Edge-preserving Image Restoration Research Articles

Combined total variation of first and fractional orders for Poisson noise removal in digital images

Introduction: Many methods have been proposed to handle the image restoration problem with Poisson noise. A popular approach to Poissonian image reconstruction is the one based on Total Variation. This method can provide significantly sharp edges and visually fine images, but it results in piecewise-constant regions in the resulting images. Purpose: Developing an adaptive total variation-based model for the reconstruction of images contaminated by Poisson noise, and an algorithm for solving the optimization problem. Results: We proposed an effective way to restore images degraded by Poisson noise. Using the Bayesian framework, we proposed an adaptive model based on a combination of first-order total variation and fractional order total variation. The first-order total variation model is efficient for suppressing the noise and preserving the keen edges simultaneously. However, the first-order total variation method usually causes artifact problems in the obtained results. To avoid this drawback, we can use high-order total variation models, one of which is the fractional-order total variation-based model for image restoration. In the fractional-order total variation model, the derivatives have an order greater than or equal to one. It leads to the convenience of computation with a compact discrete form. However, methods based on the fractional-order total variation may cause image blurring. Thus, the proposed model incorporates the advantages of two total variation regularization models, having a significant effect on the edge-preserving image restoration. In order to solve the considered optimization problem, the Split Bregman method is used. Experimental results are provided, demonstrating the effectiveness of the proposed method. Practical relevance: The proposed method allows you to restore Poissonian images preserving their edges. The presented numerical simulation demonstrates the competitive performance of the model proposed for image reconstruction. Discussion: From the experimental results, we can see that the proposed algorithm is effective in suppressing noise and preserving the image edges. However, the weighted parameters in the proposed model were not automatically selected at each iteration of the proposed algorithm. This requires additional research.

Minimizing Sum of Truncated Convex Functions and Its Applications

ABSTRACTIn this article, we study a class of problems where the sum of truncated convex functions is minimized. In statistical applications, they are commonly encountered when ℓ0-penalized models are fitted and usually lead to NP-Hard non-convex optimization problems. In this article, we propose a general algorithm for the global minimizer in low-dimensional settings. We also extend the algorithm to high-dimensional settings, where an approximate solution can be found efficiently. We introduce several applications where the sum of truncated convex functions is used, compare our proposed algorithm with other existing algorithms in simulation studies, and show its utility in edge-preserving image restoration on real data.

An Augmented Lagrangian Method for Non-Lipschitz Nonconvex Programming

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration, and signal processing can be modelled in this form. First, we propose the concept of the Karush--Kuhn--Tucker (KKT) stationary condition for the non-Lipschitz problem and show that it is necessary for optimality under a constraint qualification called the relaxed constant positive linear dependence (RCPLD) condition, which is weaker than the Mangasarian--Fromovitz constraint qualification and holds automatically if all the constraint functions are affine. Then we propose an augmented Lagrangian (AL) method in which the augmented Lagrangian subproblems are solved by a nonmonotone proximal gradient method. Under the assumption that a feasible point is known, we show that any accumulation point of the sequence generated by our method must be a feasible point. Moreover, if RCPL...

Second Order Differences of Cyclic Data and Applications in Variational Denoising

In many image and signal processing applications, as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or color image restoration in HSV or LCh spaces the data has its range on the one-dimensional sphere $\mathbb S^1$. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for $\mathbb S^1$-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances the absolute cyclic second order differences have only values in [0,{\pi}]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm.

