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An Online Plug-and-Play Algorithm for Regularized Image Reconstruction
Yu Sun, Brendt Wohlberg, Ulugbek S. Kamilov
TL;DR
Large imaging datasets challenge batch PnP methods because they process all measurements at every iteration. The paper introduces online PnP-SGD, analyzes batch and online PnP-ISTA convergence for general denoisers, and evaluates the approach in simulated diffraction tomography. The results support scalable online reconstruction, while the study remains bounded by explicit convergence assumptions and simulated diffraction-tomography data.
Problem
Batch PnP algorithms process the full measurement set at every iteration, limiting their use on very large datasets common in 3D and dynamic imaging.
Method
The paper introduces PnP-SGD, an online PnP-ISTA extension that uses measurement subsets, and develops convergence analyses for batch and online variants with denoisers not necessarily representing proximal operators.
Results
PnP-SGD converges to the same set of fixed points as batch PnP-ISTA and PnP-ADMM, with simulations demonstrating its application to diffraction-tomography reconstruction.
Takeaways & Limitations
Online PnP is beneficial when data are too large to process jointly, and PnP-SGD can provide lower per-iteration cost while remaining sufficiently close to batch solutions in the reported experiment.
Takeaways & Limitations
The convergence analysis relies on explicit assumptions, and the experimental study uses purely simulated diffraction-tomography data rather than an experimental application.
Abstract
from arXiv · showhide
Plug-and-play priors (PnP) is a powerful framework for regularizing imaging inverse problems by using advanced denoisers within an iterative algorithm. Recent experimental evidence suggests that PnP algorithms achieve state-of-the-art performance in a range of imaging applications. In this paper, we introduce a new online PnP algorithm based on the iterative shrinkage/thresholding algorithm (ISTA). The proposed algorithm uses only a subset of measurements at every iteration, which makes it scalable to very large datasets. We present a new theoretical convergence analysis, for both batch and online variants of PnP-ISTA, for denoisers that do not necessarily correspond to proximal operators. We also present simulations illustrating the applicability of the algorithm to image reconstruction in diffraction tomography. The results in this paper have the potential to expand the applicability of the PnP framework to very large and redundant datasets.
1 Introduction
The paper motivates plug-and-play reconstruction and introduces an online PnP-ISTA extension that processes measurement subsets, with convergence analysis for batch and online variants.
- Motivation: Image reconstruction from noisy measurements is commonly formulated as an optimization problem combining data fidelity and image-prior regularization.The data-fidelity term penalizes mismatch to measurements, while the regularizer imposes prior knowledge.
- Motivation: Proximal algorithms such as ISTA and ADMM handle nondifferentiable regularizers without differentiating them.They use proximal operators, which are mathematically equivalent to regularized image denoising.
- Plug-and-play priors: PnP replaces a proximal operator with a state-of-the-art denoiser that need not correspond to any explicit regularization objective.This broadens the usable denoisers but generally removes optimization-based interpretability.
- Online limitation: Batch PnP algorithms use all measurements at every iteration, limiting their application to very large 3D and dynamic-imaging datasets.The proposed online extension uses only a subset of measurements at a time to scale to otherwise prohibitively large datasets.
- Contributions: The paper introduces PnP-SGD, analyzes convergence for batch PnP-ISTA under explicit assumptions, and demonstrates the online method on diffraction-tomography reconstruction.The analysis shows that PnP-SGD converges to the same set of fixed points as batch PnP-ISTA and PnP-ADMM.
2 Background
The background develops regularized inverse-problem formulations, proximal algorithms, and PnP variants, then positions the paper’s monotone-operator convergence analysis among prior work.
- Inverse problems in imaging: Imaging inverse problems recover an unknown image from noisy measurements modeled through a measurement operator and additive noise.For linear problems, the measurement matrix models the imaging system response.
