PaperlayerView exact revision on arXiv

Source-linked AI summary

Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization

José M. Bioucas-Dias, Mário A. T. Figueiredo

arXiv:0912.1845v2math.OCmath.NA

TL;DR

Multiplicative speckle noise complicates reflectance estimation because it is non-additive and non-Gaussian, limiting direct use of methods designed for quadratic Gaussian data terms. The paper applies logarithmic transformation, variable splitting, and augmented-Lagrangian optimization to build MIDAL. Experiments report state-of-the-art denoising performance and speed.

  • Problem

    Multiplicative noise is multiplied by the image and commonly follows non-Gaussian Rayleigh or Gamma densities, unlike the additive Gaussian setting targeted by many optimization algorithms.

  • Method

    MIDAL takes logarithms to obtain an additive formulation, transforms the optimization through variable splitting, and solves the constrained problem with ADMM.

  • Results

    MIDAL achieves lower estimation errors than in experiments 1–7 and lower MAE than in experiments 8–16, while being faster than by a factor larger than 3.

  • Takeaways & Limitations

    The proposed formulation has a unique denoised solution and guaranteed convergence under the stated convexity and ADMM conditions.

Abstract

from arXiv · show

Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of difficulties with respect to the standard Gaussian additive noise scenario: (1) the noise is multiplied by (rather than added to) the original image; (2) the noise is not Gaussian, with Rayleigh and Gamma being commonly used densities. These two features of multiplicative noise models preclude the direct application of most state-of-the-art algorithms, which are designed for solving unconstrained optimization problems where the objective has two terms: a quadratic data term (log-likelihood), reflecting the additive and Gaussian nature of the noise, plus a convex (possibly nonsmooth) regularizer (e.g., a total variation or wavelet-based regularizer/prior). In this paper, we address these difficulties by: (1) converting the multiplicative model into an additive one by taking logarithms, as proposed by some other authors; (2) using variable splitting to obtain an equivalent constrained problem; and (3) dealing with this optimization problem using the augmented Lagrangian framework. A set of experiments shows that the proposed method, which we name MIDAL (multiplicative image denoising by augmented Lagrangian), yields state-of-the-art results both in terms of speed and denoising performance.

I. INTRODUCTION

Coherent imaging produces multiplicative speckle noise, making reflectance estimation a regularized inverse problem whose data fidelity can be difficult to optimize. Logarithmic formulations yield convex data terms, while existing TV-based approaches still face nonsmooth or non-Lipschitz optimization structure.

  • A. Coherent Imaging and Multiplicative Noise: Coherent imaging systems such as ultrasound, SAR, sonar, and laser imaging are better modeled by multiplicative than additive Gaussian noise.Speckle arises from random interference among scatterers in each resolution cell.
  • A. Coherent Imaging and Multiplicative Noise: Fully developed speckle produces granular intensity images with SNR equal to one, or 0 dB.The result follows from the exponential intensity distribution's equal mean and standard deviation.
  • A. Coherent Imaging and Multiplicative Noise: M-look averaging produces Gamma-distributed observations and improves the SNR to M, at the cost of spatial-resolution tradeoffs when spatial filtering is used.Spatial averaging estimates reflectance under a local-constancy assumption.
  • C. Estimation of Reflectance: Variational Approaches: Multiplicative noise removal estimates the underlying reflectance, typically using piecewise-smooth assumptions formalized through Bayesian or total-variation regularization.Variational methods produce piecewise-smooth estimates adapted to reflectance structure.
  • C. Estimation of Reflectance: Variational Approaches: For M-look models, formulating the problem in logarithmic reflectance yields convex data fidelity, unlike the directly formulated non-convex likelihood.TV-regularized objectives remain non-quadratic and nonsmooth, and may lack a Lipschitz-continuous gradient required by guaranteed-convergent forward-backward methods.

D. Proposed Approach

The paper addresses a convex logarithmic-reflectance estimation problem by combining variable splitting with augmented-Lagrangian optimization. The resulting ADMM algorithm is MIDAL, which has a unique solution and converges under the stated conditions while achieving strong experimental performance.

  • D. Proposed Approach: The method targets the convex optimization problem obtained from the M-look multiplicative model expressed in logarithmic reflectance.This formulation is adopted for reflectance estimation with total-variation regularization.
  • D. Proposed Approach: Variable splitting transforms the original unconstrained problem into an equivalent constrained problem, which is then solved with an augmented Lagrangian method.The paper specifically uses ADMM within the augmented-Lagrangian family.
  • D. Proposed Approach: The paper elaborates earlier variable-splitting work that handled the constrained problem through split-Bregman methods, while this approach uses augmented Lagrangian optimization.The paper reports that ADMM convergence conditions are satisfied.
  • D. Proposed Approach: The multiplicative observation model is converted into an additive model by taking logarithms, with MAP estimation represented as an unconstrained objective combining likelihood and regularization.The regularizer is total variation, with first-order horizontal and vertical image differences.
  • D. Proposed Approach: The logarithmic formulation has a unique minimizer because its likelihood terms are strictly convex and coercive and the TV regularizer is convex.The corresponding formulation in the original reflectance variables is non-convex and does not generally guarantee uniqueness.

