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REVIEW 3 major objections 5 minor 49 references

Online Regularization by Denoising with Applications to Phase Retrieval

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Replacing the full-batch gradient in regularization by denoising with an unbiased random minibatch of measurements preserves a worst-case O(1/√t) fixed-point convergence rate, and in phase retrieval one random measurement per iteration…

desk verdict On-RED is a real but modest contribution, and its advertised convergence claim outstrips what Theorem 1 actually proves. read the letter →

arxiv 1909.02040 v1 pith:6S6Z5LJX submitted 2019-09-04 eess.IV cs.CV

Zihui Wu , Yu Sun , Jiaming Liu , Ulugbek S. Kamilov This is my paper · ORCID
classification eess.IVcs.CV
keywords onlineregularizationbydenoisingREDstochasticgradientmethodsphaseretrievalcodeddiffractionpatternsnonexpansiveoperatorsfixed-pointconvergenceimagereconstruction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces On-RED, an online version of regularization by denoising that updates using only a small random subset of the measurements at each iteration instead of the entire dataset. Its central theoretical claim is that, when the data-fidelity components are convex and differentiable with bounded gradient variance, the denoiser is nonexpansive, and the iterates remain bounded, the average squared fixed-point residual converges at rate O(1/√t). The paper then tests On-RED on the nonconvex problem of phase retrieval from coded diffraction patterns, where a single randomly selected measurement per iteration nearly matches the reconstruction quality of the batch algorithm that uses all measurements. This matters because it gives the RED framework a path to large datasets that are too big for batch processing, while still carrying a convergence guarantee.

What carries the argument

The central object is the operator G(x)=∇g(x)+τ(x-D_σ(x)), whose zero set is the set of fixed points that RED seeks, together with the averaged operator P=I-γG. On-RED replaces P with the stochastic operator P̂=I-γĜ, where Ĝ uses a minibatch gradient. The proof works because P̂ is an unbiased perturbation of a nonexpansive P: Proposition 1 bounds the perturbation variance by γ²ν²/B, Proposition 2 gives nonexpansiveness of P under the step-size bound, and a telescoping-sum argument converts the per-iteration descent inequality into the averaged residual bound. The minibatch gradient carries the online processing, while the nonexpansive denoiser and bounded variance keep the stochastic perturbation controlled.

What would settle it

Track the quantity max_{x* ∈ zer(G)} ‖x_k − x*‖ during an On-RED run on a convex problem with γ = 1/(L+2τ) and a nonexpansive denoiser. If the iterate distance grows without bound, Assumption 2 fails and Theorem 1 does not apply; if it stays bounded but the empirical average of ‖G(x_{k-1})‖² violates the stated O(1/√t) rate, the theorem's constants or variance bound would be suspect.

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Extended reading notes

Core claim

On-RED iterates as x_k = x_{k-1} - γ(∇̂g(x_{k-1}) + τ(x_{k-1} - D_σ(x_{k-1}))), where ∇̂g is an unbiased minibatch gradient and D_σ is a denoiser. Theorem 1 states that if the component functions are convex and L-Lipschitz differentiable, the minibatch gradient has variance at most ν²/B, the zero set of G(x)=∇g(x)+τ(x-D_σ(x)) is nonempty, the iterates stay within radius R₀ of that set, and D_σ is nonexpansive, then for γ ∈ (0, 1/(L+2τ)] the average satisfies E[(1/t)∑_{k=1}^t ‖G(x_{k-1})‖²] ≤ (L+2τ)/γ [ν²γ²/B + 2γνR₀/√B + R₀²/t]. This yields an O(1/√t) average fixed-point residual, and the analysis does not require the denoiser to correspond to an explicit regularizer. In the nonconvex phase-retrieval experiments, On-RED with B=1 outperforms GM-RED using one fixed measurement by more than 4 dB and approaches the full six-measurement batch SNR.

Load-bearing premise

The load-bearing premise is that every On-RED iterate stays within a fixed Euclidean ball of radius R₀ around the set of fixed points; this is assumed rather than derived, so if the iterates drift outside that ball the convergence bound no longer follows.

