Iterated Tikhonov regularization with a general penalty term

Combining approximate solutions for linear discrete ill-posed problems

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.

An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

Invertible smoothing preconditioners for linear discrete ill-posed problems

Electrical capacitance tomography (ECT) technology has the softfield nature and the ill-posed problems. To solve the problem of low imaging quality by standard Tikhonov regularization, a wavelet fusion algorithm based on Tikhonov regularization is proposed in this paper. Firstly, Tikhonov regularization method is used to solve ill-posed characteristic of system inverse problem, then the initial image is decomposed and fused by using wavelet transform. The simulation results show that this method can not only improve the image reconstructed by standard Tikhonov regularization, but also provides a new idea to ECT image reconstruction.

Quantitative susceptibility mapping (QSM) is used increasingly for clinical research where oblique image acquisition is commonplace, but its effects on QSM accuracy are not well understood. The QSM processing pipeline involves defining the unit magnetic dipole kernel, which requires knowledge of the direction of the main magnetic field with respect to the acquired image volume axes. The direction of is dependent on the axis and angle of rotation in oblique acquisition. Using both a numerical brain phantom and in vivo acquisitions in 5 healthy volunteers, we analyzed the effects of oblique acquisition on magnetic susceptibility maps. We compared three tilt-correction schemes at each step in the QSM pipeline: phase unwrapping, background field removal and susceptibility calculation, using the RMS error and QSM-tuned structural similarity index. Rotation of wrapped phase images gave severe artifacts. Background field removal with projection onto dipole fields gave the most accurate susceptibilities when the field map was first rotated into alignment with . Laplacian boundary value and variable-kernel sophisticated harmonic artifact reduction for phase data background field removal methods gave accurate results without tilt correction. For susceptibility calculation, thresholded k-space division, iterative Tikhonov regularization, and weighted linear total variation regularization, all performed most accurately when local field maps were rotated into alignment with before susceptibility calculation. For accurate QSM, oblique acquisition must be taken into account. Rotation of images into alignment with should be carried out after phase unwrapping and before background-field removal. We provide open-source tilt-correction code to incorporate easily into existing pipelines: https://github.com/o-snow/QSM_TiltCorrection.git.

Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log Poisson likelihood function in place of the least-squares fit-to-data function. However, if the underlying mathematical model is assumed to have the form z = Au + γ, where z is the data and A is a compact operator and γ is the background light intensity, minimizing the negative-log Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov and total variation regularization. In this article, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.

A fractional-order accumulative regularization filter for force reconstruction

A new reconstruction method called generalized vector sampled pattern matching (GVSPM) has been applied to an ill-posed inverse problem involving electrical capacitance computed tomography for solid–air two-phase flow. This new method is able to achieve stable convergence without the use of an empirical value. This accurate reconstruction is accomplished using an objective function that is calculated as the inner product calculation between the experimental capacitance and the reconstructed image capacitance. The GVSPM method is compared with the conventional Landweber (LW) and iterative Tikhonov regularization (ITR) methods in terms of capacitance residual, image error and image correlation. Overall, the accuracy is strongly dependent upon the image type and the iteration number, however the GVSPM method proved superior to the LW and the ITR methods in the case of annular pseudo particle images.

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed condictivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography.

This chapter discusses iterative methods for the solution of very large severely ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by errors. Solution methods proposed in the literature employ some form of filtering to reduce the influence of the error in the right-hand side on the computed approximate solution. The amount of filtering is determined by a parameter often referred to as the regularization parameter. We discuss how the filtering affects the computed approximate solution and consider the selection of regularization parameter. Methods in which a suitable value of the regularization parameter is determined during the computation without user intervention are emphasized. New iterative solution methods based on expanding explicitly chosen filter functions in terms of Chebyshev polynomials are presented. The properties of these methods are illustrated with applications to image restoration.

The unsupervised domain adaptation problem with covariate shift assumption is considered. Within the framework of the Reproducing Kernel Hilbert Space concept, an algorithm is constructed that is a combination of the Nyström subsampling and the iterated Tikhonov regularization. This approach allows significantly reduce the amount of computing resources involved and at the same time achieves the minimal (by order) approximation accuracy under the big data settings.

Limited-data tomography, to which electromagnetic geotomography belongs, is analyzed in this paper. In this technique, a discrete forward projection model may be expressed by a rank-deficient system of linear equations whose the nullspace is non-trivial. This means that some image components may fall into the nullspace, and hence the minimal-norm least-square solution, to which many image reconstructions methods converge, may be different from the true one. The Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstruction Technique (SIRT), or Conjugate Gradients Least Squares (CGLS) are examples of such methods. In this paper, we deal with the question of how to partially recover the missing image components. First, we analyze the advantages of using the iterative Tikhonov regularization and the Maximum A Posteriori (MAP) algorithm with Gibbs prior. Then, we conclude that the missing (nullspace) image components can be partially recovered if the MAP algorithm is implemented through a multigrid technique. The results, which are presented for synthetic noise-free and noisy data, demonstrate the validity of our assumption. The problem of estimating the regularization and scaling parameters in the MAP algorithm is also addressed.