[Paper Review] Non-Convex Weighted Lp Nuclear Norm based ADMM Framework for Image Restoration

Zhiyuan Zha, Xinggan Zhang+3 more|University of Oulu Repository (University of Oulu)|Apr 24, 2017

Sparse and Compressive Sensing Techniques30 references4 citations

TL;DR

This paper proposes a non-convex weighted ℓₚ nuclear norm minimization (NCW-NNM) framework using ADMM to improve image restoration by enabling more accurate low-rank approximation than traditional nuclear norm minimization. The method assigns adaptive weights to singular values, reducing over-shrinkage and enhancing structural sparsity and nonlocal self-similarity, leading to superior performance in deblurring, inpainting, and compressive sensing recovery across multiple benchmarks.

ABSTRACT

Since the matrix formed by nonlocal similar patches in a natural image is of low rank, the nuclear norm minimization (NNM) has been widely used in various image processing studies. Nonetheless, nuclear norm based convex surrogate of the rank function usually over-shrinks the rank components and makes different components equally, and thus may produce a result far from the optimum. To alleviate the above-mentioned limitations of the nuclear norm, in this paper we propose a new method for image restoration via the non-convex weighted Lp nuclear norm minimization (NCW-NNM), which is able to more accurately enforce the image structural sparsity and self-similarity simultaneously. To make the proposed model tractable and robust, the alternative direction multiplier method (ADMM) is adopted to solve the associated non-convex minimization problem. Experimental results on various types of image restoration problems, including image deblurring, image inpainting and image compressive sensing (CS) recovery, demonstrate that the proposed method outperforms many current state-of-the-art methods in both the objective and the perceptual qualities.

Motivation & Objective

• To address the over-shrinkage and equal treatment of singular values in traditional nuclear norm minimization (NNM), which limits low-rank approximation accuracy in image restoration.
• To improve image restoration performance by exploiting nonlocal self-similarity and structural sparsity through a more flexible low-rank regularization model.
• To develop a tractable and robust optimization framework for solving the non-convex NCW-NNM model, ensuring convergence and practical applicability.
• To demonstrate the superiority of the proposed method over existing state-of-the-art techniques in multiple image restoration tasks, including deblurring, inpainting, and compressive sensing.

Proposed method

• Proposes a non-convex weighted ℓₚ nuclear norm (NCW-NNM) that applies ℓₚ norm (0 < p < 1) to singular values with adaptive weights to better approximate matrix rank.
• Introduces a weighted singular value thresholding scheme that differentially penalizes singular values based on their magnitude, reducing over-shrinkage.
• Employs the Alternating Direction Method of Multipliers (ADMM) to decompose the non-convex optimization problem into tractable subproblems for efficient solution.
• Uses a variable splitting technique to separate the data fidelity and regularization terms, enabling iterative updates of the image estimate, singular values, and dual variables.
• Applies a soft-thresholding-like update rule in the ADMM framework tailored for the non-convex ℓₚ norm, ensuring convergence in practice.
• Employs a patch-based nonlocal similarity strategy to group similar patches and construct low-rank matrices for regularization.

Experimental results

Research questions

• RQ1 Can a non-convex weighted ℓₚ nuclear norm better approximate matrix rank than standard nuclear norm in image restoration?
• RQ2 Does assigning adaptive weights to singular values reduce over-shrinkage and improve low-rank approximation accuracy?
• RQ3 Can the ADMM framework effectively and stably solve the non-convex NCW-NNM model for image restoration tasks?
• RQ4 How does NCW-NNM compare to state-of-the-art methods in terms of PSNR, FSIM, and visual quality across diverse image restoration problems?

Key findings

• The proposed NCW-NNM method achieves a PSNR of 26.52 dB and FSIM of 0.9271 in image deblurring, significantly outperforming traditional NNM.
• In image inpainting with 80% missing pixels, NCW-NNM achieves a PSNR of 28.15 dB and FSIM of 0.9321, demonstrating superior recovery quality.
• For compressive sensing recovery with 0.1N measurements, NCW-NNM achieves a PSNR of 25.49 dB and FSIM of 0.8755, surpassing NNM and other leading methods.
• Visual comparisons show that NCW-NNM preserves finer textures and edges more effectively than NNM, especially in high-frequency regions.
• Singular value analysis confirms that NCW-NNM better approximates the ground-truth rank structure than NNM, with less over-shrinkage of significant singular values.
• ADMM converges stably and efficiently, with PSNR curves rising smoothly and stabilizing after ~100 iterations, outperforming IST in convergence speed and accuracy.