Analysis
The convergence analysis derives boundedness and vanishing successive-iterate differences under parameter and sequence assumptions. It also states that limit points satisfy the optimization problem’s KKT conditions.
4 Convergence Analysis
- The bounded-multiplier lemma assumes uniformly bounded weights and a sequence v_j uniformly bounded away from zero.These assumptions are used in the convergence analysis.Source: §4 · p. 11
- Under the stated MCP and regularization-parameter condition, the generated sequence is bounded and successive differences for x, v, g, z, and m vanish.The boundedness result also concerns the augmented Lagrangian sequence.Source: §4 · pp. 11, 15
- Every limit point x* satisfies the KKT conditions of the optimization problem.The proof also establishes strict positive definiteness of the x-subproblem Hessian and hence strong convexity of that subproblem.Source: §4 · pp. 14, 17
5 Simulations results
The simulations report acceptable restoration performance in nonblind and blind settings, including recovery of lunar mountain detail and restoration of both an image and its PSF. Across the plotted iterations, error generally decreases while PSNR approaches a constant, though parameter choice affects the output.
5 Simulations results
- The simulations assess performance using PSNR and MSSIM and benchmark the proposed method against BM3D, OGS, and FOTV.The evaluation uses Poisson-noise-corrupted test images and reports the best results across tested parameter values.Source: §5 · p. 18
- Across the nonblind examples, the algorithm has acceptable restoration performance and recovers lunar mountain detail lost to noise and blur.The tests include Girl and Shepp-Logan Phantom images blurred with Gaussian PSFs, and a Moon surface image blurred with a motion PSF.Source: §5 · pp. 19–21, Table 3 · p. 22, Figure 4 · p. 22
- Across the plotted nonblind and blind iterations, error generally decreases and PSNR approaches a constant, despite fluctuations at some peaks.Figures 5 and 6 plot the nonblind examples; Figure 9 plots image and PSF error and image PSNR for the blind example.Source: §5 · pp. 22, 24, Figure 9 · p. 25
- The sensitivity analysis finds that parameter choice affects the output, and the proposed algorithm uses grid search to select appropriate parameters.The tested parameters are β, µ, and λ.Source: §5 · p. 23, Figure 7 · p. 24
- The blind method shows acceptable performance restoring both the Satellite image and its PSF, with comparisons against EM and FOTV.The comparison uses motion and Gaussian kernels.Source: §5 · p. 24, Table 4 · p. 24, Figure 8 · p. 25
6 Conclusion
The paper proposes a Poisson image-denoising model using MCP and a reweighted ℓ1 norm, extends it to blind deconvolution, and evaluates its accuracy and convergence behavior numerically.
6 Conclusion
- The proposed Poisson image-denoising model combines the minimax concave penalty with a reweighted ℓ1 norm.Source: §6 · p. 25
- The model extends to blind deconvolution, uses ADMM for numerical solution, and includes theoretical convergence analysis.Source: §6 · p. 25
- Numerical experiments confirm the proposed method's performance in both accuracy and convergence behavior.Source: §6 · p. 26