Accelerated Edge-Preserving Image Restoration Without Boundary Artifacts

To reduce blur in noisy images, regularized image restoration methods have been proposed that use nonquadratic regularizers (like l1 regularization or total-variation) that suppress noise while preserving edges in the image. Most of these methods assume a circulant blur (periodic convolution with a blurring kernel) that can lead to wraparound artifacts along the boundaries of the image due to the implied periodicity of the circulant model. Using a noncirculant model could prevent these artifacts at the cost of increased computational complexity. In this paper, we propose to use a circulant blur model combined with a masking operator that prevents wraparound artifacts. The resulting model is noncirculant, so we propose an efficient algorithm using variable splitting and augmented Lagrangian (AL) strategies. Our variable splitting scheme, when combined with the AL framework and alternating minimization, leads to simple linear systems that can be solved noniteratively using fast Fourier transforms (FFTs), eliminating the need for more expensive conjugate gradient-type solvers. The proposed method can also efficiently tackle a variety of convex regularizers, including edge-preserving (e.g., total-variation) and sparsity promoting (e.g., l1-norm) regularizers. Simulation results show fast convergence of the proposed method, along with improved image quality at the boundaries where the circulant model is inaccurate.

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 with the Kullbach Leibler divergence as a data fidelity function. We propose an iterative method, based on an alternating extragradient scheme, which is able to solve, in a numerically stable way, the primal–dual formulation of both total variation and hypersurface regularization problems. In this method, tailored for general smooth saddle-point problems, the stepsize parameter can be adaptively computed so that the convergence of the scheme is proved under mild assumptions. In the numerical experience, we focus the attention on the artificial smoothing parameter that makes different the total variation and hypersurface regularization. A set of experiments on image denoising and deblurring problems is performed in order to evaluate the influence of this smoothing parameter on the stability of the proposed method and on the features of the restored images.

A Majorize–Minimize Strategy for Subspace Optimization Applied to Image Restoration

This paper proposes accelerated subspace optimization methods in the context of image restoration. Subspace optimization methods belong to the class of iterative descent algorithms for unconstrained optimization. At each iteration of such methods, a stepsize vector allowing the best combination of several search directions is computed through a multidimensional search. It is usually obtained by an inner iterative second-order method ruled by a stopping criterion that guarantees the convergence of the outer algorithm. As an alternative, we propose an original multidimensional search strategy based on the majorize-minimize principle. It leads to a closed-form stepsize formula that ensures the convergence of the subspace algorithm whatever the number of inner iterations. The practical efficiency of the proposed scheme is illustrated in the context of edge-preserving image restoration.

Block-Triangular Preconditioners for Systems Arising from Edge-Preserving Image Restoration

Signal and image restoration problems are often solved by minimizing a cost function consisting of an l(2) data-fidelity term and a regularization term. We consider a class of convex and edge-preserving regularization functions. In specific, half-quadratic regularization as a fixed-point iteration method is usually employed to solve this problem. The main aim of this paper is to solve the above-described signal and image restoration problems with the half-quadratic regularization technique by making use of the Newton method. At each iteration of the Newton method, the Newton equation is a structured system of linear equations of a symmetric positive definite coefficient matrix, and may be efficiently solved by the preconditioned conjugate gradient method accelerated with the modified block SSOR preconditioner. Our experimental results show that the modified block-SSOR preconditioned conjugate gradient method is feasible and effective for further improving the numerical performance of the half-quadratic regularization approach.

Satellite image blind restoration based on surface fitting and multivariate model

Owing to the blurring effect from atmosphere and camera system in the satellite imaging, a blind image restoration algorithm is proposed which includes the modulation transfer function (MTF) estimation and the image restoration. In the MTF estimation stage, based on every degradation process of satellite imaging-chain, a combined parametric model of MTF is given and used to fit the surface of normalized logarithmic amplitude spectrum of degraded image. In the image restoration stage, a maximum a posteriori (MAP) based edge-preserving image restoration method is presented which introduces multivariate Laplacian model to characterize the prior distribution of wavelet coefficients of original image. During the image restoration, in order to avoid solving high nonlinear equations, optimization transfer algorithm is adopted to decompose the image restoration procedure into two simple steps: Landweber iteration and wavelet thresholding denoising. In the numerical experiment, the satellite image restoration results from SPOT-5 and high resolution camera (HR) of China & Brazil earth resource satellite (CBERS-02B) ane compared, and the proposed algorithm is superior in the image edge preservation and noise inhibition.

