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Regularization by Denoising via Fixed-Point Projection (RED-PRO)

Regev Cohen, Michael Elad, Peyman Milanfar

arXiv:2008.00226v2eess.IVcs.CV

TL;DR

PnP and RED have shown strong results with denoisers, but their theoretical account is incomplete for general denoisers. This paper reformulates RED as fixed-point projection for demicontractive denoisers and relates the method to PnP. Experiments report competitive and improved performance over RED in image deblurring and super-resolution.

  • Problem

    For general denoisers, the optimization interpretation and global-minimum convergence guarantees of PnP and RED remain incomplete.

    Source: §1 · pp. 2–3

  • Method

    RED-PRO formulates a convex problem using projection onto the fixed-point set of demicontractive denoisers and relates PnP proximal gradient to this framework.

    Source: §1 · p. 3

  • Results

    Experiments in image deblurring and super-resolution show competitive and even improved performance compared with the original RED framework.

    Source: §6 · p. 22

  • Takeaways & Limitations

    RED-PRO bridges RED and PnP while providing convergence guarantees and flexibility in denoiser choice within the stated framework.

    Source: §6 · pp. 21–22

    Relaxed RED-PRO broadens the search domain but costs more computation because it uses multiple denoiser activations per iteration.

    Source: §4.2 · p. 16

Abstract

from arXiv · show

Inverse problems in image processing are typically cast as optimization tasks, consisting of data-fidelity and stabilizing regularization terms. A recent regularization strategy of great interest utilizes the power of denoising engines. Two such methods are the Plug-and-Play Prior (PnP) and Regularization by Denoising (RED). While both have shown state-of-the-art results in various recovery tasks, their theoretical justification is incomplete. In this paper, we aim to bridge between RED and PnP, enriching the understanding of both frameworks. Towards that end, we reformulate RED as a convex optimization problem utilizing a projection (RED-PRO) onto the fixed-point set of demicontractive denoisers. We offer a simple iterative solution to this problem, by which we show that PnP proximal gradient method is a special case of RED-PRO, while providing guarantees for the convergence of both frameworks to globally optimal solutions. In addition, we present relaxations of RED-PRO that allow for handling denoisers with limited fixed-point sets. Finally, we demonstrate RED-PRO for the tasks of image deblurring and super-resolution, showing improved results with respect to the original RED framework.

1. Introduction.

PnP and RED use denoisers to regularize inverse problems, but their optimization interpretations and convergence guarantees remain incomplete for general denoisers. The paper introduces RED-PRO to connect the frameworks through fixed-point projection and reports improved results over RED in image deblurring and super-resolution.

1. Introduction.

  • PnP replaces proximal operators with denoisers, while RED uses denoising residuals to form an explicit regularizer.Both approaches build on denoising engines as a regularization mechanism.Source: §1 · pp. 1–2
  • For general denoisers beyond proximal maps, PnP lacks an explicit minimized objective, despite fixed-point convergence results under several assumptions.An explicit objective supports parameter tuning, line search, acceleration, and analysis of convergence and solutions.Source: §1 · p. 2
  • RED's objective and convergence justification rely on differentiability, local homogeneity, and a symmetric Jacobian, assumptions that exclude several popular denoisers.Existing alternatives discussed by the authors establish fixed-point convergence, but not convergence to global minima of an explicit energy function.Source: §1 · pp. 2–3
  • RED-PRO reformulates RED using projection onto the fixed-point set of demicontractive denoisers, connects it to PnP proximal gradient, and gives global-optimum convergence guarantees.The paper also proposes relaxed variants for denoisers with narrow fixed-point sets, at higher computational load.Source: §1 · p. 3

2. Preliminaries.

The preliminaries frame image recovery as optimization with a likelihood term and a prior, then review how PnP and RED use denoisers. They also set out the assumptions and theoretical gaps that motivate the paper's fixed-point formulation.