Editorial extensions

If this is right

  • For convex data-fidelity, On-RED provides a worst-case O(1/√t) convergence guarantee for the average fixed-point residual, matching the rate of stochastic gradient methods while retaining the RED regularization operator.
  • The per-iteration cost of On-RED scales with the minibatch size B rather than the total number of measurements I, so the method can be applied when the full dataset cannot be loaded or processed in one gradient step.
  • Because the theorem does not require the denoiser D_σ to come from an explicit regularizer, the same guarantee covers learned CNN denoisers such as DnCNN*, provided they are nonexpansive.
  • The phase-retrieval experiments show that cycling over random measurements with B=1 recovers image details lost when a single fixed measurement is used, and approaches the SNR of the full-batch algorithm, indicating that online measurement diversity is practically valuable even in a nonconvex problem.
  • The numerical trend that smaller step size and larger minibatch improve convergence accuracy extends the theorem's qualitative predictions to the nonconvex coded-diffraction-pattern setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would replace the assumed bounded-iterate condition with coercivity or strong convexity of g, yielding a version of Theorem 1 in which R₀ no longer appears as an unexplained constant; the paper does not make this derivation.
  • The same minibatch-perturbation proof strategy likely transfers to variance-reduced gradient estimators such as SVRG or SAGA, which could replace the ν²/B term by a term that decays with iteration count and improve the rate beyond 1/√t.
  • The empirical success at B=1 suggests that random measurement diversity itself acts as an implicit regularizer in phase retrieval; quantifying SNR as a function of measurement diversity at fixed computation would be a direct testable extension.
  • On-RED could also be applied to computed tomography or Fourier ptychography, where large measurement sets are common, but the paper's convergence analysis covers only uniform random sampling and would need adaptation for ordered-subset or data-adaptive sampling strategies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes On-RED, an online (minibatch) variant of the Regularization by Denoising framework for imaging inverse problems. At each iteration the algorithm uses a random subset of measurements to form an unbiased gradient estimate, combined with a nonexpansive denoiser. The main theoretical result, Theorem 1, bounds the expected average squared norm of the fixed-point residual G over t iterations under convexity, bounded variance, uniform boundedness of iterates, and nonexpansiveness assumptions. The paper also presents phase-retrieval experiments on coded diffraction patterns, showing that with B=1 On-RED approaches the SNR of full-batch GM-RED while using the same per-iteration cost as a fixed single-measurement batch method.

Significance. If the advertised convergence claim held as stated, On-RED would be a useful scalable alternative to batch RED for large datasets, and the paper would provide a transparent stochastic-gradient analysis for a denoiser-based regularizer. The paper has genuine strengths: the assumptions are stated explicitly, the proof algebra is straightforward to follow, and the experiments cover multiple denoisers (TV, BM3D, DnCNN*) and clearly demonstrate that online processing of measurements improves over a fixed single-measurement batch baseline. However, the central theoretical claim is stronger than what Theorem 1 actually proves, and the numerical validation metric does not directly match the theorem's bound. These issues are load-bearing because the paper's contribution is precisely the claimed O(1/sqrt(t)) fixed-point convergence, and they need to be addressed before publication.

