Plug-and-Play Phase Retrieval

• Plug-and-play phase retrieval is a family of methods that recover images from intensity-only measurements by alternately enforcing data fidelity and applying modular denoisers.
• These techniques integrate algorithms like ADMM, alternating projection, and HIO with learned or preset denoising operators to handle complex imaging models.
• Empirical studies demonstrate significant improvements in reconstruction quality, robustness, and speed across diverse modalities such as ptychography and Fourier phase retrieval.

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1. Definition, scope, and problem setting

Phase retrieval seeks recovery of a signal or image from magnitude or intensity measurements with missing phase. The cited works use closely related forward models, including

$I = |\mathbf{A}u|^2 + \omega,$

$y^2 = |Ax|^2 + w,$

and, for the ptychographic setting,

$y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$

with $u$ or $x$ complex-valued in the general case and with noise models that are Gaussian, shot-noise-like, or normal approximations to Poisson statistics (Chang et al., 2021, Wei et al., 2020, Denker et al., 2024, Isil et al., 2024).

Within this setting, “plug-and-play” refers primarily to the replacement of an explicit proximal map or regularizer by a denoiser-like module that is inserted into the iterative reconstruction loop. The central architectural separation is between a measurement-consistency operator driven by the forward physics and a prior operator driven by image statistics. In the LPR framework, the solver and prior are described as an alternating projection step and an enhancing neural network step; in TFPnP, the explicit proximal of the regularizer is replaced by a denoiser; in Deep Plug-and-Play HIO, the prior update is approximated by a deep denoiser $D(x^{k+1},\sigma_k)$ (Chang et al., 2021, Wei et al., 2020, Isil et al., 2024).

The term has also been used in a different, software-centric sense. PhasePack describes a MATLAB phase-retrieval library with a common interface through which algorithms and initializers can be swapped by changing configuration fields, and characterizes this as a “plug-and-play” experimental environment (Chandra et al., 2017). This establishes an important terminological distinction: in phase-retrieval research, “plug-and-play” may denote either denoiser-based iterative regularization or modular solver interchangeability.

2. Canonical variational decomposition

The shared mathematical template is a regularized inverse problem in which the data-fidelity term encodes the phaseless measurements and the prior term encodes image structure. Representative forms in the literature are

$$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$

and

$\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$

where $f$ or $\mathcal{D}$ enforces consistency with intensity-only measurements and $y^2 = |Ax|^2 + w,$0 or $y^2 = |Ax|^2 + w,$1 represents the prior (Wei et al., 2020, Chang et al., 2021).

In classical proximal splitting, the prior would enter through an explicit proximal operator. Plug-and-play methods replace that proximal map by a denoiser. In the PnP-ADMM formulation used for inverse imaging and instantiated for phase retrieval, the iterations are written as

$y^2 = |Ax|^2 + w,$2

$y^2 = |Ax|^2 + w,$3

$y^2 = |Ax|^2 + w,$4

so the regularizer proximal is replaced by the denoiser $y^2 = |Ax|^2 + w,$5 (Wei et al., 2020).

The same decomposition is expressed differently in generalized alternating projection and half-quadratic splitting. LPR rewrites phase retrieval in GAP form as

$y^2 = |Ax|^2 + w,$6

with alternating updates

$y^2 = |Ax|^2 + w,$7

where $y^2 = |Ax|^2 + w,$8 is the phase-retrieval solver and $y^2 = |Ax|^2 + w,$9 is the enhancing neural network (Chang et al., 2021). Deep Plug-and-Play HIO uses half-quadratic splitting,

$y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$0

followed by alternating minimization in a data step and a denoising step (Isil et al., 2024). Ptychographic PnP-HQS introduces one auxiliary variable per probe position and derives explicit Fourier-domain updates for each $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$1 (Denker et al., 2024).

This suggests that plug-and-play phase retrieval is best understood not as a single optimizer but as a decomposition principle: the nonlinear measurement model is handled by a physics-aware operator, and the prior is delegated to a modular image-restoration operator.

3. Algorithmic families

The literature contains several distinct plug-and-play phase-retrieval families that differ mainly in the data-consistency mechanism, the way the prior enters, and the degree of theoretical control.