2.1. Inverse Problems.

  • Inverse-problem recovery combines a data likelihood with a prior because the likelihood alone can yield a non-unique or unstable solution.For linear degradation with white Gaussian noise, the likelihood is a squared-error term; other noise models change its form.Source: §2.1 · pp. 4–5

2. Preliminaries.

  • The preliminaries position denoising as a source of regularization because effective denoisers are available and have been used in inverse problems.The paper identifies deblurring and super-resolution among the applications where PnP and RED have shown state-of-the-art performance.Source: §2.1 · p. 5

2.2. Plug and Play Prior (PnP).

  • PnP extends proximal-gradient and ADMM solvers by replacing proximal operators for the prior with general denoisers, which need not arise from explicit regularizers.For general denoisers, PnP lacks an explicit minimized objective, although fixed-point convergence has been shown under different conditions.Source: §2.2 · p. 6

2.3. Regularization by Denoising (RED).

  • RED defines an explicit denoiser-based regularizer whose gradient is the denoising residual under its stated conditions.Those conditions include differentiability, local homogeneity, and Jacobian symmetry; the authors note that many popular denoisers do not satisfy the symmetry condition.Source: §2.3 · p. 7

3. Demicontractivity.

The section develops demicontractivity as a condition suited to fixed-point regularization: its fixed-point set is closed and convex, and the condition encompasses or relates to assumptions used in prior convergence analyses. It then connects demicontractive denoisers to RED and PnP through residual properties and external strength control.

3.2. Demicontractive Denoisers.

  • Demicontractive mappings include quasi-nonexpansive mappings, which include nonexpansive operators, and the paper relates demicontractivity to weaker Lipschitz assumptions.The authors state that demicontractivity is hard to verify for general denoisers, while covering a broader range than common assumptions.Source: §3.1 · p. 8, §3.2 · pp. 10–11

3.1. Fixed-Point Theory.

  • For a demicontractive mapping, the fixed-point set is closed and convex, providing the set structure used by the paper's later formulation.The paper assumes throughout that the fixed-point set is nonempty.Source: §3.1 · pp. 8–9

3.2. Demicontractive Denoisers.

  • Demicontractivity corresponds to a residual inequality restricted to fixed points and is broader than co-coercivity, allowing the analysis to go beyond common optimization schemes.The residual is defined as x−f(x), and its zero set coincides with the denoiser's fixed-point set.Source: §3.2 · p. 10
  • Averaging a demicontractive denoiser provides external control of its strength, and a specified iteration-dependent averaging choice ensures convergence of the cited PnP-ADMM variant.This approach does not require the denoiser's internal strength parameter, which may be unavailable.Source: §3.2 · p. 11
  • The section motivates RED-PRO by reformulating RED as convex minimization over a demicontractive denoiser's fixed-point set and relating the resulting scheme to PnP.The authors also consider broader domains than the fixed-point set to expand which denoisers can be used.Source: §4 · p. 11

4. RED-PRO: RED via Fixed-Point Projection.

RED-PRO replaces denoiser-based regularization with optimization over a denoiser’s fixed-point set. The framework is convex for demicontractive denoisers, connects PnP-PGD to this formulation, and extends it with relaxed fixed-point domains.

4.1. Regularization by Projection.

  • For a demicontractive denoiser, constraining recovery to Fix(f) yields a convex minimization problem using a well-behaved alternative to the generally inaccessible natural-image manifold.The formulation searches for a minimizer of the data-fidelity term over Fix(f).Source: §4.1 · p. 12
  • The hybrid steepest descent method uses one denoiser activation per iteration, and its iterates converge to an optimal RED-PRO solution under the stated step-size conditions.The paper gives convergence results for both diminishing and constant step sizes.Source: §4.1 · pp. 13–14

4.1.1. PnP and RED as a special case.

  • Setting qk=0 and α=1 makes the stated RED variant coincide with PnP-PGD, whose convergence is linked to a convex feasibility problem under the stated conditions.The paper interprets PnP-PGD as a special case of RED-PRO and provides convergence results when the theorem conditions hold.Source: §4.1.1 · pp. 14–15
  • Because the fixed-point equation can have multiple solutions, the iteration may select an arbitrary one; targeting u instead yields PFix(T)(u), including a minimum-norm solution when u=0.Different PnP versions sharing the same fixed-point equation may converge to different solutions.Source: §4.1.1 · p. 15

4.2. Relaxed RED-PRO.

  • The relaxed RED-PRO objective penalizes distance from Fix(f), with λ controlling that distance; sufficiently large λ recovers the hard fixed-point constraint.The relaxed objective remains a convex minimization when the denoiser is demicontractive.Source: §4.2 · p. 15
  • The approximate fixed-point set Fixε(f) includes “almost clean” images, but it is generally nonconvex, so projection onto it is not well-defined and convergence guarantees are unavailable.The paper notes this issue for demicontractive or even nonexpansive functions.Source: §4.2 · pp. 16–17
  • The dilated set Bδ(f) is closed and convex and can lie within Fixε(f); relaxed RED-PRO uses it to widen the search, with δ=0 returning the prior relaxed problem.This approach can yield stable inverse solutions with limited fixed-point sets, but costs more denoiser activations and uses a small number of inner iterations in practice.Source: §4.2 · pp. 16–18

5. Experiments.

Experiments evaluate RED-PRO in deblurring and super-resolution, reporting improved results over RED in both tasks. Convergence behavior is also examined across denoisers.

