Problem Of Dynamic Reconstruction Research Articles

Weak* Approximations to the Solution of a Dynamic Reconstruction Problem

Weak* Approximations to the Solution of a Dynamic Reconstruction Problem

To the dynamic reconstruction problem with non-convex geometrical restrictions on the controls

To the dynamic reconstruction problem with non-convex geometrical restrictions on the controls

Using the Navier-Cauchy equation for motion estimation in dynamic imaging

<p style='text-indent:20px;'>Tomographic image reconstruction is well understood if the specimen being studied is stationary during data acquisition. However, if this specimen changes its position during the measuring process, standard reconstruction techniques can lead to severe motion artefacts in the computed images. Solving a dynamic reconstruction problem therefore requires to model and incorporate suitable information on the dynamics in the reconstruction step to compensate for the motion.</p><p style='text-indent:20px;'>Many dynamic processes can be described by partial differential equations which thus could serve as additional information for the purpose of motion compensation. In this article, we consider the Navier-Cauchy equation which characterizes small elastic deformations and serves, for instance, as a simplified model for respiratory motion. Our goal is to provide a proof-of-concept that by incorporating the deformation fields provided by this PDE, one can reduce the respective motion artefacts in the reconstructed image. To this end, we solve the Navier-Cauchy equation prior to the image reconstruction step using suitable initial and boundary data. Then, the thus computed deformation fields are incorporated into an analytic dynamic reconstruction method to compute an image of the unknown interior structure. The feasibility is illustrated with numerical examples from computerized tomography.</p>

Weak* Solution to a Dynamic Reconstruction Problem

Weak* Solution to a Dynamic Reconstruction Problem

Weak* Solution to a Dynamic Reconstruction Problem

Рассматривается задача динамической реконструкции управлений для детерминированных управляемых аффинных систем. Реконструкция производится в реальном времени по известным дискретным неточным замерам наблюдаемой траектории системы, порождаемой неизвестным измеримым управлением со значениями из заданного компактного множества. Приводится корректная постановка задачи реконструкции в слабом со звездой смысле, и предлагается ее решение с помощью вариационного подхода, развиваемого авторами. Этот подход использует вспомогательные вариационные задачи с выпукло-вогнутым лагранжианом, регуляризированным по Тихонову. При этом решение задачи реконструкции сводится к интегрированию гамильтоновых систем обыкновенных дифференциальных уравнений. Приведены условия согласования параметров аппроксимации (параметров точности и частоты замеров траектории, а также вспомогательного регуляризирующего параметра). Показано, что при выполнении этих условий реконструированные управления ограничены, а порождаемые ими траектории динамической системы равномерно сходятся к наблюдаемой траектории.

Dynamic Right-Hand Side Reconstruction Problem for a System of Fractional Differential Equations

For a controlled dynamical system described by nonlinear fractional differential equations, we consider the problem of dynamic reconstruction of an unknown disturbance from approximate measurements of the trajectory. For this problem, we construct an algorithm based on Krasovskii's extremal shift method and find its convergence rate.

The dynamical discrepancy method in problems of reconstructing unknown characteristics of a second-order system

This paper considers two problems of dynamical reconstruction of unknown characteristics of a system of nonlinear equations describing the process of innovation diffusion through inaccurate measurements of phase states. A dynamical variant for solving these problems is designed. The system is assumed to operate on a given finite time interval. The evolution of the system's phase state, i.e., the solution of the system, is determined by an unknown input. A precise reconstruction of the real input (acting on the system) is, generally speaking, impossible due to inaccurate measurements. Therefore, some approximation to this input is constructed which provides an arbitrary smallness to the real input if the measurement errors and the step of incoming information are sufficiently small. Based on the dynamical version of the discrepancy method, two algorithms for solving the problems in question are specified. One of them is oriented to the case of measuring all coordinates of the phase vector, and the other, to the case of incomplete measurements. The algorithms suggested are stable with respect to informational noises and computational errors. Actually, they are special regularizing algorithms from the theory of dynamic inverse problems.

Hamiltonian Systems in Dynamic Reconstruction Problems

Abstract We consider dynamic reconstruction (DR) problems for controlled dynamical systems linear in controls and nonlinear in state variables as inaccurate current information about real motions is known. A solution of this on-line inverse problem is obtained with the help of auxiliary problems of calculus of variations (CV) for integral discrepancy functionals. Key elements of the constructions are solutions of hamiltonian systems obtained with the help of optimality conditions for the CV problems. An illustrating example is exposed.

Deep reconstruction model for dynamic PET images.

Accurate and robust tomographic reconstruction from dynamic positron emission tomography (PET) acquired data is a difficult problem. Conventional methods, such as the maximum likelihood expectation maximization (MLEM) algorithm for reconstructing the activity distribution-based on individual frames, may lead to inaccurate results due to the checkerboard effect and limitation of photon counts. In this paper, we propose a stacked sparse auto-encoder based reconstruction framework for dynamic PET imaging. The dynamic reconstruction problem is formulated in a deep learning representation, where the encoding layers extract the prototype features, such as edges, so that, in the decoding layers, the reconstructed results are obtained through a combination of those features. The qualitative and quantitative results of the procedure, including the data based on a Monte Carlo simulation and real patient data demonstrates the effectiveness of our method.

Reconstruction of Undersampled Big Dynamic MRI Data Using Non-Convex Low-Rank and Sparsity Constraints.

