Tikhonov Method Research Articles

A nested primal–dual iterated Tikhonov method for regularized convex optimization

AbstractProximal–gradient methods are widely employed tools in imaging that can be accelerated by adopting variable metrics and/or extrapolation steps. One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal–dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal–dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal–gradient step with extrapolation by performing a prefixed number of primal–dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. Furthermore, we investigate the numerical performance of our proposed method by equipping it with a scaling matrix inspired by the Iterated Tikhonov method. The numerical results show that the combination of such scaling matrices and Nesterov-like extrapolation parameters yields an effective acceleration towards the solution of the problem.

Particle swarm optimization method for parameter selecting in Tikhonov regularization method for solving inverse problems

In this article the Tikhonov method used as a regularization technique for solving the inverse problem liner operator equation of the first kind with noisy and noise-free data. We also provide the Tikhonov method's essential analysis for tackling inverse problems. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Particle Swarm Optimization (PSO) technique. The effectiveness of combining these two approaches particle swarm optimization and Tikhonov is demonstrated to be highly practicable.

A fast identification method of surface heat flux based on numerical pre-calibration and regularization

Abstract Identification of surface heat flux is a technique of retrieving heat flux from the measured temperature in an object. The inverse heat conduction problem abbreviated as IHCP is ill-posed, and its highly ill-conditioned characteristics lead to unstable numerical calculation. In this paper, we build an identification equation with the theory of linear superposition. The matrix of pulse sensitivity coefficient is established by formula for one-dimensional model. And a finite element pre-calibration approach is proposed to obtain the matrix of pulse sensitivity coefficient for complex three-dimensional structures. Regularization methods including Tikhonov method and algebraic reconstruction technique are utilized to reduce the sensitivity of measurement error. Numerical simulation shows that this method is efficient and accurate.

Deep denoiser prior driven relaxed iterated Tikhonov method for low-count PET image restoration

Objective. Low-count positron emission tomography (PET) imaging is an efficient way to promote more widespread use of PET because of its short scan time and low injected activity. However, this often leads to low-quality PET images with clinical image reconstruction, due to high noise and blurring effects. Existing PET image restoration (IR) methods hinder their own restoration performance due to the semi-convergence property and the lack of suitable denoiser prior. Approach. To overcome these limitations, we propose a novel deep plug-and-play IR method called Deep denoiser Prior driven Relaxed Iterated Tikhonov method (DP-RI-Tikhonov). Specifically, we train a deep convolutional neural network denoiser to generate a flexible deep denoiser prior to handle high noise. Then, we plug the deep denoiser prior as a modular part into a novel iterative optimization algorithm to handle blurring effects and propose an adaptive parameter selection strategy for the iterative optimization algorithm. Main results. Simulation results show that the deep denoiser prior plays the role of reducing noise intensity, while the novel iterative optimization algorithm and adaptive parameter selection strategy can effectively eliminate the semi-convergence property. They enable DP-RI-Tikhonov to achieve an average quantitative result (normalized root mean square error, structural similarity) of (0.1364, 0.9574) at the stopping iteration, outperforming a conventional PET IR method with an average quantitative result of (0.1533, 0.9523) and a state-of-the-art deep plug-and-play IR method with an average quantitative result of (0.1404, 0.9554). Moreover, the advantage of DP-RI-Tikhonov becomes more obvious at the last iteration. Experiments on six clinical whole-body PET images further indicate that DP-RI-Tikhonov successfully reduces noise intensity and recovers fine details, recovering sharper and more uniform images than the comparison methods. Significance. DP-RI-Tikhonov’s ability to reduce noise intensity and effectively eliminate the semi-convergence property overcomes the limitations of existing methods. This advancement may have substantial implications for other medical IR.

A Generalized Iterated Tikhonov Method in the Fourier Domain for Determining the Unknown Source of the Time-Fractional Diffusion Equation

In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method.

Application of locally regularized extremal shift to the problem of realization of a prescribed motion

Abstract A controlled system of differential equations under the action of an unknown disturbance is considered. The problem discussed in the paper consists in constructing algorithms for forming a control that provides the realization of a prescribed motion for any admissible disturbance. Namely these algorithms should provide the closeness in the metric of the space of differentiable functions of a phase trajectory of a given controlled system and some etalon trajectory of an analogous system functioning when any outer actions are absent. As the space of admissible disturbances, we take the space of measurable square integrable (with respect to the Euclidean norm) functions. The cases of inaccurate measurements of phase trajectories of both systems at all times and at discrete times are under study. Two computer oriented algorithms for solving the problem are designed. The algorithms are based on the (well-known in the theory of guaranteed control) method of extremal shift. In the process, its local (at each time of control correction) regularization is performed by the method of smoothing functional (the Tikhonov method). In addition, estimates for algorithm’s convergence rate are presented.

