[Paper Review] Averaged Deep Denoisers for Image Regularization

Pravin Nair, Kunal N. Chaudhury|arXiv (Cornell University)|Jul 15, 2022

Image and Signal Denoising Methods4 citations

TL;DR

This paper proposes a novel framework to construct provably averaged (contractive) deep denoisers for image regularization using Plug-and-Play (PnP) and Regularization by Denoising (RED) algorithms. By unfolding ISTA and ADMM iterations for wavelet-based denoising, the method ensures convergence guarantees through contractivity, matching the performance of CNN denoisers while providing theoretical stability.

ABSTRACT

Plug-and-Play (PnP) and Regularization-by-Denoising (RED) are recent paradigms for image reconstruction that leverage the power of modern denoisers for image regularization. In particular, they have been shown to deliver state-of-the-art reconstructions with CNN denoisers. Since the regularization is performed in an ad-hoc manner, understanding the convergence of PnP and RED has been an active research area. It was shown in recent works that iterate convergence can be guaranteed if the denoiser is averaged or nonexpansive. However, integrating nonexpansivity with gradient-based learning is challenging, the core issue being that testing nonexpansivity is intractable. Using numerical examples, we show that existing CNN denoisers tend to violate the nonexpansive property, which can cause PnP or RED to diverge. In fact, algorithms for training nonexpansive denoisers either cannot guarantee nonexpansivity or are computationally intensive. In this work, we construct contractive and averaged image denoisers by unfolding splitting-based optimization algorithms applied to wavelet denoising and demonstrate that their regularization capacity for PnP and RED can be matched with CNN denoisers. To our knowledge, this is the first work to propose a simple framework for training contractive denoisers using network unfolding.

Motivation & Objective

• To address the instability of PnP and RED algorithms when using non-expansive denoisers, which can lead to divergence.
• To develop a training framework that guarantees the denoiser is contractive (averaged), ensuring convergence in PnP and RED iterations.
• To provide a simple, efficient alternative to existing non-expansive denoiser training that avoids computationally expensive checks or constraints.
• To demonstrate that wavelet-based denoisers constructed via unfolding can achieve regularization performance comparable to state-of-the-art CNN denoisers.
• To establish a theoretical foundation for the contractivity and averaged nature of the proposed denoisers using operator theory and averaging properties.

Proposed method

• Unfolds the Iterative Shrinkage-Thresholding Algorithm (ISTA) and Alternating Direction Method of Multipliers (ADMM) for total-variation-based wavelet denoising into a deep network architecture.
• Constructs a denoiser as a learned, parameterized operator by stacking multiple unfolding layers of ISTA/ADMM iterations.
• Proves that if the base wavelet denoiser (patch-level) is contractive or averaged, the resulting global denoiser is also contractive or averaged via averaging of local operators.
• Uses a patch-based averaging scheme over overlapping patches, where the global denoiser is expressed as a uniform average of local denoising operators.
• Employs the property that averaging contractive or averaged operators preserves contractivity and averaging, ensuring theoretical convergence in PnP and RED.
• Demonstrates that the resulting denoiser is provably non-expansive and contractive, enabling convergence guarantees for PnP and RED without requiring expensive non-expansivity checks.

(a) BM3D.

Experimental results

Research questions

• RQ1 Can a deep denoiser be constructed such that it is provably contractive (averaged) to ensure convergence in PnP and RED frameworks?
• RQ2 Can the performance of CNN-based denoisers in PnP/RED be matched using a theoretically grounded, non-expansive denoiser architecture?
• RQ3 Is it possible to train a denoiser via unfolding that inherits the contractivity of the underlying optimization algorithm (e.g., ISTA/ADMM) without imposing hard constraints?
• RQ4 How does the averaging structure of the unfolded network preserve contractivity across overlapping patches?
• RQ5 What is the theoretical relationship between the contractivity of a patch-level denoiser and the global denoiser formed via averaging?

Key findings

• The proposed method constructs deep denoisers via unfolding of ISTA and ADMM for wavelet-based denoising, ensuring provable contractivity.
• The resulting denoiser is shown to be contractive (i.e., averaged) if the base patch-level denoiser is contractive, via a mathematical proof using operator averaging.
• The method guarantees convergence of PnP and RED algorithms, unlike existing CNN denoisers that often violate non-expansivity and cause divergence.
• Numerical experiments confirm that the proposed denoisers achieve reconstruction performance comparable to state-of-the-art CNN denoisers in PnP and RED settings.
• The framework is computationally efficient and avoids the intractable checks required to verify non-expansivity in gradient-based training.
• The work establishes the first end-to-end training framework for provably averaged denoisers using unfolding, filling a critical gap in theoretical convergence for PnP and RED.

(b) DnCNN.