A. Variable Splitting

Variable splitting introduces an auxiliary variable for a composed term and imposes an equality constraint, preserving equivalence while enabling simpler alternating subproblems. Quadratic-penalty implementations can become ill-conditioned as the penalty grows, motivating augmented-Lagrangian alternatives.

  • A. Variable Splitting: Variable splitting introduces a new variable v as the argument of f2 and constrains it to equal g(u).The resulting constrained problem is equivalent to the original unconstrained problem on its feasible set.
  • A. Variable Splitting: The reformulation is useful because the constrained problem may be easier to solve than the original unconstrained problem.Alternating minimization can separate otherwise difficult components.
  • A. Variable Splitting: Quadratic-penalty methods alternate over u and v while increasing µ so the penalized solution approaches the equivalent constrained problem.This continuation process forces constraint satisfaction asymptotically.
  • A. Variable Splitting: As µ becomes very large, intermediate minimization problems become increasingly ill-conditioned and can cause numerical problems.This is a practical drawback of the quadratic-penalty approach.
  • A. Variable Splitting: For linear constraints, split-Bregman iterations are equivalent to augmented-Lagrangian methods, linking the alternative formulation to the paper's optimization perspective.The paper subsequently uses ADMM to solve the split constrained problem.

B. Augmented Lagrangian

The augmented Lagrangian method solves equality-constrained optimization by alternating minimization of an augmented objective and multiplier updates. Its alternative form and convergence properties motivate its use in the paper’s algorithms.

  • The augmented Lagrangian adds a quadratic penalty for constraint violations to the original objective and includes Lagrange multipliers.
  • ALM minimizes the augmented Lagrangian with respect to the primal variable while holding multipliers fixed, then updates the multipliers.
  • ALM does not require the penalty parameter µ to approach infinity to converge to the constrained problem’s solution.
  • Completing the square produces an equivalent ALM form with a single quadratic term, enabling the version used later.
  • The ALM formulation is equivalent to the Bregman iterative method and relates to proximal point methods applied to the dual problem.
  • The ALM sequence converges in the dual variables, while cluster points of the primal sequence solve the constrained problem.

C. Augmented Lagrangian for Variable Splitting

Applying ALM to a split convex problem yields alternating updates for two variables and a scaled dual variable. Under standard convexity and rank assumptions, approximate subproblem solutions can still preserve convergence.

  • ALM is applied when the constrained problem has an identity linear coupling between the split variables.
  • The u,v subproblem can be difficult because its quadratic terms are generally non-separable and potentially nonsmooth.
  • The ADMM iteration alternates minimization over u and v, followed by an update of the scaled dual variable d.
  • Convergence is guaranteed for closed, proper, convex functions when G has full column rank, including the MIDAL case G = I.
  • If the constrained problem has a solution, the sequence {u_k} converges to a solution; otherwise, at least one of {v_k} or {d_k} diverges.
  • Exact minimization of the ADMM subproblems is unnecessary when the optimization-error sequences are absolutely summable.

IV. PROPOSED APPROACH

The proposed approach splits the logarithmic image variable into decoupled data and TV-regularization subproblems, then applies ADMM to obtain MIDAL. Its updates use scalar Newton solves and an inexact TV proximity computation while retaining convergence guarantees.

  • Variable splitting converts the original image variable z into paired variables (z,u) whose objective terms are decoupled.
  • ADMM applied to the split constrained problem produces the proposed multiplicative image denoising by augmented Lagrangian algorithm, MIDAL.
  • The z-update consists of n independent scalar convex minimizations, solved efficiently with a few Newton iterations instead of the Lambert W closed form.
  • The u-update is an ℓ2-TV denoising problem with regularization parameter λ/µ, equivalently a Moreau proximity-operator evaluation.
  • The TV proximity computation may be inexact, provided the Euclidean norms of its errors are summable across ADMM iterations.
  • MIDAL convergence is guaranteed when the z- and u-subproblem errors are absolutely summable because the relevant functions are closed, proper, convex and G = I has full column rank.

V. COMMENTS ON RELATED WORK

The paper contrasts MIDAL with alternative splitting, penalty, flow, and nonconvex approaches. Its discussion emphasizes simpler subproblems, avoidance of ill-conditioning, and differences in convergence or initialization behavior.