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

THREE-COMPONENT VERSION OF THE TIKHONOV REGULARIZATION METHOD FOR OPERATOR EQUATIONS OF THE FIRST KIND

t An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented.

Inverse Calculation and Regularization Process for the Solar Aspect System (SAS) of HXI Payload on ASO-S Spacecraft

For the ASO-S/HXI payload, the accuracy of the flare reconstruction is reliant on important factors such as the alignment of the dual grating and the precise measurement of observation orientation. To guarantee optimal functionality of the instrument throughout its life cycle, the Solar Aspect System (SAS) is imperative to ensure that measurements are accurate and reliable. This is achieved by capturing the target motion and utilizing a physical model-based inversion algorithm. However, the SAS optical system’s inversion model is a typical ill-posed inverse problem due to its optical parameters, which results in small target sampling errors triggering unacceptable shifts in the solution. To enhance inversion accuracy and make it more robust against observation errors, we suggest dividing the inversion operation into two stages based on the SAS spot motion model. First, the as-rigid-as-possible (ARAP) transformation algorithm calculates the relative rotations and an intermediate variable between the substrates. Second, we solve an inversion linear equation for the relative translation of the substrates, the offset of the optical axes, and the observation orientation. To address the ill-posed challenge, the Tikhonov method grounded on the discrepancy criterion and the maximum a posteriori (MAP) method founded on the Bayesian framework are utilized. The simulation results exhibit that the ARAP method achieves a solution with a rotational error of roughly ±3.″5 (1/2-quantile); both regularization techniques are successful in enhancing the stability of the solution, the variance of error in the MAP method is even smaller—it achieves a translational error of approximately ±18 μm (1/2-quantile) in comparison to the Tikhonov method’s error of around ±24 μm (1/2-quantile). Furthermore, the SAS practical application data indicates the method’s usability in this study. Lastly, this paper discusses the intrinsic interconnections between the regularization methods.

A new second-order dynamical method for solving linear inverse problems in Hilbert spaces

A new second-order dynamic method (SODM) is proposed for solving ill-posed linear inverse problems in Hilbert spaces. The SODM can be viewed as a combination of Tikhonov regularization and second-order asymptotical regularization methods. As a result, a double-regularization-parameter strategy is adopted. The regularization properties of SODM are demonstrated under both a priori and a posteriori stopping rules. In the context of time discretization, we propose several iterative schemes with different choices of damping parameters. A truncated discrepancy principle is employed as the stop criterion. Finally, numerical experiments are performed to show the efficiency of the SODM: on the whole, compared with the classical Tikhonov method and the first-order dynamical-system method, the SODM leads to more-accurate approximate solutions while requiring fewer-iterative numbers.

Chebyshev polynomials to Volterra-Fredholm integral equations of the first kind

Numerous methods have been studied and discussed for solving ill-posed Volterra integral equations and ill-posed Fredholm integral equations, but rarely for both simultaneously. In this study, we focus on numerically solving the ill-posed Volterra-Fredholm integral equation of the first kind by replacing it with its perturbed counterpart. We employ Chebyshev polynomials of the first kind to solve the perturbed equation. Our findings suggest that this technical approach is superior to the regularization method of Tikhonov. It is simpler, less cumbersome, and this simplicity is demonstrated through various examples.

A velocity evaluation method for in-pipe metallic flow through inversion of magnetic field perturbation measured surrounding the pipe

It is important to measure the global and/or local velocity of an in-pipe metallic flow to control its running state in applications such as a Tokamak fusion reactor. The magnetic field outside the pipe wall will be perturbed by the motion induced eddy current when the liquid metal flows across an applied static magnetic field. This phenomenon gives a possibility to evaluate the in-pipe velocity from the measured magnetic field perturbation signals. In this paper, a non-intrusive velocity evaluation method is proposed accordingly for measuring the velocity of liquid metal through measurement and inversion of the magnetic field surrounding the pipe. An efficient forward simulation method to calculate the magnetic field near a metallic flow in a static environmental magnetic field is developed at first. An inversion scheme based on the singular value decomposition and the L-curve method is then proposed to reconstruct the velocity distribution at a pipe cross-section with the linear equations correlating the flow velocity and the magnetic field regulated using the Tikhonov method. The reconstruction results of pipe flows of different velocity modes verified the feasibility and efficiency of the proposed velocity measurement method for in-pipe metallic flows.