- Inverse problems in imaging: Ill-posed reconstruction is regularized with data-fidelity terms such as least squares and image priors such as ℓ1 sparsity or total variation.The regularizers use λ > 0, with total variation involving the discrete gradient operator D.
- Proximal algorithms: ISTA and ADMM address nonsmooth regularizers through proximal operators, but ISTA uses the data-fidelity gradient whereas ADMM uses its proximal operator.For some inverse problems, the gradient is easier to evaluate; ADMM can require expensive matrix inversion.
- Using denoisers as priors: PnP substitutes a controllable-strength denoiser for the proximal operator, with denoising strength often scaled as σ = √γλ.ISTA and FISTA differ through the acceleration sequence {qk}, while the PnP versions retain the corresponding iterative structure.
- PnP variants: The paper focuses on PnP-ISTA and PnP-ADMM, extending PnP beyond its original ADMM formulation while excluding AMP because of its instability for general matrices.PnP can also be combined with other proximal algorithms, but the paper restricts its analysis to ISTA- and ADMM-based variants.
- Convergence theory: Prior convergence results impose conditions such as nonexpansiveness, symmetric denoiser gradients, or boundedness, whereas this paper uses monotone operator theory for broader denoiser analysis.The framework targets denoisers that do not necessarily correspond to proximal operators of an explicit objective.
3 Batch Algorithm
Batch PnP-ISTA is analyzed through fixed points under averaged-denoiser assumptions, extending beyond proximal operators while identifying convergence guarantees and limitations.
- Algorithm and assumptions: PnP-ISTA applies a denoiser after a gradient step and analyzes convergence to the fixed-point set of the resulting operator.The data-fidelity function is assumed convex with an L-Lipschitz continuous gradient.
- Convergence results: When the denoiser is proxγr, fixed points of PnP-ISTA coincide with minimizers of f = d + r.This recovers the traditional optimization interpretation as a special case.
- Algorithm and assumptions: The analysis generalizes denoisers beyond proximal operators by assuming the denoiser is averaged.Averaged operators include proximal operators and can include broader denoiser classes.
- Convergence results: Under the stated assumptions and γ ∈ (0, 1/L], PnP-ISTA iterates approach fix(P) at rate O(1/t).The convergence statement is expressed in terms of fixed-point distance rather than objective minimization.
- Convergence results: Under the same assumptions, PnP-ISTA and PnP-ADMM have the same fixed-point solutions, while PnP-ISTA does not require the symmetric-gradient assumption.The equivalence is stated in terms of the shared set fix(P).
- Limitations: Bounded denoisers alone do not guarantee PnP-ISTA convergence: a permitted configuration can diverge when σ > γ/√c.This limitation is established by a specific counterexample.
4 Online Algorithm
PnP-SGD is an online PnP-ISTA variant that uses minibatch gradients, enabling large-scale reconstruction while converging under explicit assumptions toward fixed points.
- Online algorithm: PnP-SGD approximates the full data-fidelity gradient using an average of B ≪ I randomly sampled component gradients at each iteration.The minibatch size B controls how many gradient components are used per iteration.
- Assumptions: The convergence analysis assumes convex differentiable component functions with common Lipschitz constant L, averaged denoisers, and unbiased bounded-variance gradient estimates.These conditions define the assumptions used to analyze the online algorithm.
- Convergence: The expected convergence error is proportional to γ and inversely proportional to minibatch size B, so parameter selection controls the attainable convergence rate.The analysis explicitly relates step size and minibatch size to convergence behavior.
- Empirical illustration: Figure 2 compares distance to a fixed point across iteration number, three step sizes, and basic versus accelerated PnP-SGD for B = 30.The caption states that smaller γ improves accuracy and that the empirical behavior is consistent with Proposition 5.
- Convergence: With a particular selection of B and γ, PnP-SGD can approach the fixed-point set at worst-case rate O(1/t) in expectation.This statement is made under Assumption 2.
