Source-linked AI summary
Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients
Sylvain Durand, Jalal Fadili, Mila Nikolova
TL;DR
The paper addresses multiplicative noise removal, for which standard filtering works well mainly under moderate or weak noise. It develops an efficient algorithm with a specialized minimization scheme, and reports encouraging numerical results, proved consistency, and convergence.
Problem
Images representing an unknown image are corrupted by multiplicative noise that severely degrades image quality.
Method
The method combines a specialized hybrid criterion with a Douglas/Peaceman-Rachford splitting algorithm for minimization.
Results
The obtained numerical results outperform the most recent methods, while the algorithm's consistency and convergence are proved.
Takeaways & Limitations
The work proposes an original, efficient, and fast method for multiplicative noise removal with encouraging numerical performance.
Takeaways & Limitations
Filtering methods work well basically when the noise is moderate or weak, i.e. when K is large, while visible outliers raise questions about PSNR relevance.
Abstract
from arXiv · showhide
We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods.
1 Introduction
Multiplicative noise degrades active-imaging data, motivating methods that transform the problem into log space and combine coefficient shrinkage with variational restoration. The proposed approach uses thresholded frame coefficients, an ℓ1–TV criterion, Douglas–Rachford minimization, and bias correction before returning to the image domain.
- Multiplicative noise can severely degrade images, while adaptive filtering methods work mainly for moderate or weak noise when K is large.
- The observation model uses multiple measurements Sk = S0 ηk + nk, combining multiplicative noise ηk with additive noise nk, usually white Gaussian.
- Taking logarithms converts multiplicative noise into additive noise through v = log S = log S0 + log η = u0 + n.
- The restored log-image is exponentiated to obtain the image, with a bias correction; minimization uses a Douglas–Rachford splitting scheme specially adapted to the criterion.
- The method applies hard-thresholding to log-data frame coefficients, then minimizes an ℓ1 fitting term to those coefficients plus Total Variation regularization in the image domain.
2 Restoration of the log-image
The paper reviews shrinkage and variational restoration of the log-image, then combines under-thresholded frame coefficients with an edge-preserving image-domain regularizer. The resulting criterion has a minimizer, although the minimizer need not be unique.
- Drawbacks of shrinkage restoration and variational methods: Shrinkage methods can create Gibbs-like oscillations, remove texture, erode spiky features, and leave outliers when noisy coefficients survive thresholding.These artifacts arise because small coefficients may carry fine-detail information, while retained noisy coefficients generate frame-shaped artifacts.
- Drawbacks of shrinkage restoration and variational methods: Variational methods balance data fidelity against regularity, but quadratic regularization smooths images and TV can remove textures through stair-casing.The discussion also notes that edge-preserving potentials may underestimate edge amplitudes and erode spiky regions.
- A specialized hybrid criterion: Direct hard-thresholding is unsatisfactory because it combines frame-shaped artifacts with loss of fine-detail information, regardless of the threshold choice.The displayed thresholded log-image restorations are described as unsatisfactory for all tested choices of T.
- A specialized hybrid criterion: The proposed hybrid method restores under-thresholded frame coefficients by combining coefficient-domain data fitting with an edge-preserving regularization of the log-image.The preprocessing keeps selected higher-SNR coefficients intact and hard-thresholds the others before the hybrid restoration.
- A specialized hybrid criterion: The specialized criterion has a minimizer for the stated frame setting, while the minimizer is not necessarily unique.The paper further states that minimizers may produce very similar images sharing the same level lines.
3 Minimization for the log-image
The log-image minimization uses an ℓ1 fidelity term on thresholded frame coefficients and TV regularization, solved with specialized splitting methods. The Douglas-Rachford iterations converge weakly to a minimizer under stated stepsize conditions.
- Criterion: The criterion combines weighted ℓ1 fidelity to thresholded coefficients with TV regularization after synthesis into the image domain.The functionals are Ψ(x) = ∥Λ(x − yTH)∥1 and Φ(x) = ∥f_Wx∥TV.
- Splitting method: Douglas-Rachford splitting separates the resolvents of the two convex terms instead of evaluating their combined resolvent directly.The separate resolvents are easier to evaluate for the proposed functionals.
- Splitting method: Forward-backward splitting is not adapted to the functional because its usual differentiability and stepsize restrictions are not met.The method typically requires one differentiable term with Lipschitz-continuous gradient and stepsizes constrained by its continuity modulus.
- Discrete implementation: The discrete TV proximal computation uses a projected gradient, or forward-backward, iteration that converges weakly to a minimizer under a stepsize bound.The bound is expressed as sup_t β_t < 2/δ^2.
- Discrete implementation: The proposed TV projection algorithm is new relative to Chambolle’s projection algorithm and permits a sharper upper bound through forward-backward splitting.The comparison concerns the projection algorithm used for the TV-norm.