Framework	Data-consistency mechanism	Prior mechanism
TFPnP / PnP-ADMM	ADMM-style proximal splitting for nonlinear inverse problems	Denoiser $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$2 with learned policy over $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$3, and stopping
LPR / PNP-GAP	Alternating projection in nonlinear complex space	FFDNet enhancing network
Deep Plug-and-Play HIO	Magnitude update plus inner HIO iterations	Pretrained CNN denoiser
deepECpr	EC likelihood exploitation with Laplace approximation	Deep denoiser in EC message passing
PnP-HQS for ptychography	Closed-form Fourier-domain $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$4-updates per probe position	TV, WCRR, or DRUNet
PnP-iBPDCA	Inertial Bregman DC step	Gaussian gradient step denoiser as Bregman proximal map

TFPnP treats the internal parameters of a PnP solver as a sequential decision problem and places a tuning-free control layer on top of standard PnP optimization. Its contribution is not a new phase-retrieval likelihood but automated control of denoising strength, ADMM penalty, and stopping time in a nonlinear inverse problem (Wei et al., 2020).

LPR extends PNP-GAP from real-valued image restoration to nonlinear complex-valued phase retrieval. It uses AP as the phase-retrieval solver because it is described as simple, efficient, modality-general, and low-memory, while FFDNet provides the enhancement prior (Chang et al., 2021).

Deep Plug-and-Play HIO begins from the same denoise-and-project logic but replaces the ER-like object update by HIO iterations in order to reduce stagnation in local minima. The resulting method performs a magnitude blending step, several inner HIO updates, and then denoising (Isil et al., 2024).

deepECpr departs from direct projection or proximal splitting. It uses expectation-consistent approximation to estimate the latent measurement vector $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$5, explicitly tracks approximate posterior variances, and introduces stochastic damping on the measurement-exploitation side. The paper emphasizes that this is not merely alternating denoising and projection, but EC message passing with extrinsic means and variances (Shastri et al., 2024).

PnP-iBPDCA embeds phase retrieval in a difference-of-convex formulation with Bregman geometry and an inertial step. Its distinctive claim is that the Gaussian gradient step denoiser is equivalent to a Bregman proximal operator of an implicit weakly convex functional, which permits convergence statements for the PnP variant under KL assumptions (Chow et al., 2024).

4. Learned hyperparameter control and tuning-free PnP

A recurrent difficulty in plug-and-play phase retrieval is the sensitivity of performance to internal parameters such as denoising strength, penalty parameters, and termination time. TFPnP addresses this by recasting parameter selection as a sequential decision problem. In its formulation, the state $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$6 contains optimization variables and auxiliary information, the action $y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$7 consists of internal parameters together with a stop/continue decision, and the policy is trained to maximize the expected return

$y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$8

The reward is

$y_\ell^2 = |F(A_\ell x)|^2,\qquad \ell=1,\dots,L,$9

where $u$0 is the PSNR at step $u$1, so the policy is rewarded for PSNR improvement and penalized for failing to terminate (Wei et al., 2020).

The policy network is split into two sub-policies: a stochastic termination policy $u$2, which samples a boolean action from a two-class categorical distribution, and a deterministic continuous policy $u$3, which predicts the optimization hyperparameters. Their gradients are learned by a mixed RL strategy in which the binary stopping decision is trained model-free and the continuous parameters are trained model-based by differentiating through the environment dynamics (Wei et al., 2020).

For phase retrieval, this control problem is more difficult than in linear inverse problems because the forward model is nonlinear and the proximal data-fidelity step is not generally available in closed form. The cited work therefore uses a differentiable environment model and a nonlinear data-fidelity proximal step or gradient-based approximation when needed, and specifies that, for nonlinear inverse problem—phase retrieval, the HIO algorithm is used as the initialization function (Wei et al., 2020). The earlier version of the method further specifies that the action space includes the penalty parameter $u$4, denoising strength $u$5, and terminal time $u$6, with rollout decisions taken every $u$7 PnP iterations and with a maximum of $u$8 decision steps (Wei et al., 2020).

A plausible implication is that “tuning-free” in this context refers to test-time control of solver internals rather than elimination of learned components. The same source explicitly notes that the method still requires a trained denoiser prior and a learned policy network (Wei et al., 2020).

5. Forward models, modalities, and complex-valued structure

Plug-and-play phase retrieval is now applied across several optical and computational-imaging modalities. In LPR, the framework is explicitly modality-agnostic in the sense that only the forward operator $u$9 changes. The paper instantiates this on coherent diffraction imaging, coded diffraction pattern imaging, and Fourier ptychographic microscopy. The corresponding forward models are

$x$0

$x$1

and

$x$2

respectively (Chang et al., 2021).