major comments (3)
  1. [Theorem 1 and Introduction (contribution bullet 1)] Theorem 1 bounds only the Cesàro average (and hence the minimum) of ||G(x_{k-1})||² over k=1,...,t. For any fixed minibatch size B and fixed step size γ, the right-hand side has the positive limit (L+2τ)/γ [ν²γ²/B + 2γνR0/√B] as t→∞. Therefore the theorem does not establish convergence of the iterates to an element of zer(G), and the Introduction's statement that 'On-RED converges to a fixed point at the worst-case rate of O(1/√t)' is not supported by the theorem. The O(1/√t) statement is obtained only by taking B=t, a growing minibatch that is not what is run in Section 5 (B=1 or B in {10,20,30}). Please revise the advertised claim to a bound on the average fixed-point residual with a positive noise floor, and clearly separate the fixed-B regime from the growing-B regime.
  2. [Assumption 2 and the proof of Theorem 1 (Section 7, Eq. (15)-(16))] Assumption 2 postulates that all iterates lie within a fixed R0-ball around every point in zer(G), but this is neither derived from the algorithm nor verified in the nonconvex phase-retrieval experiments. In the proof, the Cauchy-Schwarz step requires ||P(x_{k-1})-P(x*)|| ≤ R0 for every k, so the bound genuinely depends on this unverified condition. However, taking the conditional expectation before applying Cauchy-Schwarz makes the cross term vanish because E[hatP(x)|x]=P(x); the telescoping sum then needs only ||x0-x*|| ≤ R0. This shows that Assumption 2 is either avoidable and should be removed from the theorem, or, if kept, must be justified or numerically checked. As written, it cannot support the empirical convergence claims in the nonconvex setting.
  3. [Section 5.2, definition of Norm. Acc. and Table 2] The quantity plotted and tabulated as 'Norm. Acc.' is ||G(x_k)||²/||G(x_0)||² at the final iteration (or at each k), whereas Theorem 1 bounds E[(1/t)Σ_{k=1}^t ||G(x_{k-1})||²], the average over iterations. The observed improvement with smaller γ and larger B is therefore not a direct empirical verification of the theorem's quantitative bound. Please either report the running average of the residuals (the quantity actually bounded) or clarify that the plots illustrate a related heuristic rather than a direct validation of Theorem 1.
minor comments (5)
  1. [Abstract and Conclusion] The phrase 'We establish the theoretical convergence of On-RED in convex settings' is too strong given the noise-floor issue in Theorem 1; suggest 'we establish a fixed-point residual bound' or similar.
  2. [Proposition 2 (Section 7)] The proof of Proposition 2 is entirely imported from the supplementary material of [39] by setting U=UT=I and Gi=G; since this result is central to the nonexpansiveness of P, please include a self-contained derivation or state the exact result from [39] in enough detail to make the proof readable without consulting another paper's supplement.
  3. [Assumption 2] The wording 'the distance between the farthest point in zer(G) and the sequence {x_k}' is confusing; it should state clearly that all iterates satisfy ||x_k - x*||₂ ≤ R0 for every x* in zer(G) and every k ≥ 0.
  4. [Table 3] The column layout of Table 3 is hard to follow: SGM has no denoiser, and GM-RED (fixed 6) is listed only for DnCNN*, making it unclear which columns are comparable; please restructure the table to make the denoiser and algorithm for each column explicit.
  5. [Section 5.2] The theorem's first inequality E[min_k ||G(x_{k-1})||²] ≤ E[(1/t)Σ||G(x_{k-1})||²] is correct, but the min form can mislead readers into thinking a particular iterate converges; consider omitting it or adding a remark that the bound is on the average residual, not on the final iterate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is derived from stated assumptions, and the only same-group citation supplies an independent deterministic-RED lemma.

full rationale

The claimed derivation chain is not circular. Theorem 1 bounds the average squared norm of G(x_{k-1}) by writing the On-RED update as x_k = P̂(x_{k-1}) with P̂ = I − γĜ. Proposition 1 derives E[P̂(x)] = P(x) and the variance bound γ²ν²/B directly from Assumption 1's unbiasedness and bounded variance, with no fitted constants. Proposition 2 (P nonexpansive for γ ≤ 1/(L+2τ)) is taken from Sun et al. [39], but it is a lemma about the full-gradient RED operator, obtained by specializing U = I and Gi = G; it does not contain the online/minibatch claim, and its proof is parameter-free under Assumption 3. The telescoping inequality is then obtained in the paper from these two ingredients plus Assumption 2's bounded-radius bound, and the theorem follows by averaging and taking expectation. The O(1/√t) statement is derived by explicitly choosing B = t; the fixed-B bound retains a positive variance floor, which is a limitation of the theorem rather than a circular identification. The empirical section tunes τ against ground-truth SNR, but no theoretical result is obtained from those tuned values. Assumption 2 is assumed rather than verified in the nonconvex experiments, but an unverified assumption is a correctness risk, not circularity. No equation is equivalent to its inputs by construction, and no self-citation is the sole support for the central claim.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The theoretical contribution depends on the three stated assumptions plus imported lemmas from the authors' own BC-RED paper. The empirical SNR claims depend on τ chosen with oracle access to test ground truth and on an unreleased trained DnCNN* model. No new physical or mathematical entities are introduced.