5. Experiments.

  • For comparisons, shared RED and RED-PRO parameters follow values from [77], while additional RED-PRO parameters are tuned manually to improve PSNR.The deblurring experiments include uniform and Gaussian blur; the super-resolution setup uses a Gaussian kernel and 3× downsampling per axis.Source: §5 · p. 18, §5.2 · p. 19, §5.3 · p. 20

5.1. Convergence.

  • Across denoisers, fidelity errors decline and solutions approach fixed points; BM3D shows a notably greater decrease, with convergence rates varying by denoiser.The authors caution that these convergence measures may not accurately reflect restored-image quality.Source: §5.1 · p. 19, Figure 5.1 · p. 19

5.2. Image Deblurring.

  • RED-PRO improves on RED for NLM deblurring, especially with uniform blur; for TNRD, the relaxed framework's improvement is small.The reported deblurring comparisons use PSNR, with higher values better.Source: §5.2 · p. 20, Table 5.2 · p. 21

5.3. Super-Resolution.

  • In super-resolution, RED-PRO yields slightly better results than RED, while TNRD outperforms NLM in both quantitative and visual comparisons.The experiments blur and downsample images by a factor of three at each axis before restoration.Source: §5.3 · pp. 20–21

6. Conclusion.

The conclusion presents RED-PRO as a fixed-point regularization framework with convergence guarantees and a connection to PnP. Deblurring and super-resolution experiments show competitive or improved performance over RED, while practical questions remain about denoiser conditions and perceptual quality.

6. Conclusion.

  • RED-PRO uses fixed-point sets of demicontractive denoisers to formulate inverse problems as convex regularized problems and derives iterative methods with convergence to a global optimum.The framework connects RED and PnP, and its relaxed variants use approximated fixed-point sets that are broader but not generally convex.Source: §6 · pp. 21–22
  • Experiments in image deblurring and super-resolution show competitive or improved performance compared with the original RED framework.The supplied table captions cover Gaussian-blur deblurring with NLM and TNRD, and super-resolution with NLM and TNRD; they do not provide cell values.Source: §6 · p. 22, Table 5.4 · p. 22, Table 5.5 · p. 23
  • Whether an arbitrary denoiser is demicontractive remains an open question, making efficient tests for that property a practical research need.Nonexpansive denoisers are demicontractive, but verifying nonexpansiveness for a general function is unclear.Source: §6 · p. 23
  • RED-PRO uses denoisers optimized for distortion measures such as PSNR, which may result in low perceptual quality.The authors identify perceptual-loss denoisers as a research direction for addressing the perception-distortion tradeoff.Source: §6 · pp. 23–24

A.1. Proof of Proposition 3.7.

The proof passage considers bounded pairs of points in [a, b]^n and then states that the proof is complete.

A.1. Proof of Proposition 3.7.

  • The proof considers any bounded pair x, z in [a, b]^n.The passage then ends by stating that this completes the proof.Source: §A.2 · p. 25, §A.1 · p. 25

A.2. Proof of Proposition 3.10.

The proof bounds the displacement under f_α using distance to a fixed point, then derives a bound over the box [a, b]^n.

A.2. Proof of Proposition 3.10.

  • For x in [a, b]^n and a fixed point z, the proof bounds the displacement ∥f_α(x) − x∥² in terms of ∥x − z∥².The stated inequality includes a factor involving α and d.Source: §A.2 · p. 25
  • The proof next bounds the distance-dependent expression by a quantity involving n, α, d, and the box width b − a.The passage then states that the proof is complete.Source: §A.2 · p. 25
  • Theorem 5 of proves convergence of Algorithm 4.1 to a point, but the supplied passage ends before identifying the set containing that point.Source: §A.3 · p. 25

A.3. Proof of Theorem 4.3.

Under the stated conditions, the sequence generated by Algorithm 4.1 is guaranteed to converge to an optimal solution of (4.2).

