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Image Restoration using Total Variation with Overlapping Group Sparsity
Jun Liu, Ting-Zhu Huang, Ivan W. Selesnick, Xiao-Guang Lv, Po-Yu Chen
TL;DR
Image restoration is ill-posed, and classical TV can produce staircase artifacts despite preserving edges. The paper proposes an overlapping group sparsity TV model with an efficient ADMM–MM algorithm. Compared with three state-of-the-art methods, it achieves the highest PSNR and averages a 0.3 ∼1 dB improvement while using less time and fewer iterations than FastTV and LLT-ALM in most cases.
Problem
Classical TV preserves edges but produces staircase artifacts, while HTV can oversmooth restored images.
Method
The paper uses an overlapping group sparsity total variation regularizer and solves its minimization problem with OGSATV-ADM4, combining ADMM with MM.
Results
0.3 ∼1 dB average PSNR improvement: the proposed method achieves the highest PSNR against three state-of-the-art methods and generally uses less time and fewer iterations than FastTV and LLT-ALM.
Takeaways & Limitations
The numerical results support avoiding staircase effects while preserving edges, with effective and efficient restoration reported for the proposed method.
Takeaways & Limitations
The method’s extension to real applications such as compressed sensing, blind deconvolution, and image enhancement remains future work.
Abstract
from arXiv · showhide
Image restoration is one of the most fundamental issues in imaging science. Total variation (TV) regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase-like artifacts. Usually, the high-order total variation (HTV) regularizer is an good option except its over-smoothing property. In this work, we study a minimization problem where the objective includes an usual $l_2$ data-fidelity term and an overlapping group sparsity total variation regularizer which can avoid staircase effect and allow edges preserving in the restored image. We also proposed a fast algorithm for solving the corresponding minimization problem and compare our method with the state-of-the-art TV based methods and HTV based method. The numerical experiments illustrate the efficiency and effectiveness of the proposed method in terms of PSNR, relative error and computing time.
I. INTRODUCTION
Image restoration is ill-posed because acquisition, transmission, and storage degrade observations, motivating regularized recovery. The paper focuses on OGS-TV to address TV staircasing while retaining edge-preserving restoration.
- Image restoration is ill-posed, so regularization is used to obtain stable and accurate approximations of the true image.
- The standard l2 regularizer is simple but oversmooths important edges, whereas TV was introduced to preserve them.
- TV can create staircase artifacts and false edges by converting smooth ramps into piecewise-constant regions.
- HTV uses higher-order differential operators to produce piecewise-linear solutions that better fit smooth intensity changes, but the paper instead studies OGS-TV.
- The proposed work extends OGS-TV from one-dimensional denoising to two-dimensional image restoration and develops an efficient optimization method.
A. OGS-TV
The OGS-TV construction applies overlapping groups to image differences, with a K × K group window defining the regularization structure.
- A two-dimensional K × K group is formed from neighboring pixels of the vectorized image.
- The overlapping group sparsity functional is defined by stacking each local K × K image block into a vector.
- The regularization functional uses group size K × K, which controls the local window used by OGS-TV.
- When K = 1, the functional becomes the commonly used anisotropic TV functional, called OGS-ATV in the paper.
B. ADMM
ADMM solves a constrained separable convex problem by alternating between subproblems for separate variable blocks and updating the multiplier.
- ADMM targets a constrained separable convex optimization problem with two convex objectives, linear transforms, convex domains, and a shared constraint.
- The augmented Lagrangian adds a multiplier term and a quadratic penalty for the linear constraint.
- ADMM alternately minimizes the augmented Lagrangian with respect to each variable while keeping the other variable and multiplier fixed.
- The algorithm updates the variables and scaled multiplier iteratively until a stopping criterion is satisfied.
- The method exploits separability by decomposing the augmented-Lagrangian subproblem into two alternating subproblems.
C. MM
MM replaces a difficult optimization with a sequence of easier surrogate problems that majorize the original objective. Here it is used to construct an efficient OGS-TV subproblem solver.
- MM minimizes a sequence of easier surrogate functionals instead of directly minimizing the difficult cost functional.
- Each majorizer lies above the objective and touches it at the current iterate; for convex objectives, the iterates converge under mild conditions.
- A useful majorizer can avoid large matrix inversions, linearize optimization, separate parameters, handle constraints, or smooth nondifferentiability.
- For the proposed subproblem, MM majorizes the OGS-TV functional while retaining the simple quadratic data term.
- The resulting matrix inversion can be computed component-wise, yielding an efficient iterative algorithm.
III. PROPOSED ALGORITHM
The proposed algorithm reformulates the constrained image-restoration problem with auxiliary variables and solves the resulting subproblems through ADMM, MM, projection, and FFT-based linear algebra. Its convergence is retained under absolutely summable errors in the inexact inner updates.
- Model and splitting: The model imposes a box constraint so restored 8-bit image pixels remain in the dynamic range [0, 255].The constraint is implemented through an orthogonal projection onto the feasible interval.