The TFPnP phase-retrieval experiments use coded diffraction patterns with four measurements,

$x$3

where the masks $x$4 have entries uniformly drawn from the unit circle. The observed data are intensity-only, and the noise is approximated by

$x$5

with $x$6 controlling the sigma-to-noise ratio (Wei et al., 2020).

Deep Plug-and-Play HIO focuses on Fourier phase retrieval, where $x$7, but also states that the same logic can be adapted to coded diffraction patterns. Its main derivation produces a closed-form magnitude update

$x$8

followed by an object-domain update and a denoising step (Isil et al., 2024).

Ptychographic PnP-HQS further generalizes the setting by introducing one auxiliary variable per probe position. The local operators satisfy

$x$9

and the measurements obey

$D(x^{k+1},\sigma_k)$0

This yields explicit $D(x^{k+1},\sigma_k)$1-updates in the Fourier domain for each probe position and a global prior step for the object $D(x^{k+1},\sigma_k)$2 (Denker et al., 2024).

Complex-valued priors require additional care. In ptychography, the cited work discusses two strategies for using real-image denoisers on complex-valued objects: denoising real and imaginary parts separately, or denoising magnitude and phase separately. In the reported experiments, pre-trained denoisers are applied to real and imaginary parts independently (Denker et al., 2024). This indicates that the “plugged” prior need not be inherently complex-valued, provided the interface between complex physics and real-valued denoising is defined.

6. Reported empirical behavior

The empirical literature reports improvements in reconstruction quality, robustness, scalability, and, in some cases, runtime. TFPnP evaluates phase retrieval on noisy intensity-only CDP measurements from twelve images used in prDeep and compares against HIO, WF, DOLPHIn, SPAR, BM3D-prGAMP, and prDeep. The reported PSNR values are best in all tested noise regimes:

• $D(x^{k+1},\sigma_k)$3: $D(x^{k+1},\sigma_k)$4 for the proposed method, versus $D(x^{k+1},\sigma_k)$5 for BM3D-prGAMP and $D(x^{k+1},\sigma_k)$6 for prDeep;
• $D(x^{k+1},\sigma_k)$7: $D(x^{k+1},\sigma_k)$8 versus $D(x^{k+1},\sigma_k)$9 and $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$0;
• $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$1: $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$2 versus $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$3 and $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$4 (Wei et al., 2020).

LPR reports broad gains across modalities. On CDI it achieves up to $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$5 dB PSNR improvement and $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$6 SSIM improvement over AP; on CDP with five modulations it achieves up to $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$7 dB PSNR improvement and $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$8 SSIM improvement; on CDP with single modulation it reports as much as $$\mathop{\mathrm{min}_{x \in \mathbb{R}^N}} ~ \mathcal{D}(x)+ \lambda \mathcal{R}(x),$$9 dB PSNR improvement and about $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$0 SSIM improvement; and on FPM simulation it gives nearly $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$1 dB improvement over AP. The same paper also reports ultra-large-scale $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$2K phase retrieval at $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$3 per color channel, with $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$4 million pixels, on CDP with five modulations and $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$5 dB input SNR, completed in minute-level time (Chang et al., 2021).

Deep Plug-and-Play HIO compares HIO, prDeep, iterative DNN-HIO, Plug-and-Play PR, and Plug-and-Play HIO on a $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$6-image test set. At $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$7, the reported averaged results are $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$8 dB / $\hat{u}=\arg \min _{u} f(u)+\lambda g(u),$9 for HIO, $f$0 dB / $f$1 for prDeep, $f$2 dB / $f$3 for iterative DNN-HIO, $f$4 dB / $f$5 for Plug-and-Play PR, and $f$6 dB / $f$7 for Plug-and-Play HIO. The same work states that Plug-and-Play HIO is about $f$8 faster than prDeep while achieving the best reconstruction quality (Isil et al., 2024).

deepECpr evaluates oversampled Fourier and coded diffraction pattern measurements on color, natural grayscale, and out-of-distribution unnatural grayscale images. Representative numbers include $f$9 dB / $\mathcal{D}$0 SSIM on FFHQ oversampled Fourier at $\mathcal{D}$1, compared with prDeep’s $\mathcal{D}$2 dB / $\mathcal{D}$3, and $\mathcal{D}$4 dB / $\mathcal{D}$5 on FFHQ CDP at $\mathcal{D}$6. The paper also reports roughly a $\mathcal{D}$7 reduction in denoiser calls versus prDeep, Deep-ITA, and DPS, together with convergence about an order of magnitude faster in iterations (Shastri et al., 2024).