Image restoration using L1 norm penalty function

The process of estimating an original image from a given blurred and noisy image is known as image restoration. It is an ill-posed inverse problem, since one of the ways of solving it requires finding a solution to a Fredholm integral equation of convolution type in two-dimensional space. The focus of the article is to achieve a quality edge preserving image restoration using a less expensive (fast) regularization technique with L1 norm penalty function. L1 norm based approaches do not penalize edges or high frequency contents in the restored image compared to L2 norm based approaches. Total variation (TV) is an established L1 norm regularization approach that performs edge preserving image restoration, but at a high computational cost. TV regularization requires linearization of a highly nonlinear penalty term, which increases the restoration time considerably for large scale images. In order to reduce the computational cost, we extend least absolute shrinkage and selection operator (LASSO), an L1 norm minimization statistical modeling technique to image restoration. The penalty function of LASSO is an identity matrix so it is computationally fast. The metrics, like, residual error, peak signal to noise ratio (PSNR), restoration time, edge map of the restored image, and subjective visual evaluation are used to assess the performances of both methods. Based on our experimental results, we show that LASSO achieves similar quality of edge preserving restoration as TV regularization, and is approximately two times faster in computation compared to TV regularization on the same set of images. We also analyze the impact of the different degree of blurring caused by point spread functions (PSFs) corrupted by different signal to noise ratios (SNRs) on image restoration.

Fast edge-preserving image restoration based on phase information

We propose a fast method for linearizing the edge-preserving regularization. The regularization operator is decomposed to the sum of some linear operators. The phase-only image is used in place of the estimated image. Those linear operators are computed from the phase-only image. They are the approximation of the true regularization operator. The new discontinuity maps do not need to be computed from the last image estimate at every step of the algorithm. This fast method can reduce much algorithm processing time.

Efficient Huber-Markov Edge-Preserving Image Restoration

The regularization of the least-squares criterion is an effective approach in image restoration to reduce noise amplification. To avoid the smoothing of edges, edge-preserving regularization using a Gaussian Markov random field (GMRF) model is often used to allow realistic edge modeling and provide stable maximum a posteriori (MAP) solutions. However, this approach is computationally demanding because the introduction of a non-Gaussian image prior makes the restoration problem shift-variant. In this case, a direct solution using fast Fourier transforms (FFTs) is not possible, even when the blurring is shift-invariant. We consider a class of edge-preserving GMRF functions that are convex and have nonquadratic regions that impose less smoothing on edges. We propose a decomposition-enabled edge-preserving image restoration algorithm for maximizing the likelihood function. By decomposing the problem into two subproblems, with one shift-invariant and the other shift-variant, our algorithm exploits the sparsity of edges to define an FFT-based iteration that requires few iterations and is guaranteed to converge to the MAP estimate.

Mistral: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images

Deconvolution is a necessary tool for the exploitation of a number of imaging instruments. We describe a deconvolution method developed in a Bayesian framework in the context of imaging through turbulence with adaptive optics. This method uses a noise model that accounts for both photonic and detector noises. It additionally contains a regularization term that is appropriate for objects that are a mix of sharp edges and smooth areas. Finally, it reckons with an imperfect knowledge of the point-spread function (PSF) by estimating the PSF jointly with the object under soft constraints rather than blindly (i.e., without constraints). These constraints are designed to embody our knowledge of the PSF. The implementation of this method is called MISTRAL. It is validated by simulations, and its effectiveness is illustrated by deconvolution results on experimental data taken on various adaptive optics systems and telescopes. Some of these deconvolutions have already been used to derive published astrophysical interpretations.