free parameters (5)
  • τ (regularization strength) = 0.2 for Section 5.2; per-image optimized for Section 5.3
    Selected to maximize reconstruction SNR against ground-truth test images, making the empirical comparisons oracle-tuned.
  • σ (denoiser input noise level) = 5
    Chosen by hand; affects denoiser strength but is not fitted to the target measurement data.
  • Step size γ = 1/(L+2τ) multiplied by 1, 1/3, or 1/9 in Section 5.2; 1/(L+2τ) in Section 5.3
    Experimental choices within the theoretical range, used to show the predicted convergence trends.
  • Minibatch size B and total measurements I = B in {10,20,30} with I=40 in Section 5.2; B=1 with I=6 in Section 5.3
    Set to demonstrate online behavior; the I=6 scenario is small and does not stress the large-data regime the paper motivates.
  • DnCNN* trained weights = Not released; trained on BSD400 with data augmentation
    The neural denoiser is learned externally and is central to the phase-retrieval results, but no model artifact is provided.
assumptions (7)
  • domain assumption Assumption 1: the component functions g_i are convex and differentiable with a common Lipschitz constant L, and the minibatch gradient is unbiased with variance bounded by ν²/B.
    Needed to make the stochastic gradient recursion track the full gradient; standard but unverifiable in the phase-retrieval problem, where g is nonconvex.
  • domain assumption Assumption 2: all On-RED iterates stay within distance R0 of the fixed-point set zer(G).
    Used in the telescoping step of the proof; no algorithmic mechanism enforces this bound.
  • domain assumption Assumption 3: the denoiser Dσ is nonexpansive.
    Ensures P is nonexpansive via the cited BC-RED result; the DnCNN* used in experiments is not shown to be nonexpansive.
  • domain assumption Lemma from [39]: with U=U^T=I and Gi=G, the operator P is nonexpansive for γ in (0, 1/(L+2τ)].
    Imported without proof; this is the key step connecting nonexpansiveness to the descent bound.
  • domain assumption Bound (14) from the proof of Theorem 1 in the supplementary material of [39].
    Adapted by setting U=U^T=I and Gi=G; the paper does not reproduce the bound, so the proof depends on the correctness of the earlier preprint.
  • standard math Krasnosel'skii-Mann theorem and monotone operator theory.
    Background framework; accepted.
  • domain assumption CDP phase retrieval forward model y_i = |F M_i x| with additive Gaussian noise.
    Used for the experiments; the nonlinearity makes g nonconvex, placing the application outside Theorem 1.

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Pith. "Pith review of Online Regularization by Denoising with Applications to Phase Retrieval." pith.science (2026). https://pith.science/paper/6S6Z5LJX

@misc{pith2026190902040,
  author       = {Pith},
  title        = {Pith review of: Online Regularization by Denoising with Applications to Phase Retrieval},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6S6Z5LJX}},
  note         = {Machine review of arXiv:1909.02040}
}
read the original abstract

Regularization by denoising (RED) is a powerful framework for solving imaging inverse problems. Most RED algorithms are iterative batch procedures, which limits their applicability to very large datasets. In this paper, we address this limitation by introducing a novel online RED (On-RED) algorithm, which processes a small subset of the data at a time. We establish the theoretical convergence of On-RED in convex settings and empirically discuss its effectiveness in non-convex ones by illustrating its applicability to phase retrieval. Our results suggest that On-RED is an effective alternative to the traditional RED algorithms when dealing with large datasets.

Figures

Figures reproduced from arXiv: 1909.02040 by the authors.

