Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications (1605.01710v2)

Abstract: Alternating direction method of multiplier (ADMM) is a widely used algorithm for solving constrained optimization problems in image restoration. Among many useful features, one critical feature of the ADMM algorithm is its modular structure which allows one to plug in any off-the-shelf image denoising algorithm for a subproblem in the ADMM algorithm. Because of the plug-in nature, this type of ADMM algorithms is coined the name "Plug-and-Play ADMM". Plug-and-Play ADMM has demonstrated promising empirical results in a number of papers. However, it is unclear under what conditions and by using what denoising algorithms would it guarantee convergence. Also, since Plug-and-Play ADMM uses a specific way to split the variables, it is unclear if fast implementation can be made for common Gaussian and Poissonian image restoration problems. In this paper, we propose a Plug-and-Play ADMM algorithm with provable fixed point convergence. We show that for any denoising algorithm satisfying an asymptotic criteria, called bounded denoisers, Plug-and-Play ADMM converges to a fixed point under a continuation scheme. We also present fast implementations for two image restoration problems on super-resolution and single-photon imaging. We compare Plug-and-Play ADMM with state-of-the-art algorithms in each problem type, and demonstrate promising experimental results of the algorithm.

• The paper establishes fixed point convergence of the ADMM framework when using bounded denoisers under a continuation scheme.
• It integrates off-the-shelf image denoisers into ADMM, enabling efficient restoration tasks like super-resolution and photon imaging.
• Extensive experiments confirm state-of-the-art performance, demonstrating practical benefits over traditional restoration methods.

Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications

The paper primarily addresses developing and analyzing a variant of the Alternating Direction Method of Multipliers (ADMM) tailored for image restoration tasks, termed as "Plug-and-Play ADMM". This approach leverages the modular structure of ADMM, which allows for the integration of off-the-shelf image denoising algorithms into the ADMM framework. The significant contributions of this paper include establishing the fixed point convergence of the Plug-and-Play ADMM under specific conditions and demonstrating its application to super-resolution and single photon imaging problems.

Motivation and Background

ADMM is a powerful algorithm for solving constrained optimization problems, particularly in image restoration. The classical ADMM formulation allows the algorithm to decompose the optimization problem into smaller, more manageable subproblems. The essential innovation in Plug-and-Play ADMM is the substitution of the proximal operator corresponding to the image prior in ADMM with a generic image denoiser. This substitution simplifies the implementation since the denoising operation can be performed using advanced, off-the-shelf algorithms without explicitly defining the prior distribution.

However, this substitution introduces questions regarding the convergence properties of the modified ADMM. Traditional ADMM ensures convergence under certain conditions, such as convexity and properness of the objective functions. The paper seeks to establish under what conditions Plug-and-Play ADMM guarantees convergence, particularly focusing on the properties of the denoising algorithms used.

Key Results and Contributions

1. Theoretical Analysis of Convergence:
   • The authors propose a modified Plug-and-Play ADMM algorithm incorporating a continuation scheme, where a parameter $\rho$ is progressively increased.
   • They establish that for a broad class of denoisers termed "bounded denoisers", the algorithm converges to a fixed point. A bounded denoiser ensures that the noise residue tends to zero as the parameter $\rho$ increases, ensuring the stability of the denoising step.
   • This fixed point convergence provides a weaker but practically significant form of convergence compared to global convergence, which is generally hard to guarantee for non-convex denoisers.
2. Applications and Implementation Techniques:
   • Image Super-Resolution:

   The paper demonstrates the application of Plug-and-Play ADMM to image super-resolution, where the goal is to reconstruct a high-resolution image from a low-resolution observation. The authors design a polyphase decomposition-based method to efficiently solve the subproblems in ADMM, enabling closed-form solutions without iterative solvers. This approach significantly reduces computational overhead compared to conjugate gradient methods. - Single Photon Imaging:

   For imaging tasks using quanta image sensors (QIS), the paper utilizes Plug-and-Play ADMM to address the challenges posed by the Poisson nature of the photon arrivals. The authors detail a practical implementation that exploits the separable structure of the problem, thus streamlining the restoration process.

3. Experimental Validation:

   • Extensive experiments were conducted on standard datasets for both the super-resolution and single photon imaging tasks. The empirical results validate the theoretical findings, showing that Plug-and-Play ADMM achieves state-of-the-art performance across various configurations and surpasses several existing methods in terms of peak signal-to-noise ratio (PSNR).

Practical and Theoretical Implications

The implications of the proposed Plug-and-Play ADMM are manifold. Practically, it provides a modular and flexible framework for image restoration tasks, leveraging sophisticated denoising techniques without explicitly requiring the formulation of complex priors. This modularity is crucial as it facilitates easy adaptation and integration of new and improved denoising algorithms.

Theoretically, the establishment of fixed point convergence under the bounded denoiser condition provides a foundation for further analysis and potentially stronger convergence guarantees. This work opens up avenues for exploring convergence properties with other classes of denoisers and extending the framework to other inverse problems beyond image restoration.

Future Directions

Future work could explore the following:

• Extending the convergence analysis to more general or non-symmetric denoisers.
• Investigating the integration of learning-based denoisers within the Plug-and-Play ADMM framework, potentially exploiting deep learning models.
• Exploring the application of Plug-and-Play ADMM to more diverse image processing and computer vision tasks, such as video restoration and medical imaging.

In conclusion, the paper provides a significant step forward in leveraging the modular nature of ADMM for practical image restoration tasks, balancing theoretical rigor with practical applicability. The insights gained from this work have substantial implications for both academic research and practical implementations in the field of image processing.