- Model and splitting: Auxiliary variables vx, vy, and z convert the original constrained minimization into an equivalent ADMM-compatible problem.The variables decouple parts of the optimization into separately solvable subproblems.
- Subproblem solution: The f-subproblem is a least-squares problem whose normal equation is efficiently computed with FFTs under periodic boundary conditions.The relevant operators form block circulant matrices with circulant blocks.
- Algorithm: OGSATV-ADM4 alternates updates of f, vx, vy, z, and the auxiliary multipliers until a stopping criterion is satisfied.The algorithm initializes the primal variables at g and performs the listed updates iteratively.
- Convergence: Convergence is guaranteed for exact subproblem minimization and remains uncompromised for inexact vx and vy updates when their successive errors are absolutely summable.The paper identifies OGSATV-ADM4 as an instance of ADMM under exact updates.
IV. NUMERICAL RESULTS
The numerical section evaluates OGSATV-ADM4 on image-restoration problems using standard quality metrics and examines parameter-selection considerations. Experiments span test images of varying sizes and use a common stopping criterion for comparisons.
- Experimental setup: The experiments use test images ranging from 256 × 256 to 1024 × 1024.The experiments were conducted in Matlab on a Windows 7 32-bit desktop with an Intel Core i3-2130 CPU and 4 GB of RAM.
- Evaluation metrics: Restoration quality is measured with relative error and PSNR, while BSNR describes the noise added to blurred images.Higher PSNR indicates higher image quality.
- Parameter studies: The study examines group window size K, inner iterations N, and penalty parameter σ before comparing restoration performance.The authors report using the empirical rule σ = λ/3 in all experiments.
A. Study on some parameters
Parameter studies examine how group size and inner MM iterations affect OGSATV-ADM4. The experiments select K = 3 and N = 5 as practical settings based on restoration quality and computational cost.
- Group size K: K = 3 gives satisfying restoration results, so the authors empirically choose K = 3 for subsequent experiments.The tested group-size range is K = 1 to 21; K = 1 corresponds to classic anisotropic TV.
- Inner iterations N: Except for N = 1, different inner-iteration counts produce almost the same PSNR and RelErr but substantially different computational times.Larger N increases computational cost.
- Inner iterations N: A small number of MM iterations is sufficient to achieve satisfying results, leading the authors to set N = 5.The paper reports this choice after testing N = 1, 5, 20, 200, and 1000.
B. Comparison with other state-of-the-art methods
The proposed OGSATV-ADM4 is evaluated against TV-, HTV-, and related restoration methods for denoising and deblurring. It generally achieves strong image quality while reducing staircase artifacts and preserving edges, though some competitors are faster in particular settings.
- Comparison setup: OGSATV-ADM4 is compared with state-of-the-art methods for image denoising and deblurring under a common stopping criterion.The comparisons use the same stopping criterion with ϵ = 1 × 10^-5.
- Image denoising: OGSATV-ADM4 reaches the lowest converged RelErr in denoising within reasonable time, while Split Bregman converges fastest and uses the least CPU time.Chambolle’s method is reported as slowest in both iterations and elapsed time.
- Image denoising: TV methods show staircase artifacts in denoising, whereas LLT-ALM and OGSATV-ADM4 reduce them; OGSATV-ADM4 also avoids LLT-ALM’s local over-smoothing.The comparison covers Boats and Man image fragments corrupted with Gaussian noise at δ = 30.
- Image deblurring: OGSATV-ADM4 obtains the highest PSNR among the four deblurring methods and improves PSNR by 0.3 ∼ 1 dB on average.It generally needs less computing time and fewer iterations than FastTV and LLT-ALM, except for the Jellyfish case.
- Image deblurring: In deblurring examples, OGSATV-ADM4 restores edges without staircase artifacts and avoids the over-smoothing observed for LLT-ALM.The reported examples use Gaussian and average blur with BSNR = 40.
V. CONCLUSION
The paper proposes OGSATV-ADM4 for image restoration with an overlapping group sparsity total variation regularizer, combining ADMM and MM to solve the minimization problem. Numerical comparisons report effectiveness and efficiency, with reduced staircase artifacts and preserved edges, while extensions to several real applications remain future work.
- Method: OGSATV-ADM4 combines classic ADMM with MM to efficiently solve the minimization problem using the overlapping group sparsity total variation regularizer.The algorithm is presented as the paper’s solution method for the corresponding image-restoration problem.
- Results: Numerical comparisons with state-of-the-art methods show that the proposed method is effective and efficient.The paper reports comparisons with multiple existing methods rather than a single baseline.
- Results: The results indicate that the proposed method avoids staircase effects while preserving edges in restored images.These are the two principal restoration qualities highlighted in the conclusion.
- Scope: The authors identify compressed sensing, blind deconvolution, and image enhancement as future real-application extensions.These applications are described as ongoing work rather than demonstrated results of the current study.