Figure 1
Figure 1. Conceptual illustration of online regularization by denois￾ing (On-RED). The proposed algorithm uses a random subset of noisy measurements at every iteration to reconstruct a high-quality image using a convolutional neural network (CNN) denoser. remarkable flexibility in choosing image priors, but also com￾plicates its analysis due to the lack of an explicit objective function. An alternative strategy for leveraging… view at source ↗
Figure 2
Figure 2. Test images used in the experiments. From left to right: [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Illustration of the influence of γ and B on the convergence of On-RED for phase retrieval under DnCNN∗ . The left plot shows the convergence results of On-RED for three different step sizes with a fixed minibatch size B = 10 and the right plot shows the results of On-RED for three different minibatch sizes with a fixed step size γ = 1 L+2τ . Both experiments draw random samples from a total of I = 40 measurements. T… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Visual examples of recontructed Barbara, Parrot, and Pepper images by GM-RED (1), On-RED (1), and GM-RED (6) with BM3D and DnCNN∗ denoisers. The original images are displayed in the first column. The second and the third columns show the results of batch GM-RED using 1…

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Reference graph

Works this paper leans on

49 extracted references · 48 canonical work pages

  1. [39]

    Y . Sun, J. Liu, and U. S. Kamilov. Block coordinate regular- ization by denoising. InProc. Advances in Neural Information Processing Systems 32, Vancouver, BC, Canada, Dec. 2019. 1, 3, 4, 7

    work page 2019

  2. [1]

    M. V . Afonso, J. M.Bioucas-Dias, and M. A. T. Figueiredo. Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process., 19(9):2345–2356, September 2010. 1

    work page 2010

  3. [2]

    H. H. Bauschke and P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces . Springer, 2 edition, 2017. 2, 3

    work page 2017

  4. [3]

    Beck and M

    A. Beck and M. Teboulle. Fast gradient-based algorithm for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process., 18(11):2419–2434, November 2009. 1

    work page 2009

  5. [4]

    J. Bect, L. Blanc-Feraud, G. Aubert, and A. Chambolle. A 𝓁1-unified variational framework for image restoration. In Proc. Euro. Conf. Comp. Vis. (ECCV), volume 3024, pages 1–13, New York, 2004. 1

    work page 2004

  6. [5]

    Bernstein, Y .-X

    J. Bernstein, Y .-X. Wang, K. Azizzadenesheli, and A. Anand- kumar. signSGD: Compressed optimization for non-convex problems. In Proc. 35th Int. Conf. Machine Learning (ICML), volume 80, pages 560–569, Stockholm, Sweden, July 2018. 4

    work page 2018

  7. [6]

    Bottou and O

    L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Proc. Advances in Neural Information Processing Systems (NIPS), pages 161–168, Vancouver, BC, Canada, December 3-6, 2007. 1

    work page 2007

  8. [7]

    S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Dis- tributed optimization and statistical learning via the alternat- ing direction method of multipliers. Foundations and Trends in Machine Learning, 3(1):1–122, 2011. 1

    work page 2011

Show all 49 references
  1. [8]

    Brifman, Y

    A. Brifman, Y . Romano, and M. Elad. Turning a denoiser into a super-resolver using plug and play priors. In Proc. IEEE Int. Conf. Image Proc. (ICIP), pages 1404–1408, Phoenix, AZ, USA, September 25-28, 2016. 1

    2016

  2. [9]

    G. T. Buzzard, S. H. Chan, S. Sreehari, and C. A. Bouman. Plug-and-play unplugged: Optimization free reconstruc- tion using consensus equilibrium. SIAM J. Imaging Sci. , 11(3):2001–2020, September 2018. 2

    2001

  3. [10]

    E. J. Cand`es, J. Romberg, and T. Tao. Robust uncertainty prin- ciples: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory, 52(2):489– 509, February 2006. 2

    2006

  4. [11]

    E. J. Cand`es, T. Strohmer, and V . V oroninski. PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming. Communications on Pure and Applied Mathematics, 66(8):1241–1274, 2013. 1, 2

    2013

  5. [12]

    S. H. Chan, X. Wang, and O. A. Elgendy. Plug-and-play ADMM for image restoration: Fixed-point convergence and applications. IEEE Trans. Comp. Imag., 3(1):84–98, March

  6. [13]

    Dabov, A

    K. Dabov, A. Foi, V . Katkovnik, and K. Egiazarian. Image denoising by sparse 3-D transform-domain collaborative filter- ing. IEEE Trans. Image Process., 16(16):2080–2095, August

  7. [14]

    Daubechies, M

    I. Daubechies, M. Defrise, and C. D. Mol. An iterative thresh- olding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math., 57(11):1413–1457, November 2004. 1

    2004

  8. [15]

