LLT Model Research Articles

A fractional variational image denoising model with two-component regularization terms

Image denoising is to recover true image from noisy image. Many image deonising models are proposed during the last decades. Some models preserve the margin of tissue, i.e., TV model, while the others, i.e., LLT model, prefer smooth solutions. By decomposing true image into cartoon part and texture part, we propose a fractional image denoising model with two-component regularization terms. Setting some appropriate parameters, the proposed model can deal with both smooth and non-smooth image denosing problems. The existence and uniqueness of solution for the variational model are proved. Moreover, a Split-Bregman(S-B) based numerical algorithm to solve this model is also proposed to validate the theoretical results. Numerical tests show that the proposed model can produce competitive denoising result to the other three published models.

A new anisotropic fourth-order diffusion equation model based on image features for image denoising

<p style='text-indent:20px;'>Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.</p>

A novel high-order functional based image registration model with inequality constraint

In this paper, a novel variational image registration model using a second-order functional as regularizer is presented. The main motivation for the new model stems from the LLT model (see Lysaker, 2003). In order to avoid mesh folding, inequality constraint on the determinant of the Jacobian matrix J of the transformation is also proposed. Furthermore, a fast solver is provided for numerical implementation of registration model with inequality constraints. Numerical experiments are illustrated to show the good performance of our new model according to the registration quality.

Combination of LLT Model and TV Model for Image Denoising

Developing a variational model that is capable of restoring both smooth (no edges) and non-smooth (with edges) images is still a valid challenge at the image processing. In this paper, we present two methods for image denoising problems based on the use of the LLT model (see [14]) and TV model (see [20]). The idea of our methods is, add the texture which is separated from the cartoon and noisy, back to the original noisy image or the texture plus noisy part, and the sum then processed. In order to obtain the texture, we first separate texture plus noise from cartoon by LLT model, and then use TV model to remove some noisy from texture. Numerical experiments show our method is able to maintain some important information such as small details in the image, and at the same time to get a better visualization.

Image Denoising Combining the P-M Model and the LLT Model

In this paper, we present a noise removal technique by combining the P-M model with the LLT model. The combined technique takes full use of the advantage of both filters which is able to preserve edges and simultaneously overcomes the staircase effect. We use a weighting function in our model, and compare this model with the P-M model as well as other fourth-order functional both in theory and numerical experiment.

New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter

We consider simultaneously estimating the restored image and the spatially dependent regularization parameter which mutually benefit from each other. Based on this idea, we refresh two well-known image denoising models: the LLT model proposed by Lysaker et al. (2003) and the hybrid model proposed by Li et al. (2007). The resulting models have the advantage of better preserving image regions containing textures and fine details while still sufficiently smoothing homogeneous features. To efficiently solve the proposed models, we consider an alternating minimization scheme to resolve the original nonconvex problem into two strictly convex ones. Preliminary convergence properties are also presented. Numerical experiments are reported to demonstrate the effectiveness of the proposed models and the efficiency of our numerical scheme.

Nonlinear multigrid method for solving the LLT model

Our goal in this paper is to study a nonlinear multigrid method for solving the LLT model. In order to improve the convergence rate of the proposed multigrid method, an improved dual iteration is proposed as its smoother. Furthermore, we give the local Fourier analysis (LFAs) of the Chambolle’s dual iterations and a modified smoother for solving the LLT model, respectively. Numerical results illustrate the efficiency of the proposed method and indicate that such a multigrid method is more suitable to deal with large-sized images.

Nonlinear multigrid method for solving the anisotropic image denoising models

In this paper, we study a nonlinear multigrid method for solving a general image denoising model with two L 1-regularization terms. Different from the previous studies, we give a simpler derivation of the dual formulation of the general model by augmented Lagrangian method. In order to improve the convergence rate of the proposed multigrid method, an improved dual iteration is proposed as its smoother. Furthermore, we apply the proposed method to the anisotropic ROF model and the anisotropic LLT model. We also give the local Fourier analysis (LFAs) of the Chambolle's dual iterations and a modified smoother for solving these two models, respectively. Numerical results illustrate the efficiency of the proposed method and indicate that such a multigrid method is more suitable to deal with large-sized images.