  • The proposed splitting converts a nonquadratic data term plus TV regularization into a decoupled minimization and a quadratic-data TV denoising problem.
  • The alternative splitting in creates a non-separable, nonquadratic intermediate problem requiring an inner Douglas-Rachford iteration.
  • The splitting-and-penalty method approaches the target minimizers only as α approaches infinity, but large α causes severe ill-conditioning and numerical difficulty.
  • The splitting-and-penalty method therefore uses moderate α in practice, producing solutions that are not minima of the regularized negative log-likelihood.
  • The nonconvex formulation discussed from lacks guaranteed uniqueness, and its estimate depends critically on initialization.
  • The Douglas-Rachford perspective has been applied to multiplicative-noise restoration, but not to the Gamma distribution considered here.

VI. EXPERIMENTS

The experiments evaluate MIDAL on synthetic multiplicative-noise data across 16 setups, comparing it with recent state-of-the-art methods. Regularization-parameter selection is deliberately outside the paper’s scope.

  • Experimental setup: The evaluation uses 16 synthetic experimental setups spanning original images, image sizes, pixel ranges, SNR-related M values, and regularization parameters.Experiments 1–7 reproduce setups from and, while experiments 8–16 follow.
  • Experimental setup: The experiments use synthetic observations generated from the multiplicative-noise model, with most observed images having very low SNR values.The low-SNR regime is described as usual in applications involving multiplicative noise.
  • Experimental setup: The reported implementation uses MATLAB 7.5 on a 3.0GHz Intel Core2Extreme PC with 4Gb of RAM, with m = 4 in experiments 1–7 and m = 2 in experiments 8–16.Relative timing comparisons therefore depend on the specified hardware and implementation settings.
  • Experimental setup: MIDAL’s regularization parameter λ is selected using the lowest mean squared error against the true image, rather than automatically.Automatic selection of λ is explicitly out of scope because the paper focuses mainly on algorithm speed.
  • Experimental setup: The experiments use µ = λ for the augmented-Lagrangian penalty parameter because this setting was experimentally found to give good convergence results.MIDAL is guaranteed to converge for any µ > 0 when Theorem 1’s conditions hold, but µ affects convergence speed.

A. Computing the TV Proximity Operator

MIDAL computes the TV proximity operator with Chambolle’s fixed-point algorithm and reuses dual variables from the previous run. The experiments show improved errors and speed relative to the compared methods, while objective and constraint measures decrease during iterations.

  • A. Computing the TV Proximity Operator: Initializing Chambolle’s TV-proximity iterations with dual variables from the previous run is intended to accelerate convergence as successive inputs become closer.The alternative is a fixed initialization, such as setting the dual variables to zero.
  • A. Computing the TV Proximity Operator: Figure 2 compares objective-function evolution when Chambolle’s dual variables are initialized from the previous iteration versus initialized to zero.The previous-iteration initialization supports the summable minimization-error condition used in the convergence analysis.
  • B. Results: In experiments 1–7, MIDAL always achieves lower estimation errors than and runs faster in every experiment by a factor larger than 3.The error gain is larger for images with lower SNR, corresponding to more difficult problems.
  • B. Results: In experiments 8–16, MIDAL achieves lower MAE in all cases and outperforms in relative error in 5 of 9 cases.Relative error ties occur in two cases, while performs slightly better in two cases.
  • B. Results: For Experiment 1, the constraint residual decreases by approximately 7 orders of magnitude over 21 MIDAL iterations, showing practical satisfaction of the constraint.Figure 7 tracks both the objective function and the constraint function during the iterations.

VII. CONCLUDING REMARKS

The paper proposes MIDAL, combining logarithmic reformulation, variable splitting, and ADMM for multiplicative-noise estimation. The resulting method has a unique solution, guaranteed convergence, and state-of-the-art estimation performance and speed in the reported experiments.

  • VII. CONCLUDING REMARKS: MIDAL transforms multiplicative-noise estimation into an equivalent constrained problem through variable splitting and solves it with an augmented Lagrangian ADMM method.The formulation targets variational or MAP estimation with total-variation regularization, while allowing other priors.
  • VII. CONCLUDING REMARKS: The experiments compare observed noisy images with image estimates across setups varying the number of looks M.Figures 3 and 4 report M values for Experiments 1–7, while Figures 5 and 6 show estimates for Experiments 8–13 after a display-only nonlinear transformation.
  • VII. CONCLUDING REMARKS: Strict convexity of the logarithmic data term combined with convex total variation makes the denoised image unique and guarantees MIDAL convergence.The original multiplicative model does not have this strict-convexity property.
  • VII. CONCLUDING REMARKS: MIDAL exhibited state-of-the-art estimation performance and speed in experiments comparing it with recent methods.The experiments used synthetic data and included 16 experimental setups; Table II reports iterations, relative error, mean absolute-deviation error, and time.
  • VII. CONCLUDING REMARKS: The reported extension plans include linear observation operators, Poissonian observations, and frame-based regularizers.The current work focuses on total-variation regularization and does not yet cover these extensions.
Loading 0912.1845v2…