5 Numerical Simulations
The simulations evaluate PnP-SGD for diffraction tomography, showing how step and minibatch sizes affect convergence and how online processing improves efficiency against batch methods.
- 5 Numerical Simulations: The experiments evaluate PnP-SGD with TV, BM3D, and TNRD for simulated diffraction-tomography image reconstruction.Diffraction tomography is highly data-intensive, motivating the focus on online processing.
- 5.2 Convergence of PnP-SGD: TV provides a reference case because its proximal operator is 1/2-averaged and satisfies Proposition 5’s assumptions.Figure 4 examines the influence of step and minibatch sizes under TV.
- 5.2 Convergence of PnP-SGD: Under BM3D, smaller step sizes and larger minibatches improve final fixed-point accuracy, while acceleration improves convergence.The experiments use γ ∈ {1/L, 1/(4L), 1/(16L)} and B ∈ {10, 20, 30}.
- 5.3 Benefits of online processing: For fixed runtime, PnP-SGD uses B = 10 illuminations per iteration and has lower per-iteration costs than batch PnP-FISTA and PnP-ADMM.Average iteration times were 8.86, 44.94, and 382.83 seconds for PnP-SGD, PnP-FISTA, and PnP-ADMM, respectively.
- 5.3 Benefits of online processing: Under fixed measurement budgets of 10 or 30 illuminations, PnP-SGD significantly outperforms batch counterparts at the same per-iteration cost.With B = 10 and B = 30, final SNR performance is comparable, although B = 30 converges faster.
- 5.3 Benefits of online processing: The average SNR difference between PnP-SGD with B = 10 and B = 30 is within 0.2 dB, and smaller minibatches nearly match batch image quality.The paper contrasts this mild image-quality difference with a larger fixed-point-distance difference.
6 Conclusion
The paper concludes that online PnP is beneficial for large-scale image reconstruction when data cannot be processed jointly, while its theory and experiments support the approach.
- 6 Conclusion: Online PnP is beneficial for large-scale image reconstruction when the data are too large to process jointly.The conclusion combines theoretical convergence analysis with experiments demonstrating higher efficiency in representative settings.
- 6 Conclusion: The paper develops convergence theory for both batch and online PnP-ISTA and identifies future work on other tasks, relaxed assumptions, ADMM, and FISTA.The stated future directions extend the theoretical and application scope of the work.
7 Appendix
The appendix develops operator-theoretic preliminaries for analyzing PnP iterations, then proves convergence-related results for batch and online operators and gives a divergence counterexample.
- 7.1 Preliminaries: Denoisers are modeled as functions Dσ: R^n → R^n, while Gγ ≜ I − γ∇d denotes the gradient-step operator.The framework assumes all operators are defined everywhere on R^n.
- 7.1 Preliminaries: Nonexpansive operators have Lipschitz constant 1, and composing operators multiplies their Lipschitz constants.Therefore, composing two nonexpansive operators remains nonexpansive.
- 7.1 Preliminaries: An α-averaged operator is a convex combination of the identity and a nonexpansive operator, and damped iterations of such operators converge to fixed points under the stated theorem.The appendix also records that proximal operators and suitable gradient steps are averaged, and compositions of averaged operators remain averaged.
- 7.3 Proof of Proposition 2: The appendix uses averaged-operator theory and Krasnosel’skii–Mann iteration to establish a convergence result for PnP-ISTA.PnP-ISTA is analyzed through the composition of the denoiser and gradient-step operators.
- 7.4 Proof of Proposition 3: The fixed-point analysis of PnP-ADMM shows that its fixed-point variables satisfy z∗ = x∗ and x∗ ∈ fix(P).The derivation uses continuity, smoothness of d, and the proximal characterization.
- 7.5 Counterexample: For any σ > γ/√c and any initial z0, the constructed bounded-denoiser example makes PnP-ISTA diverge.The data-fidelity function is convex with Lipschitz-gradient constant L = 1, so boundedness alone does not guarantee convergence in this example.