On inertial iterated Tikhonov methods for solving ill-posed problems

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: (i) a 2D inverse potential problem, (ii) an image deblurring problem; the computational efficiency of the method is compared with standard implementations of the iT method.

Application of Regularization Methods in the Sky Map Reconstruction of the Tianlai Cylinder Pathfinder Array

The Tianlai cylinder pathfinder is a radio interferometer array to test 21 cm intensity mapping techniques in the post-reionization era. It works in passive drift scan mode to survey the sky visible in the northern hemisphere. To deal with the large instantaneous field of view and the spherical sky, we decompose the drift scan data into m-modes, which are linearly related to the sky intensity. The sky map is reconstructed by solving the linear interferometer equations. Due to incomplete uv coverage of the interferometer baselines, this inverse problem is usually ill-posed, and regularization method is needed for its solution. In this paper, we use simulation to investigate two frequently used regularization methods, the Truncated Singular Value Decomposition (TSVD), and the Tikhonov regularization techniques. Choosing the regularization parameter is very important for its application. We employ the generalized cross validation method and the L-curve method to determine the optimal value. We compare the resulting maps obtained with the different regularization methods, and for the different parameters derived using the different criteria. While both methods can yield good maps for a range of regularization parameters, in the Tikhonov method the suppression of noisy modes are more gradually applied, produce more smooth maps which avoids some visual artefacts in the maps generated with the TSVD method.

Simultaneous determination of 2-D distributions of temperature and absorption coefficient in large-scale pulverized-coal-fired boilers by flame images processing

This paper proposes a new algorithm that can simultaneously measure the two-dimensional temperature distribution and absorption coefficient distribution of a pulverized-coal-fired (PCF) boiler furnace. On the base of optimal estimation of the uniform absorption and scattering coefficients as well as 2-D temperature distribution from the color radiation images of the flame by the Tikhonov regularization algorithm, the product of the absorption coefficient and the absolute blackbody emission intensity forms a new unknown distribution of emission source, which is also reconstructed from the color radiation images of the flame by the Tikhonov method. Then the distribution of absorption coefficient can be obtained from the new distribution of emission source as the 2-D temperature distribution has been known. Experimental investigation has been conducted on a 660 MWe four corner, tangentially firing, PCF boiler. In the flame detectors, a three-band pass filter is added in the CCD industrial camera to improve the monochromaticity of the original flame image. The temperature and absorption coefficient distributions of the cross-sections at three detection layers under a load of 330 MW and at the highest layer under a load of 500 MW are obtained. In order to verify the reliability of the algorithm, the flame radiation intensity images calculated from the 2-D distributions of temperature and absorption coefficient reconstructed at the highest layer under the load of 330 MW are taken as measurement data, the algorithm is repeated and the results show that the 2-D temperature distribution reconstructed is similar to the input one, with relative error less than 1% between their mean temperatures, and relative error less than 10% between the reconstructed and input mean values of absorption and scattering coefficients. It demonstrates that the algorithm and technology proposed in this paper have important application prospects in the monitoring, diagnosis and control of PCF combustion processes.

Integrating Learning-based Priors with Physics-based Models in Ultrasound Elasticity Reconstruction.

Ultrasound elastography images which enable quantitative visualization of tissue stiffness can be reconstructed by solving an inverse problem. Classical model-based methods are usually formulated in terms of constrained optimization problems. To stabilize the elasticity reconstructions, regularization techniques such as Tikhonov method are used with the cost of promoting smoothness and blurriness in the reconstructed images. Thus, incorporating a suitable regularizer is essential for reducing the elasticity reconstruction artifacts while finding the most suitable one is challenging. In this work, we present a new statistical representation of the physical imaging model which incorporates effective signal-dependent colored noise modeling. Moreover, we develop a learning-based integrated statistical framework which combines a physical model with learning-based priors. We use a dataset of simulated phantoms with various elasticity distributions and geometric patterns to train a denoising regularizer as the learning-based prior. We use fixed-point approaches and variants of gradient descent for solving the integrated optimization task following learning-based plug-and-play (PnP) prior and regularization by denoising (RED) paradigms. Finally, we evaluate the performance of the proposed approaches in terms of relative mean square error (RMSE) with nearly 20% improvement for both piece-wise smooth simulated phantoms and experimental phantoms compared to the classical model-based methods and 12% improvement for both spatially-varying breast-mimicking simulated phantoms and an experimental breast phantom, demonstrating the potential clinical relevance of our work. Moreover, the qualitative comparisons of reconstructed images demonstrate the robust performance of the proposed methods even for complex elasticity structures that might be encountered in clinical settings.