Preconditioned edge-preserving image deblurring and denoising

Preconditioned conjugate gradient (PCG) algorithms have been successfully used to significantly reduce the number of iterations in Tikhonov regularization techniques for image restoration. Nevertheless, in many cases Tikhonov regularization is inadequate, in that it produces images that are oversmoothed across intensity edges. Edge-preserving regularization can overcome this inconvenience but has a higher complexity, in that it involves non-convex optimization. In this paper, we show how the use of preconditioners can improve the computational performance of edge-preserving image restoration as well. In particular, we adopt an image model which explicitly accounts for a constrained binary line process, and a mixed-annealing algorithm that alternates steps of stochastic updating of the lines with steps of preconditioned conjugate gradient-based estimation of the intensity. The presence of the line process requires a specific preconditioning strategy to manage the particular structure of the matrix of the equivalent least squares problem. Experimental results are provided to show the satisfactory performance of the method, both with respect to the quality of the restored images and the computational saving.

Noise-adaptive edge-preserving image restoration algorithm

Noise-adaptive edge-preserving image restoration algorithm

A neural architecture for fully data driven edge-preserving image restoration

This paper proposes a neural architecture, based on two Hopfield nets interconnected with a Boltzmann Machine, for a completely data driven edge-preserving restoration of blurred and noisy images. Solving this restoration problem entails the joint es

Robust edge-preserving image restoration in the presence of non-Gaussian noise

An approach to image restoration is presented which combines the properties of classical regularised iterative algorithms and M-estimation. The method is based on a penalised maximum likelihood estimation incorporating a generalised robust objective function which takes into account non-Gaussian noise and edge-preserving image priors. Results are presented which demonstrate the effectiveness of the method with low-resolution noisy images of simulated landmines.

Unsupervised edge-preserving image restoration via a saddle point approximation

This paper proposes a fast method to estimate the Gibbs hyperparameters of an MRF image model with explicit lines during the restoration process. It consists of a mixed-annealing algorithm for the maximization of the posterior distribution with respect to the image field, periodically interrupted to compute, via ML estimation, a new set of parameters. We first consider the weak membrane model and show that, by adopting a saddle point approximation for the partition function, these new parameters are defined as those that maximize the conditional prior distribution of the lines given the intensities, evaluated on the current estimate of the whole image field. In this way the computation of the expectations involved in the ML estimation can be performed by analytical summation over the binary line elements alone, with a strong reduction of the computational complexity. The approach can be extended to the general case of self-interacting line models, by substituting the analytical computations with a binary, short-range Gibbs sampler.

Sigmoidal approximations for self-interacting line processes in edge-preserving image restoration

Image restoration is formulated as the problem of minimizing a non-convex cost function E( f, l) in which a binary self-interacting line process is introduced. Each line element is then approximated by a sigmoidal function of the local intensity gradient, which depends on a parameter T, thus obtaining a sequence of functions F T ( f) converging to a function F( f) that implicitly refers to the line process. In the case of a non-interacting line process, function F( f) coincides with the one derived for the weak membrane problem. The minimum of F( f) is computed through a GNC-type algorithm which minimizes in sequence the various F T ( f)'s using gradient descent techniques. When generalized to the case of self-interacting line elements, the method is flexible in introducing any kind of constraint on the configurations of the discontinuity field. The results of simulations highlight that the method improves the quality of the reconstruction when constraints on the line process are introduced, without any increase in the computational costs with respect to the case where there are no self-interactions between lines.

Edge-preserving restoration of low-light-level microscope images

The low light inmtensity of fluorescence has always been a serious limitation to its routine use in biology and biomedicine. Since average users in digital microscopy usually possess a commercial TV camera, which is not always a sophisticated or very expensive camera, procedures for software image restoration can represent an alternative and valid solution for studying ‘ in vivo’ biological events. In this paper we present a novel algorithm for edge-preserving image restoration, and show the results of its application on a fluorescence test image. The quality of the restored image is comparable to the quality of the image acquired by a high quality camera.