4 Bias correction to recover the sought-after image
The estimator obtained from the log-image is biased, and exponentiation propagates that bias into the restored image. A variance-based correction is used to obtain nearly unbiased estimates of both the log-image and the original image.
- Bias problem: The Douglas-Rachford log-image estimator satisfies E[û] = u0 − b_û, so it is prone to bias.The section identifies correction of this bias as necessary before inverse transformation.
- Bias problem: Exponentiation makes the bias consequential because E[exp û] depends on both the estimator bias and its variance.A Taylor expansion gives E[exp û] = S0 exp(−b_û)(1 + Var[û]/2 + R2).
- Bias correction: For nearly symmetric û, the correction b_û ≈ log(1 + Var[û]/2) provides nearly unbiased log-image estimates.The variance can be estimated from ψ1(K).
- Bias correction: The corresponding restored-image estimate is exp(û)(1 + Var[û]/2), preserving the mean of the original image approximately.The correction is presented as a finite-sample, nearly unbiased estimate of S0.
- Bias correction: The proposed variance-based correction is equivalent to an alternative correction based on the log-noise mean when K is reasonably large.The two correction factors are reported to be very close for reasonably large K.
5 Full algorithm to suppress multiplicative noise
The full denoising algorithm thresholds curvelet coefficients of the log-image, solves the specialized variational problem, and applies the inverse curvelet transform with bias correction. Its transform operations use fast implicit implementations.
- Algorithm: The method then uses Douglas-Rachford outer iterations and forward-backward inner iterations with specified relaxation, projection, and regularization parameters.The parameters include NDR, NFB, µ_t, β_t, and γ.
- Algorithm: The algorithm first computes v = log S, transforms it with W, and hard-thresholds the coefficients to obtain yTH.The threshold is applied to all coefficients except those in the designated index set I*.
- Algorithm: The reconstructed log-image is obtained through inverse curvelet synthesis after the optimization stage.The inverse transform is written as u(t) = f_Wx(t).
- Computational cost: The bulk of computation applies the frame transform and its pseudoinverse as fast implicit analysis and synthesis operators.For a second-generation curvelet transform, each application costs O(N log N).
- Computational cost: With NDR outer iterations and NFB forward-backward subiterations, the denoising complexity is of order NDRNFB2N log N.The stated complexity combines the iteration counts with the curvelet-transform cost.
6 Experiments
Experiments on three 256 × 256 grayscale images under multiplicative noise show that the proposed denoiser outperforms competing methods visually and quantitatively. Its PSNR improvement reaches up to 4 dB on Shepp-Logan and approximately 1 dB on Lena and Boat.
- Compared methods: The comparison included AA, Stein-Block curvelet shrinkage, and a naive method, with competitors tuned for best performance on each noisy realization.Stein-Block thresholds curvelet coefficients by blocks rather than individually.
- Metric caveat: Although some persistent outliers remain visible, the experiments also question whether PSNR or MAE adequately measures perceptual restoration quality.This caveat concerns the relationship between numerical metrics and visual quality.
- Experimental setup: The algorithms were tested on Shepp-Logan phantom, Lena, and Boat images, each sized 256 × 256 and corrupted with multiplicative noise using µ = 1 and K = 10.Performance was measured using PSNR and MAE.
- Overall results: The proposed denoiser clearly outperformed its competitors both visually and quantitatively according to PSNR and MAE.The results are reported across Figures 3–5.
- Quantitative results: Up to 4 dB PSNR improvement occurred on the Shepp-Logan phantom, while the improvement was approximately 1 dB for Lena and Boat.The reported comparison concerns the competing methods evaluated in the experiments.
- Qualitative comparison: The AA algorithm systematically tends to lose important details and retain a low-frequency ghost visible in the error maps for Lena and Boat.These artifacts are identified in the third-row error maps of Figures 4 and 5.
7 Conclusions
The paper proposes a fast multiplicative-noise removal method that combines log-domain preprocessing, frame-coefficient thresholding, an ℓ1-plus-TV restoration criterion, and bias-corrected exponentiation. Its consistency and convergence are proved, and numerical results outperform recent methods.
- Problem: The work addresses multiplicative noise removal, a difficult problem arising in applications including laser, ultrasound, and SAR imaging.Multiplicative noise introduces inherent difficulties that restrict major restoration algorithms.
- Method: The method restores a log-image by hard-thresholding tight-frame coefficients, then uses ℓ1 coefficient fitting with TV regularization in the log-image domain.The final image is obtained through an exponential transform with bias correction.
- Method: The sought image is recovered by exponentiating the optimized log-image and applying a pertinent bias correction.This completes the transition from the restored log-image to the image domain.
- Results: The resultant algorithm is fast, with consistency and convergence proved, and its numerical results outperform the most recent methods.The conclusion characterizes the numerical results as encouraging.