    D. L. Donoho. Compressed sensing. IEEE Trans. Inf. Theory, 52(4):1289–1306, April 2006. 2

    2006

  9. [16]

    Eckstein and D

    J. Eckstein and D. P. Bertsekas. On the Douglas-Rachford splitting method and the proximal point algorithm for max- imal monotone operators. Mathematical Programming , 55:293–318, 1992. 1

    1992

  10. [17]

    Elad and M

    M. Elad and M. Aharon. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process., 15(12):3736–3745, December 2006. 1

    2006

  11. [18]

    M. A. T. Figueiredo and R. D. Nowak. Wavelet-based image estimation: An empirical Bayes approach using Jeffreys’ non- informative prior. IEEE Trans. Image Process., 10(9):1322– 1331, September 2001. 1

    2001

  12. [19]

    M. A. T. Figueiredo and R. D. Nowak. An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process., 12(8):906–916, August 2003. 1

    2003

  13. [20]

    A. K. Fletcher, P. Pandit, S. Rangan, S. Sarkar, and P. Schniter. Plug-in estimation in high-dimensional linear inverse prob- lems: A rigorous analysis. In Proc. Advances in Neural Information Processing Systems (NIPS), pages 7451–7460, Montreal, QC, Canada, December 2-8, 2018. 2

    2018

  14. [21]

    Ghadimi and G

    S. Ghadimi and G. Lan. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Math. Program. Ser. A, 156(1):59–99, March 2016. 4

    2016

  15. [22]

    U. S. Kamilov. A parallel proximal algorithm for anisotropic total variation minimization. IEEE Trans. Image Process., 26(2):539–548, February 2017. 2

    2017

  16. [23]

    U. S. Kamilov, H. Mansour, and B. Wohlberg. A plug-and- play priors approach for solving nonlinear imaging inverse problems. IEEE Signal. Proc. Let., 24(12):1872–1876, De- cember 2017. 1, 2

    2017

  17. [24]

    D. Kim, D. Pal, J. Thibault, and J. A. Fessler. Accelerating ordered subsets image reconstruction for X-ray CT using spatially nonuniform optimization transfer. IEEE Trans. Med. Imag., 32(11):1965–1978, Nov 2013. 1

    1965

  18. [25]

    Mataev, M

    G. Mataev, M. Elad, and P. Milanfar. Deepred: Deep image prior powered by RED. 2019. arXiv:1903.10176 [cs.CV]. 2

    2019 arXiv

  19. [26]

    Meinhardt, M

    T. Meinhardt, M. Moeller, C. Hazirbas, and D. Cremers. Learning proximal operators: Using denoising networks for regularizing inverse imaging problems. In Proc. IEEE Int. Conf. Comp. Vis. (ICCV) , pages 1799–1808, Venice, Italy, October 22-29, 2017. 1, 2

    2017

  20. [27]

    Metzler, P

    C. Metzler, P. Schniter, A. Veeraraghavan, and R. Baraniuk. prDeep: Robust phase retrieval with a flexible deep network. In Proc. 35th Int. Conf. Machine Learning (ICML) , pages 3501–3510, Stockholmsm¨assan, Stockholm Sweden, 10–15 Jul 2018. 1, 3, 4

    2018

  21. [28]

    C. A. Metzler, A. Maleki, and R. G. Baraniuk. From denoising to compressed sensing. IEEE Trans. Inf. Theory, 62(9):5117– 5144, September 2016. 2

    2016

  22. [29]

    J. J. Moreau. Proximit ´e et dualit´e dans un espace Hilbertien. Bull. Soc. Math. France, 93:273–299, 1965. 2

    1965

  23. [30]

    M. K. Ng, P. Weiss, and X. Yuan. Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. , 32(5):2710–2736, August 2010. 1

    2010

  24. [31]

    S. Ono. Primal-dual plug-and-play image restoration. IEEE Signal. Proc. Let., 24(8):1108–1112, 2017. 2

    2017

  25. [32]

    Parikh and S

    N. Parikh and S. Boyd. Proximal algorithms. Foundations and Trends in Optimization, 1(3):123–231, 2014. 4

    2014

  26. [33]

