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In recent years, image denoising methods based on total variational regularization have attracted extensive attention. However, the traditional total variational regularization method is an approximate solution based on convex method, and does not consider the particularity of the region with rich details. In this paper, the adaptive total-variation and nonconvex low-rank model for image denoising is proposed, which is a hybrid regularization model. First, the image is decomposed into sparse terms and low rank terms, and then the total variational regularization is used to denoise. At the same time, an adaptive coefficient based on gradient is constructed to adaptively judge the flat area and detail texture area, slow down the denoising intensity of detail area, and then play the role of preserving detail information. Finally, by constructing a nonconvex function, the optimal solution of the function is obtained by using the alternating minimization method. This method not only effectively removes the image noise, but also retains the detailed information of the image. The experimental results show the effectiveness of the proposed model, and SNR and SSIM of the denoised image are improved.

In this paper, we apply an inexact dual alternating direction method of multipliers (dADMM) to an image decomposition model. In this model, an image is divided into two meaningful components, i.e., a cartoon part and a texture part. The dADMM presents not only the cartoon part and the texture part of an image but also the restored image (cartoon $\mathrm{\text{part}}+\mathrm{\text{texture}}$ part). We also summarize the global convergence for the algorithm. Numerical experiments demonstrate the efficiency and robustness of the inexact dADMM. Furthermore, we can obtain relatively higher signal-to-noise ratio (SNR) comparing to other algorithms.

In this paper, we consider the problem of image restoration with box-constraints. Image restoration problem is ill-conditioned and the regularization approach has widely been used to stabilize the solution. The restored image highly depends on the choice of the regularization parameter. The regularization parameter is generally determined by trial-and-error method when no true original image is available. Obviously, it is time consuming. The main aim in this paper is to develop an algorithm to choose the regularization parameter automatically when the box-constraints are imposed. In the proposed algorithm, the regularization parameter is adaptively determined by the previous iterative solution. Numerical simulations are used to demonstrate the performance of the proposed method.

Image denoising as a part of pre-processing in image analysis is a challenging area of research since noise removal and image detail preservation need a tradeoff. For classical denoising models, the convex total variation (TV) or some nonconvex regularizers are used to achieve the tradeoff. However, the denoising performance of classical models is still inadequate. To overcome this problem, this paper proposes a new variational model for image restoration, where a weighted regularizer is designed to protect more geometric structural details of images from over-smoothing and to remove much noise simultaneously. To solve the model efficiently, a novel algorithm based on Chambolle’s dual projection method and the iteratively reweighting method is presented. Numerical results prove that the proposed denoising method can show better performance than the classical TV-based and the nonconvex regularizer-based denoising methods.

In order to discover the difference among dual strategies, we propose an alternating primal-dual algorithm (APDA) that can be considered as a general version for minimizing problem which is multiple-summed separable convex but not necessarily smooth. First, the original multiple-summed problem is transformed into two subproblems. Second, one subproblem is solved in the primal space and the other is solved in the dual space. Finally, the alternating direction method is executed between the primal and the dual part. Furthermore, the classical alternating direction method of multipliers (ADMM) is extended to solve the primal subproblem which is also multiple summed, therefore, the extended ADMM can be seen as a parallel method for the original problem. Thanks to the flexibility of APDA, different dual strategies for image restoration are analyzed. Numerical experiments show that the proposed method performs better than some existing algorithms in terms of both speed and accuracy.

In this paper, we present a novel method for blind image inpainting, which can restore images with missing or corrupted pixels, or images where the location of the damaged pixels is unknown. The method applies weighted nonlocal Laplacian to address the problem of blind image inpainting using low-dimensional manifold model (LDMM) regularization, and uses semi-local blocks instead of point integrals to implement constraints in LDMM. This solves the problem of low solution efficiency caused by the asymmetry of the linear equations solved by point integration, and the problem of the high iteration count to get good restoration effect. Experiments show that our method is competitive with latest methods in terms of both repairing images with large missing pixels rate and inpainting speed.

The linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end is a useful model for studying chaotic behavior in distributed parameter system. In this paper, we study the dynamics of the Riemann invariants (u, v) of the wave equation by means of the total variations of the snapshots on the spatial interval. Our main contributions here are the classification of the growth of total variations of the snapshots of u and v in long-time horizon, namely, there are three cases when a certain parameter enters a different regime: the growth

(i) remains bounded;

(ii) is unbounded (but nonexponential);

(iii) is exponential,

for a large class of initial conditions with finite total variations. In particular, case (iii) corresponds to the onset of chaos. The results here sharpen those in an earlier work [Chen et al., 2001].

In this paper, boundary anticontrol of spatiotemporal chaos for 1D hyperbolic equations is studied. Firstly, a new definition of chaotic vibrations for PDEs is given in terms of the growth rate of the total variations of the solutions with respect to the spatial variable as t→∞. Then, a boundary feedback controller is designed as composing with a sawtooth function, which can drive the originally nonchaotic linear or nonlinear dynamical system chaotic. Finally, as applications, anticontrol of chaos for 1D linear wave equations with linear or nonlinear boundary conditions is discussed.

The question of spectral analysis for deterministic chaos is not well understood in the literature. In this paper, using iterates of chaotic interval maps as time series, we analyze the mathematical properties of the Fourier series of these iterates. The key idea is the connection between the total variation and the topological entropy of the iterates of the interval map, from where special properties of the Fourier coefficients are obtained. Various examples are given to illustrate the applications of the main theorems.

