Efficient Iterative Regularization Method for Total Variation-Based Image Restoration

Total variation (TV) regularization has received much attention in image restoration applications because of its advantages in denoising and preserving details. A common approach to address TV-based image restoration is to design a specific algorithm for solving typical cost function, which consists of conventional ℓ2 fidelity term and TV regularization. In this work, a novel objective function and an efficient algorithm are proposed. Firstly, a pseudoinverse transform-based fidelity term is imposed on TV regularization, and a closely-related optimization problem is established. Then, the split Bregman framework is used to decouple the complex inverse problem into subproblems to reduce computational complexity. Finally, numerical experiments show that the proposed method can obtain satisfactory restoration results with fewer iterations. Combined with the restoration effect and efficiency, this method is superior to the competitive algorithm. Significantly, the proposed method has the advantage of a simple solving structure, which can be easily extended to other image processing applications.

IPGM: Inertial Proximal Gradient Method for Convolutional Dictionary Learning

Inspired by the recent success of the proximal gradient method (PGM) and recent efforts to develop an inertial algorithm, we propose an inertial PGM (IPGM) for convolutional dictionary learning (CDL) by jointly optimizing both an ℓ2-norm data fidelity term and a sparsity term that enforces an ℓ1 penalty. Contrary to other CDL methods, in the proposed approach, the dictionary and needles are updated with an inertial force by the PGM. We obtain a novel derivative formula for the needles and dictionary with respect to the data fidelity term. At the same time, a gradient descent step is designed to add an inertial term. The proximal operation uses the thresholding operation for needles and projects the dictionary to a unit-norm sphere. We prove the convergence property of the proposed IPGM algorithm in a backtracking case. Simulation results show that the proposed IPGM achieves better performance than the PGM and slice-based methods that possess the same structure and are optimized using the alternating-direction method of multipliers (ADMM).

Fractional-Order Adaptive P -Laplace Equation-Based Art Image Edge Detection

In recent years, with the rapid development of image processing research, the study of nonstandard images has gradually become a research hotspot, for example, fabric images, remote sensing images, and gear images. Some of the remote sensing images have a complex background and low illumination compared with standard images and are easy to be mixed with noise during acquisition; some of the fabric images have rich texture information, which adds difficulty to the related processing, and are also easy to be mixed with noise during acquisition. In this paper, we propose a fractional-order adaptive P -Laplace equation image edge detection algorithm for the problem of image edge detection in which the edge and texture information of the image is lost. The algorithm can apply for the order adaptively to filter the noise according to the noise distribution of the image, and the adaptive diffusion factor is determined by both the fractional-order curvature and fractional-order gradient of the iso-illumination line and combined with the iterative approach to realize the fine-tuning of the noisy image. The experimental results demonstrate that the algorithm can remove the noise while preserving the texture and details of the image. A fractional-order partial differential equation image edge detection model with a fractional-order fidelity term is proposed for Gaussian noise. The model incorporates a fractional-order fidelity term because this fidelity term smoothes out the rougher parts of the image while preserving the texture in the original image in greater detail and eliminating the step effect produced by other models such as the Perona-Malik (PM) and Rudin-Osher-Fatemi (ROF) models. By comparing with other algorithms, the image edge detection effect is measured with the help of evaluation metrics such as peak signal-to-noise ratio and structural similarity, and the optimal value is selected iteratively so that the image with the best edge detection result is retained. A convolutional mask image edge detection model based on adaptive fractional-order calculus is proposed for the scattered noise in medical images. The adaption is mainly reflected in the model algorithm by constructing an exponential parameter relation that is closely related to the image, which can dynamically adjust the parameter values, thus making the model algorithm more practical. The model achieves the scattering noise removal in four steps.

An image super-resolution reconstruction model based on fractional-order anisotropic diffusion equation

<abstract><p>The image denoising model based on anisotropic diffusion equation often appears the staircase effect while image denoising, and the traditional super-resolution reconstruction algorithm can not effectively suppress the noise in the image in the case of blur and serious noise. To tackle this problem, a novel model is proposed in this paper. Based on the original diffusion equation, we propose a new method for calculating the adaptive fidelity term and its coefficients, which is based on the relationship between the image gradient and the diffusion function. It is realized that the diffusion speed can be slowed down by adaptively changing the coefficient of the fidelity term, and it is proved mathematically that the proposed fractional adaptive fidelity term will not change the existence and uniqueness of the solution of the original model. At the same time, washout filter is introduced as the control item of the model, and a new model of image super-resolution reconstruction and image denoising is constructed. In the proposed model, the order of fractional differential will be determined adaptively by the local variance of the image. And we give the numerical calculation method of the new model in the frequency domain by the method of Fourier transform. The experimental results show that the proposed algorithm can better prevent the staircase effect and achieve better visual effect. And by introducing washout filter to act as the control of the model, the stability of the system can be improved and the system can converge to a stable state quickly.</p></abstract>

Exact solutions for the total variation denoising problem of piecewise constant images in dimension one

A method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

A Constrained Convex Optimization Approach to Hyperspectral Image Restoration with Hybrid Spatio-Spectral Regularization

We propose a new constrained optimization approach to hyperspectral (HS) image restoration. Most existing methods restore a desirable HS image by solving some optimization problems, consisting of a regularization term(s) and a data-fidelity term(s). The methods have to handle a regularization term(s) and a data-fidelity term(s) simultaneously in one objective function; therefore, we need to carefully control the hyperparameter(s) that balances these terms. However, the setting of such hyperparameters is often a troublesome task because their suitable values depend strongly on the regularization terms adopted and the noise intensities on a given observation. Our proposed method is formulated as a convex optimization problem, utilizing a novel hybrid regularization technique named Hybrid Spatio-Spectral Total Variation (HSSTV) and incorporating data-fidelity as hard constraints. HSSTV has a strong noise and artifact removal ability while avoiding oversmoothing and spectral distortion, without combining other regularizations such as low-rank modeling-based ones. In addition, the constraint-type data-fidelity enables us to translate the hyperparameters that balance between regularization and data-fidelity to the upper bounds of the degree of data-fidelity that can be set in a much easier manner. We also develop an efficient algorithm based on the alternating direction method of multipliers (ADMM) to efficiently solve the optimization problem. We illustrate the advantages of the proposed method over various HS image restoration methods through comprehensive experiments, including state-of-the-art ones.

Automatic Binary Data Classification Using a Modified Allen–Cahn Equation

In this paper, we propose an automatic binary data classification method using a modified Allen–Cahn (AC) equation. The modified AC equation was originally developed for image segmentation. The equation consists of the AC equation with a fidelity term which enforces the solution to be the given data. In the proposed method, we start from a coarse grid and refine the grid until the accuracy of the data classification reaches a given tolerance. Therefore, we can avoid a laborious trial and error procedure. For a numerical method for the modified AC equation, we use a recently developed explicit hybrid scheme. We perform several 2D and 3D computational tests to demonstrate the performance of the proposed method. The computational results confirm that the proposed algorithm is automatic.

A derivation of Griffith functionals from discrete finite-difference models

We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step $$\delta $$ δ is smaller than the ellipticity parameter $$\varepsilon $$ ε , we show the $$\varGamma $$ Γ -convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no $$L^p$$ L p fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.