Dynamic magnetic resonance imaging (MRI) has been extensively utilized for enhancing medical living environment visualization, however, in clinical practice it often suffers from long data acquisition times. Dynamic imaging essentially reconstructs the visual image from raw (k,t)-space measurements, commonly referred to as big data. The purpose of this work is to accelerate big medical data acquisition in dynamic MRI by developing a non-convex minimization framework. In particular, to overcome the inherent speed limitation, both non-convex low-rank and sparsity constraints were combined to accelerate the dynamic imaging. However, the non-convex constraints make the dynamic reconstruction problem difficult to directly solve through the commonly-used numerical methods. To guarantee solution efficiency and stability, a numerical algorithm based on Alternating Direction Method of Multipliers (ADMM) is proposed to solve the resulting non-convex optimization problem. ADMM decomposes the original complex optimization problem into several simple sub-problems. Each sub-problem has a closed-form solution or could be efficiently solved using existing numerical methods. It has been proven that the quality of images reconstructed from fewer measurements can be significantly improved using non-convex minimization. Numerous experiments have been conducted on two in vivo cardiac datasets to compare the proposed method with several state-of-the-art imaging methods. Experimental results illustrated that the proposed method could guarantee the superior imaging performance in terms of quantitative and visual image quality assessments.

An algorithm for dynamic reconstruction of a disturbance in a linear system

We study the problem of dynamic reconstruction of an unknown disturbance in a linear system whose states are measured with some error. We propose a solution algorithmfor this problem under the assumption that the observed time interval is sufficiently large. The algorithm is based on the method of auxiliary positional control models.

Radial-basis-function level-set-based regularized Gauss–Newton-filter reconstruction scheme for dynamic shape tomography

The dynamic reconstruction problem in tomographic imaging is encountered in several applications, such as species determination, the study of blood flow through arteries/veins, motion compensation in medical imaging, and process tomography. The reconstruction method of choice is the Kalman filter and its variants, which, however, are faced by issues of filter tuning. In addition, since the time-propagation models of physical parameters are typically very complex, most of the time, a random walk model is considered. For geometric deformations, affine models are typically used. In our work, with the objectives of minimizing tuning issues and reconstructing time-varying geometrically deforming features of interest with affine in addition to pointwise-normal scaling motions, a novel level-set-based reconstruction scheme for ray tomography is proposed for shape and electromagnetic parameters using a regularized Gauss-Newton-filter-based scheme. We use an implicit Hermite-interpolation-based radial basis function representation of the zero level set corresponding to the boundary curve. Another important contribution of the paper is an evaluation of the shape-related Frechet derivatives that does not need to evaluate the pointwise Jacobian (the ray-path matrix in our ray-tomography problem). Numerical results validating the formulation are presented for a straight ray-based tomographic reconstruction. To the best of our knowledge, this paper presents the first tomographic reconstruction results in these settings.

Dynamic reconstruction for nonlinear equations describing the process of innovation diffusion

The problem of dynamic reconstruction of an unknown coordinate in a system of nonlinear equations describing the process of innovation diffusion from results of observations of phase state is considered. An algorithm for solving this problem, which is stable with respect to informational noises and computational errors, is suggested. The algorithm is based on the principle of auxiliary models with adaptive controls. Mathematics Subject Classification: 34A55, 49N45

On reconstructing an unknown coordinate of a nonlinear system of differential equations

The paper discusses a method of auxiliary controlled models and the application of this method to solving problems of dynamical reconstruction of an unknown coordinate in a nonlinear system of dierential equations. The solving algorithm, which is stable with respect to informational noises and computational errors, is presented.

On an algorithm for dynamic reconstruction of the input

We consider the problem of dynamic reconstruction of the input in a system described by a vector differential equation and nonlinear in the state variable. We indicate an algorithm that is stable under information noises and computational errors and is aimed at infinite system operation time. The algorithm is based on the dynamic regularization method.

A New Homotopy Perturbation Method for Solving an Ill-Posed Problem of Multi-Source Dynamic Loads Reconstruction

A New Homotopy Perturbation Method for Solving an Ill-Posed Problem of Multi-Source Dynamic Loads Reconstruction

On dynamic reconstruction of input in the second-order linear systems

Consideration was given to the problem of dynamic reconstruction of input in the second-order linear systems. Procedures for its real-time solution were specified. They feature dynamicity, which means that the current values of the approximated input are generated in real time, and stability, which means that for a sufficiently small measurement error approximation is arbitrarily precise.

On reconstruction of boundary controls in a parabolic equation

A problem of dynamical reconstruction of boundary controls in a nonlinear parabolic equation is considered. In the case when a control is concentrated in the Neumann boundary conditions, a solving algorithm which is stable with respect to informational noises and computational errors is described. The algorithm is based on the ideas of the theory of feedback control.

On one algorithm of input action reconstruction for linear systems

The problem of dynamic reconstruction of the variable input of a linear system is considered. In the case when observations of some characteristics of the phase trajectory of the system are subject to uncertainty, an algorithm for solving this problem, based on control method with a model, is specified. The algorithm is robust to information noise and computational errors. General constructions are illustrated by an example.

Method of controlled models in the problem of reconstructing a boundary input

The problem of dynamic reconstruction of boundary controls in a nonlinear parabolic equation is considered. In the case of a control concentrated in the Neumann boundary conditions, a solution algorithm is described, which is stable with respect to the information noise and calculation errors. The algorithm is based on the construction of feedback-controlled auxiliary models.