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.

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.

Discretization of equations Gelfand-Levitan-Krein and regularization algorithms

The paper considers the initial-boundary-value inverse problem of acoustics for onedimensional and multidimensional cases. The inverse problems are to reconstruct the coefficients using one-dimensional and multidimensional analogues of the Gelfand-Levitan-Krein integral equations. It is known that such equations are linear integral Fredholm equations of the first kind, which are ill-posed. The aim of the work is to find a numerical solution of the Gelfand-Levitan-Krein equation using iterative regularizing algorithms. Using the specifics of these equations (the kernel of the equation depends on the difference of arguments) it is possible to create highly efficient iterative regularizing algorithms. The implemented algorithms can be successfully applied in solving such problems as reconstruction of blurred and defocused images, inverse problem of gravimetric, linear programming problem with inaccurately given matrix of constraints, inverse problem of Geophysics, inverse problems of computed tomography, etc. The main results of the work are the discretization of the one-dimensional and multidimensional Gelfand-Levitan-Krein equation and the construction of iterative regularization algorithms.

Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.

Regularized inversion of aerosol hygroscopic growth factor probability density function: Application to humidity-controlled fast integrated mobility spectrometer measurements

Aerosol hygroscopic growth plays an important role in atmospheric particle chemistry and the effects of aerosol on radiation and hence climate. The hygroscopic growth is often characterized by a growth factor probability density function (GF-PDF), where the growth factor is defined as the ratio of the particle size at a specified relative humidity to its dry size. Parametric, least-square methods are the most widely used algorithms for inverting the GF-PDF from measurements of humidified tandem differential mobility analyzers (HTDMA) and have been recently applied to the GF-PDF inversion from measurements of the humidity-controlled fast integrated mobility spectrometer (HFIMS). However, these least square methods suffer from noise amplification due to the lack of regularization in solving the ill-posed problem, resulting in significant fluctuations in the retrieved GF-PDF and even occasional failures of convergence. In this study, we introduce nonparametric, regularized methods to invert aerosol GF-PDF and apply them to HFIMS measurements. Based on the HFIMS kernel function, the forward convolution is transformed into a matrix-based form, which facilitates the application of the nonparametric inversion methods with regularizations, including Tikhonov regularization and Twomey’s iterative regularization. Inversions of the GF-PDF using the nonparameteric methods with regularization are demonstrated using HFIMS measurements simulated from representative GF-PDFs of ambient aerosols. The characteristics of reconstructed GF-PDFs resulting from different inversion methods, including previously developed least-square methods, are quantitively compared. The result shows that Twomey’s method generally outperforms other inversion methods. The capabilities of the Twomey’s method in reconstructing the pre-defined GF-PDFs and recovering the mode parameters are validated.