This paper is a continuation of Part I where the authors treated the Fourier analysis of chaotic time series generated by a chaotic interval map. Here, we perform multiresolution analysis by using wavelet coefficients and characterize some necessary and sufficient conditions for the occurrence of chaos by the exponential growth with respect to the number of iterations n of certain sums of the wavelet coefficients.

In this paper, we describe a fast and efficient method for multi-modal and discontinuity-preserving image registration, implemented on graphics hardware. Multi-sensory data fusion and medical image analysis often pose the challenging task of aligning dense, non-rigid and multi-modal images. However, also optical sequences or stereo image pairs may present variable illumination conditions and noise. The above problems can be addressed by an invariant similarity measure, such as mutual information. Additionally, when using a regularized approach to deal with the ill-posedness of the problem, one has to take care of preserving discontinuities at the motion boundaries. Our approach efficiently addresses the above issues through a primal-dual convex estimation framework, using an approximated Hessian matrix that decouples pixel dependencies, while being asymptotically correct. At the same time, we achieve a high computational efficiency by means of pre-quantized kernel density estimation and differentiation, as well as a parallel implementation on the GPU. Our approach is demonstrated on ground-truth data from the Middlebury database, as well as medical and visible-infrared image pairs.

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝ^N. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L¹(ℝ^N). We are able to characterize limit solutions when the initial condition μ=h+μ_s, where h∈L¹(ℝ^N)∩L^∞(ℝ^N), and μ_s=αℋ^k⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k<N-1 we prove that the singular part of the solution does not move, it remains equal to μ_s for all t≥0. In particular, u(t)=δ₀ when u(0)=δ₀. In case k=N-1 we prove that the singular part of the limit solution is and we also characterize its absolutely continuous part. This explicit behaviour permits to characterize limit solutions. We also give an entropy condition characterization of the solution which is more satisfactory when k<N-1. Finally, we describe some distributional solutions which do not have the behaviour characteristic of limit solutions.

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to , we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed -norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler–Lagrange equation as necessary condition for minimizers. Here the degenerate expression Du/|Du| has to be replaced by a suitable vector field to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler–Lagrange equations have to be satisfied.

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a $r$-order (an)-isotropic total variation semi-norms ${\mathrm{\text{TV}}}^r$, $r\in {ℝ}^{+}$, defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative $r$, of such loss functions.

In this paper, we present a novel noise suppression and detail preservation algorithm. The test image is firstly pre-processed through a multiresolution analysis employing the discrete wavelet transform. Then, we apply a fast and robust total variation technique, incorporating a statistical representation in the style of maximum likelihood estimation. Finally, we compare this proposed approach to current state-of-the-art denoising methods using synthetic and real images. The results demonstrate encouraging performance of our algorithm.

Image inpainting is an active research area in the image processing field. The essential idea of image inpainting algorithm is to fill in the missing or damaged regions with available information from their surroundings. In this paper, we propose two image inpainting models based on the variational method. We show that the diffusion performance of the proposed models for image inpainting are superior to classical total variation (TV) inpainting model according to the physical characteristics in local coordinates. To show the effective performance of the proposed models, we apply it to restoring of scratched photos, text removal and even removal of entire objects from images.

The degraded image during the process of image analysis needs more number of iterations to restore it. These iterations take long waiting time and slow scanning, resulting in inefficient image restoration. A few numbers of measurements are enough to recuperate an image with good condition. Due to tree sparsity, a 2D wavelet tree reduces the number of coefficients and iterations to restore the degraded image. All the wavelet coefficients are extracted with overlaps as low and high sub-band space and ordered them such that they are decomposed in the tree ordering structured path. Some articles have addressed the problems with tree sparsity and total variation (TV), but few authors endorsed the benefits of tree sparsity. In this paper, a spatial variation regularization algorithm based on tree order is implemented to change the window size and variation estimators to reduce the loss of image information and to solve the problem of image smoothing operation. The acceptance rate of the tree-structured path relies on local variation estimators to regularize the performance parameters and update them to restore the image. For this, the Localized Total Variation (LTV) method is proposed and implemented on a 2D wavelet tree ordering structured path based on the proposed image smooth adjustment scheme. In the end, a reliable reordering algorithm proposed to reorder the set of pixels and to increase the reliability of the restored image. Simulation results clearly show that the proposed method improved the performance compared to existing methods of image restoration.

In this paper, we present a new class of iterative regularization methods in the setting of Besov spaces, which can be seen as generalizations of J. Xu's method. By incorporating translation invariant wavelet transform, minimizers of the new methods can be understood as the alternative to translation invariant wavelet shrinkage with weight that is dependent on the wavelet decomposition scale and the Besov smooth order. And we generalize the iterative regularization methods to a new class of nonlinear inverse scale spaces with scale and Besov smooth order dependent weight. The numerical results show an excellent denoising effect and improvement over J. Xu's method.

This paper solves the image fusion problem by using a multi-object optimization strategy and a key energy function. The energy function mainly consists of two components. One ensures injection of more correlated detailed spatial information. The detailed information is extracted from gradient representation of the image to be fused. The other one guarantees that the spectral information is preserved by a data fitting term. By minimizing the proposed energy function, the fusion result can be obtained. Moreover, a key parameter is used in the energy function to adjust the weights of the spectral and spatial information during the image fusion. In this paper, a multi-object optimization is constructed to determine such a key parameter. The image fusion performance is evaluated through visional perception and some fusion indexes. Experimental results further demonstrate advantages of the proposed technique over the conventional fusion techniques.