A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration

@article{Chan1999ANP,
  title={A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration},
  author={Tony F. Chan and Gene H. Golub and Pep Mulet},
  journal={SIAM J. Sci. Comput.},
  year={1999},
  volume={20},
  pages={1964-1977}
}
We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $|\nabla u|$ in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to… 
AN INFEASIBLE PRIMAL-DUAL ALGORITHM FOR TV-BASED INF-CONVOLUTION-TYPE IMAGE RESTORATION
In this paper, a primal-dual algorithm for TV-type image restoration is analyzed and tested. Analytically it turns out that employing a global L s-regularization, with s > 1, in the dual problem
On the Convergence of Primal–Dual Hybrid Gradient Algorithms for Total Variation Image Restoration
TLDR
The convergence of a general primal–dual method for nonsmooth convex optimization problems whose structure is typical in the imaging framework, as, for example, in the Total Variation image restoration problems, is established.
An Infeasible Primal-Dual Algorithm for Total Bounded Variation-Based Inf-Convolution-Type Image Restoration
TLDR
The globalized primal-dual algorithm introduced in this paper works with generalized derivatives, converges locally at a superlinear rate, and is stable with respect to noise in the data.
An alternating extragradient method for total variation-based image restoration from Poisson data
Variational models are a valid tool for edge-preserving image restoration from data affected by Poisson noise. This paper deals with total variation and hypersurface regularization in combination
A conditional gradient method for primal-dual total variation-based image denoising
where Ω is the domain of the image. It is assumed to be a convex, connected, and bounded Lipschitz open subset of R. One of the most popular regularization methods is Tikhonov regularization based on
A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science
TLDR
This work generalizes the primal-dual hybrid gradient (PDHG) algorithm to a broader class of convex optimization problems, and surveys several closely related methods and explains the connections to PDHG.
Parameter selection for total-variation-based image restoration using discrepancy principle
  • Y. Wen, R. Chan
  • Mathematics
    IEEE Transactions on Image Processing
  • 2012
TLDR
This paper derives a fast algorithm that simultaneously estimates the regularization parameter and restores the image and shows that it is better than some state-of-the-art methods in terms of both speed and accuracy.
Primal dual algorithms for convex models and applications to image restoration, registration and nonlocal inpainting
The main subject of this dissertation is a class of practical algorithms for minimizing convex non-differentiable functionals coming from image processing problems defined as variational models. This
Convergence analysis of primal-dual algorithms for total variation image restoration
Recently, some attractive primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation (TV) image restoration. This paper
An Efficient Primal-Dual Method for L1TV Image Restoration
TLDR
The paper ends with a report on restoration results obtained by the new algorithm for salt-and-pepper or random-valued impulse noise including blurring and a comparison with other methods is provided.
...
...

References

SHOWING 1-10 OF 42 REFERENCES
An Efficient Primal-Dual Interior-Point Method for Minimizing a Sum of Euclidean Norms
TLDR
A primal-dual interior-point algorithm for the problem of minimizing a sum of Euclidean norms is derived, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition.
Continuation method for total variation denoising problems
The denoising problem can be solved by posing it as a constrained minimization problem. The objective function is the TV norm of the denoised image whereas the constraint is the requirement that the
An Affine Scaling Algorithm for Minimizing Total Variation in Image Enhancement
TLDR
The resulting computational scheme, when viewed as an image enhancement process, has the feature that it can be used in an interactive manner in situations where knowledge of the noise level is either unavailable or unreliable.
A Multigrid Method for Total Variation-Based Image Denoising
Byimage reconstruction we mean obtaining the solution u of an operator equation of the form $$Au = z,$$ (1.1) where A is typically (but not necessarily) linear, and the dataz is assumed to
Fast Multigrid Techniques in Total Variation-Based Image Reconstruction
Existing multigrid techniques are used to effect an efficient method for reconstructing an image from noisy, blurred data. Total Variation minimization yields a nonlinear integro-differential
Recovery of Blocky Images from Noisy and Blurred Data
TLDR
The purpose of this investigation is to understand situations under which an enhancement method succeeds in recovering an image from data which are noisy and blurred, and selects one that has the least total variation.
Minimal surfaces and functions of bounded variation
I: Parametric Minimal Surfaces.- 1. Functions of Bounded Variation and Caccioppoli Sets.- 2. Traces of BV Functions.- 3. The Reduced Boundary.- 4. Regularity of the Reduced Boundary.- 5. Some
Scale-Space and Edge Detection Using Anisotropic Diffusion
TLDR
A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.
Image selective smoothing and edge detection by nonlinear diffusion. II
A new version of the Perona and Malik theory for edge detection and image restoration is proposed. This new version keeps all the improvements of the original model and avoids its drawbacks: it is
...
...