Image Restoration Based on the Hybrid Total‐Variation‐Type Model

We propose a hybrid total‐variation‐type model for the image restoration problem based on combining advantages of the ROF model with the LLT model. Since two L1‐norm terms in the proposed model make it difficultly solved by using some classically numerical methods directly, we first employ the alternating direction method of multipliers (ADMM) to solve a general form of the proposed model. Then, based on the ADMM and the Moreau‐Yosida decomposition theory, a more efficient method called the proximal point method (PPM) is proposed and the convergence of the proposed method is proved. Some numerical results demonstrate the viability and efficiency of the proposed model and methods.

Adaptive fourth-order partial differential equation filter for image denoising

To overcome the staircasing effects and simultaneously avoid edge blurring, this paper describes a fourth-order partial differential equation based edge-preserving regularization filter for noise removal. This technique is closely related to the nonlinear anisotropic diffusion. Compared results distinctly demonstrate the superiority of our proposed scheme over the LLT model, in terms of removing noise while sharply maintaining the edge features.

A two-step model for image denoising using a duality strategy and surface fitting

In order to alleviate the staircase effect or the edge blurring in the course of the image denoising, we propose a two-step model based on the duality strategy. In fact, this strategy follows the observation that the dual variable of the restored image can be looked at as the normal vector. So we first obtain the dual variable and then reconstruct the image by fitting the dual variable. Following the augmented Lagrangian strategy, we propose a projection gradient method for solving this two-step model. We also give some convergence analyses of the proposed projection gradient method. Several numerical experiments are tested to compare our proposed model with the ROF model and the LLT model.

A Compound Algorithm of Denoising Using Second-Order and Fourth-Order Partial Differential Equations

In this paper, we propose a compound algorithm for the image restoration. The algorithm is a convex combination of the ROF model and the LLT model with a parameter functionθ. The numerical experiments demonstrate that our compound al- gorithm is efficient and preserves the main advantages of the two models. In particular, the errors of the compound algorithm in L2 norm between the exact images and cor- responding restored images are the smallest among the three models. For images with strong noises, the restored images of the compound algorithm are the best in the cor- responding restored images. The proposed algorithm combines the fixed point method, an improved AMG method and the Krylov acceleration. It is found that the combination of these methods is efficient and robust in the image restoration.

Semismooth Newton method for minimization of the LLT model

In this paper, we discuss the nonsmooth second-orderregularization, suggested by Lysaker, Lundervold and Tai, and itsapplication in image denoising. A function space $BV^2$ is givenand the well-posedness of the LLT model is proved in this functionspace. By means of the Fisher-Burmeister NCP function, wereformulate the dual formula of the LLT model in discrete settingas a system of semismooth equations. Then we propose a semismoothNewton method for the LLT model to build up a Q-superlinearlyconvergent numerical scheme. The computational experiments aresupplied to demonstrate the efficiency of the proposed method.

Modeling medium-scale TEC structures, observed by Belgian GPS receivers network

GALOCAD project “Development of a Galileo Local Component for the nowcasting and forecasting of atmospheric disturbances affecting the integrity of high precision Galileo applications” aims to perform a detailed study on ionospheric small- and medium-scale structures and to assess the influence of these structures on the reliability of Galileo precise positioning applications. GPS-derived TEC (total electron content) is obtained from the Belgium Dense Network (BDN), consisting of 67 permanent GPS stations. An empirical 3-D model is developed for studying these ionospheric structures. The model, named LLT model, described temporal variations of TEC in latitude/longitude frame (46°, 52°)N and (−1°, 11°)E. The spatial variations of TEC are modeled by Tchebishev base functions, while the temporal variations are described by a trigonometric basis. To fit the model to the data, the observed area is divided into bins with (1° × 1°) geographic scale and 6 min on time axis. LLT model is made flexible, with varying number of coefficients along each axis. This allows different degree of smoothing, which is the key element of the present approach. Model runs with higher number of coefficients, capturing in details medium-scale TEC structures are subtracted from results obtained with smaller number of coefficients; the latter represent the background ionosphere. The residual structures are localized and followed as they travel across the observed area. In this way, the size, velocity, and direction of the irregular structures are obtained.

A dual algorithm for minimization of the LLT model

We apply the dual algorithm of Chambolle for the minimization of the LLT model. A convergence theorem is given for the proposed algorithm. The algorithm overcomes the numerical difficulties related to the non-differentiability of the LLT model. The dual algorithm is faster than the original gradient descent algorithm. Numerical experiments are supplied to demonstrate the efficiency of the algorithm.