PnP-iBPDCA reports the best Gaussian-noise phase-retrieval results among the compared methods across SNR $\mathcal{D}$8, $\mathcal{D}$9, and $y^2 = |Ax|^2 + w,$00, with $y^2 = |Ax|^2 + w,$01, $y^2 = |Ax|^2 + w,$02, and $y^2 = |Ax|^2 + w,$03 dB PSNR and $y^2 = |Ax|^2 + w,$04, $y^2 = |Ax|^2 + w,$05, and $y^2 = |Ax|^2 + w,$06 SSIM. Under the Poisson-style settings $y^2 = |Ax|^2 + w,$07, the paper states that there is a slight drop relative to the best supervised PnP baseline, specifically TFPnP$y^2 = |Ax|^2 + w,$08, because the model follows a phase-retrieval formulation primarily aligned with the Gaussian-noise setting of the referenced benchmark (Chow et al., 2024).

Method	Setting	Reported result
TFPnP	CDP, $y^2 = |Ax|^2 + w,$09	$y^2 = |Ax|^2 + w,$10 dB PSNR
LPR	CDP, single modulation	as much as $y^2 = |Ax|^2 + w,$11 dB PSNR and about $y^2 = |Ax|^2 + w,$12 SSIM improvement
LPR	8K CDP	$y^2 = |Ax|^2 + w,$13, minute-level time
Plug-and-Play HIO	$y^2 = |Ax|^2 + w,$14	$y^2 = |Ax|^2 + w,$15 dB / $y^2 = |Ax|^2 + w,$16 SSIM
deepECpr	FFHQ CDP, $y^2 = |Ax|^2 + w,$17	$y^2 = |Ax|^2 + w,$18 dB / $y^2 = |Ax|^2 + w,$19 SSIM
PnP-iBPDCA	Gaussian SNR $y^2 = |Ax|^2 + w,$20	$y^2 = |Ax|^2 + w,$21 dB / $y^2 = |Ax|^2 + w,$22 SSIM

These results collectively indicate that plug-and-play phase retrieval is no longer confined to modest Fourier experiments. The reported regime now includes nonlinear CDP, FPM, ptychography, out-of-distribution images, and ultra-large-scale reconstructions.

7. Limitations, misconceptions, and open issues

A common misconception is that plug-and-play phase retrieval is simply post-processing of a classical reconstruction. In the cited methods, the prior operator is embedded within the optimization loop and alternates with a measurement-consistency step at every outer iteration or message-passing round (Chang et al., 2021, Isil et al., 2024, Shastri et al., 2024). Another misconception is that plug-and-play phase retrieval is synonymous with ADMM. The current literature spans ADMM, GAP, HQS, HIO-based splitting, EC approximation, and Bregman DC optimization (Wei et al., 2020, Chang et al., 2021, Denker et al., 2024, Chow et al., 2024).

Convergence remains a central point of differentiation. The ptychography HQS paper notes that PnP methods generally do not converge in the classical sense unless restrictive conditions are imposed on the denoiser, and therefore finite-iteration reconstructions are used in practice (Denker et al., 2024). The earlier TFPnP formulation likewise states that nonconvexity remains and does not claim global optimality for phase retrieval (Wei et al., 2020). By contrast, PnP-iBPDCA is built specifically to retain subsequential and global convergence statements under the Kurdyka-Łojasiewicz framework, while proving that its Gaussian gradient step denoiser is a Bregman proximal map for an implicit weakly convex functional (Chow et al., 2024). This suggests an active fault line in the field between empirically strong but loosely analyzed PnP schemes and constructions designed for explicit convergence theory.

Training dependence is another recurring caveat. TFPnP is “tuning-free” only in the sense of automated test-time parameter selection; it still requires a trained denoiser, a trained policy network, and, for the model-based component, differentiability of the denoiser and an approximable data-consistency step (Wei et al., 2020). The same work notes that the current approach is trained on moderately sized problems and that scaling and more advanced RL remain open directions (Wei et al., 2020).

Finally, the term “plug-and-play” itself remains overloaded. PhasePack uses it to mean a common software interface for interchangeable phase-retrieval algorithms and initializers, whereas denoiser-based PnP methods use it to mean insertion of a prior operator into an iterative inverse solver (Chandra et al., 2017). Distinguishing these usages is essential for precise discussion of the field.