    E. T. Reehorst and P. Schniter. Regularization by denoising: Clarifications and new interpretations. IEEE Trans. Comput. Imag., 5(1):52–67, Mar. 2019. 1, 3

    2019

  27. [34]

    Romano, M

    Y . Romano, M. Elad, and P. Milanfar. The little engine that could: Regularization by denoising (RED). SIAM J. Imaging Sci., 10(4):1804–1844, 2017. 1, 2, 3

    2017

  28. [35]

    L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms.Physica D, 60(1–4):259–268, November 1992. 1, 2

    1992

  29. [36]

    E. K. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, and W. Yin. Plug-and-play methods provably converge with properly trained denoisers. In Proc. 36th Int. Conf. Machine Learning (ICML), pages 5546–5557, 2019. 2

    2019

  30. [37]

    Sreehari, S

    S. Sreehari, S. V . Venkatakrishnan, B. Wohlberg, G. T. Buz- zard, L. F. Drummy, J. P. Simmons, and C. A. Bouman. Plug-and-play priors for bright field electron tomography and sparse interpolation. IEEE Trans. Comput. Imaging, 2(4):408– 423, December 2016. 1, 2

    2016

  31. [38]

    J. L. Starck, E. Pantin, and F. Murtagh. Deconvolution in as- tronomy: A review. Pub. Astron. Soc. Pacific, 114(800):1051– 1069, October 2002. 1

    2002

  32. [40]

    Y . Sun, B. Wohlberg, and U. S. Kamilov. An online plug-and- play algorithm for regularized image reconstruction. IEEE Trans. Comput. Imaging, 2019. 1, 2, 3, 4

    2019

  33. [41]

    Y . Sun, S. Xu, Y . Li, L. Tian, B. Wohlberg, and U. S. Kamilov. Regularized fourier ptychography using an online plug-and- play algorithm. In Proc. IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP), pages 7665–7669, Brighton, UK, May 12-17, 2019. 1

    2019

  34. [42]

    Teodoro, J

    A. Teodoro, J. M. Bioucas-Dias, and M. Figueiredo. Scene- adapted plug-and-play algorithm with convergence guaran- tees. In Proc. IEEE Int. Workshop on Machine Learning for Signal Processing, pages 1–6, Tokyo, Japan, September 25-28, 2017. 2

    2017

  35. [43]

    A. M. Teodoro, J. M. Biocas-Dias, and M. A. T. Figueiredo. Image restoration and reconstruction using variable splitting and class-adapted image priors. In Proc. IEEE Int. Conf. Image Proc. (ICIP), pages 3518–3522, Phoenix, AZ, USA, September 25-28, 2016. 1

    2016

  36. [44]

    Tian and L

    L. Tian and L. Waller. 3D intensity and phase imaging from light field measurements in an LED array microscope. Optica, 2:104–111, 2015. 1

    2015

  37. [45]

    Tibshirani

    R. Tibshirani. Regression and selection via the lasso. J. R. Stat. Soc. Series B (Methodological), 58(1):267–288, 1996. 2

    1996

  38. [46]

    S. V . Venkatakrishnan, C. A. Bouman, and B. Wohlberg. Plug- and-play priors for model based reconstruction. InProc. IEEE Global Conf. Signal Process. and Inf. Process. (GlobalSIP), pages 945–948, Austin, TX, USA, December 3-5, 2013. 1, 2

    2013

  39. [47]

    Xu and U

    X. Xu and U. S. Kamilov. Signprox: One-bit proximal al- gorithm for nonconvex stochastic optimization. In IEEE Int. Conf. Acoustics, Speech and Signal Process. (ICASSP), pages 7800–7804, Brighton, UK, May 2019. 4

    2019

  40. [48]

    Zhang, W

    K. Zhang, W. Zuo, Y . Chen, D. Meng, and L. Zhang. Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising. IEEE Trans. Image Process., 26(7):3142–3155, July 2017. 1

    2017

  41. [49]

    Zhang, W

    K. Zhang, W. Zuo, S. Gu, and L. Zhang. Learning deep CNN denoiser prior for image restoration. In Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR) , pages 3929–3938, Honolulu, USA, July 21-26, 2017. 1

    2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.