Intelligent Particle Swarm Optimization Method for Parameter Selecting in Regularization Method for Integral Equation

We use the Tikhonov method as a regularization technique for solving the integral equation of the first kind with noisy and noise-free data. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Intelligent Piratical Swarm Optimization (IPOS) technique. The effectiveness of combining these two approaches IPOS and Tikhonov is demonstrated to be highly practicable.

Construction by the Tikhonov Method of a Nonzero Solution of the Homogeneous Cauchy Problem for one Equation with a Fractional Derivative

Construction by the Tikhonov Method of a Nonzero Solution of the Homogeneous Cauchy Problem for one Equation with a Fractional Derivative

Brain perfusion estimation by Tikhonov model-free deconvolution in a long axial field of view PET/CT scanner exploring five different PET tracers

PurposeNew total-body PET scanners with a long axial field of view (LAFOV) allow for higher temporal resolution due to higher sensitivity, which facilitates perfusion estimation by model-free deconvolution. Fundamental tracer kinetic theory predicts that perfusion can be estimated for all tracers despite their different fates given sufficiently high temporal resolution of 1 s or better, bypassing the need for compartment modelling. The aim of this study was to investigate whether brain perfusion could be estimated using model-free Tikhonov generalized deconvolution for five different PET tracers, [15O]H2O, [11C]PIB, [18F]FE-PE2I, [18F]FDG and [18F]FET. To our knowledge, this is the first example of a general model-free approach to estimate cerebral blood flow (CBF) from PET data.MethodsTwenty-five patients underwent dynamic LAFOV PET scanning (Siemens, Quadra). PET images were reconstructed with an isotropic voxel resolution of 1.65 mm3. Time framing was 40 × 1 s during bolus passage followed by increasing framing up to 60 min. AIF was obtained from the descending aorta. Both voxel- and region-based calculations of perfusion in the thalamus were performed using the Tikhonov method. The residue impulse response function was used to estimate the extraction fraction of tracer leakage across the blood–brain barrier.ResultsCBF ranged from 37 to 69 mL blood min−1 100 mL of tissue−1 in the thalamus. Voxelwise calculation of CBF resulted in CBF maps in the physiologically normal range. The extraction fractions of [15O]H2O, [18F]FE-PE2I, [11C]PIB, [18F]FDG and [18F]FET in the thalamus were 0.95, 0.78, 0.62, 0.19 and 0.03, respectively.ConclusionThe high temporal resolution and sensitivity associated with LAFOV PET scanners allow for noninvasive perfusion estimation of multiple tracers. The method provides an estimation of the residue impulse response function, from which the fate of the tracer can be studied, including the extraction fraction, influx constant, volume of distribution and transit time distribution, providing detailed physiological insight into normal and pathologic tissue.

Enhancing Kinematic Calibration Accuracy for Parallel Manipulators Based on Truncated Total Least-Square Regularization

Abstract This article proposes a novel kinematic calibration approach to address the ill-conditioned identification matrix problem and enhance the accuracy of parallel manipulators (PMs). The kinematic calibration approach applies the truncated total least-square (T-TLS) regularization method to the inverse kinematic calibration of PMs and does not have the unit inconsistency issue and configuration selection issue. The performance of the T-TLS method in the inverse kinematic calibration of PMs that have the ill-conditioned identification matrix problem is compared to that of classical regularization methods, including the truncated singular value decomposition (TSVD) method and Tikhonov method, in simulations. Then, the kinematic calibration approach is applied to a 3-PSS/S spherical PM prototype in the real world. For the 3-PSS/S spherical PM prototype, the kinematic calibration approach can reduce the absolute angular errors of the end-effector from over 1.0 deg before calibration to less than 0.1 deg after calibration.