A.3. Proof of Theorem 4.3.

  • The cited convergence result assumes (A2), quasi-nonexpansiveness and quasi-shrinking of f_α, and paramonotonicity of ∇ℓ over Fix(f).Paramonotonicity is specified by an inner-product equality condition for u, v in Fix(f).Source: §A.3 · p. 26
  • Under (A1), f_α is continuous and strongly quasi-nonexpansive, while convexity of ℓ over Fix(f) yields paramonotonicity of ∇ℓ there.The passage uses these properties to satisfy the cited theorem's conditions.Source: §A.3 · p. 26
  • The sequence {x_k} is guaranteed to converge to an optimal solution of (4.2).The guarantee follows from the conditions of Theorem 5 in.Source: §A.3 · p. 26

A.4. Proof of Theorem 4.4.

The proof establishes that the iteration is generated by a continuous strongly quasi-nonexpansive mapping, then uses Fejér monotonicity and convergence arguments to show its limit is a fixed point satisfying the stated stationarity condition.

A.4. Proof of Theorem 4.4.

  • The composition of two strongly quasi-nonexpansive mappings with a nonempty intersection of fixed-point sets is strongly quasi-nonexpansive and has that intersection as its fixed-point set.The composition constant is given by γ = γ1γ2/(γ1+γ2).Source: §A.4 · p. 26
  • The iteration mapping T is continuous and strongly quasi-nonexpansive, with Fix(T) = Fix(f) ∩ Fix(Gℓ).The proof rewrites Algorithm 4.1 as x_k+1 = T(x_k).Source: §A.4 · p. 26
  • The iterates are Fejér monotone with respect to Fix(T) and bounded.The proof derives this from a decrease inequality for the squared distance to any x* in Fix(T).Source: §A.4 · p. 26
  • The iterates converge to a point x in Fix(f) with ∇ℓ(x) = 0.The proof shows the limit is in Fix(T), which implies both stated properties.Source: §A.4 · p. 26

A.5. Proof of .

The passage sketches an update-rule rearrangement leading to (4.13) and states a strong quasi-nonexpansiveness property for the averaged operator fα.

A.5. Proof of .

  • The update rule is rewritten in two stages, then expressed with respect to vj before interchanging xj and vj to obtain (4.13).The supplied passage does not state the intermediate formulas.Source: §A.5 · pp. 26–27
  • The averaged operator fα is γ-strongly quasi-nonexpansive for γ > 1.The passage invokes Proposition 3.7 and then Proposition 3.3 to obtain a bound on ∥x − fα(x)∥.Source: §A.6 · p. 27

A.6. Proof of Theorem 4.6.

The proof concludes that x belongs to the approximate fixed-point set Fixε(f).

A.6. Proof of Theorem 4.6.

  • The preceding bound implies that x ∈ Fixε(f).The supplied passage states this implication but does not include the bound itself.Source: §A.6 · p. 27

Appendix B. RED Revisited.

Appendix B revisits RED by expressing its objective through a Rockafellar function, including for nondifferentiable denoisers. It identifies cyclic firm nonexpansiveness as a sufficient condition for convexity and convergence of the RED algorithms.

Appendix B. RED Revisited.

  • The appendix reformulates RED using the Rockafellar function to study the case of a nondifferentiable denoiser.It introduces the denoiser residual A = Id − f and the objective ℓ(x; y) + λRA(x; u).Source: Appendix B · pp. 27–28
  • The appendix presents cyclic firm nonexpansiveness as a sufficient condition for the residual mapping A = Id − f to be maximal cyclically monotone.This connection is used to establish convergence of RED algorithms.Source: Appendix B · p. 27
  • For a convex, lower-semicontinuous, proper h whose gradient is A, h is determined by A up to an additive constant.The appendix also states that the Rockafellar function is convex, lower-semicontinuous, and proper when A is maximal cyclically monotone.Source: Appendix B · p. 28
  • When f is cyclically firmly nonexpansive, problem (B.1) is convex and its objective gradient is given by (2.10).Under this condition, the residual A = Id − f is maximal cyclically monotone and the Rockafellar function has gradient x − f(x).Source: Appendix B · p. 28
  • The stated condition ensures that RED algorithms converge, even when the denoiser is not differentiable.The required maximal cyclic monotonicity of A is necessary; cyclic firm nonexpansiveness of f is a sufficient condition that may be relaxed and is satisfied by proximal operators.Source: Appendix B · p. 28
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