Continuous-time image reconstruction using differential equations for computed tomography

An approach for reconstructing tomographic images based on the idea of continuous dynamical methods is presented. The method consists of a continuous-time image reconstruction (CIR) system described by differential equations for solving linear inverse problems. We theoretically demonstrate that the trajectories converge to a least squares solution to the linear inverse problem. An implementation of its equivalent electronic circuit is significantly faster than conventional discrete-time image reconstruction (DIR) systems executed in a digital computer. Moreover, the merits of our CIR are demonstrated on a tomographic inverse problem where simulated noisy projection data are generated from a known phantom. Here, we numerically demonstrate that the CIR system does not produce unphysical negative pixel values if one starts out with positive initial values. Besides, CIR also recovers the phantom with almost the same quality as DIR images.     

Stability results for uniquely determined sets from two directions in discrete tomography

In this paper we prove several new stability results for the reconstruction of binary images from two projections. We consider an original image that is uniquely determined by its projections and possible reconstructions from slightly different projections. We show that for a given difference in the projections, the reconstruction can only be disjoint from the original image if the size of the image is not too large. We also prove an upper bound for the size of the image given the error in the projections and the size of the intersection between the image and the reconstruction.     

Multi-resolution method for binary tomography

Multi-resolution and region-growing strategies have been successfully used in several fields of image processing. In this paper we investigate how these two strategies can be applied for binary tomography. We included these strategies into a reconstruction method using simulated annealing and tested these new methods on different images.     

Performance analysis of a distributed fixed-step power control algorithm via window concept in cellular mobile systems

In cellular mobile systems, the received carrier-to-interference ratio (CIR) can be maintained within the desirable range provided that the path gain remains approximately constant over a number of consecutive power control steps. However, when channels suffer short-term fading, it is not clear whether existing power control algorithms remain convergent. This paper proposes a distributed fixed-step power control algorithm with binary feedback via window concept for cellular mobile systems. The essence of the proposed algorithm is that the power control step size can be regulated by window size. The performance of the proposed scheme is analyzed in short-term fading channels. A sufficient condition for system stability is derived using a simple received CIR model and a power control window. It is shown herein that the bound of the received CIR of each user varies as a function of the target CIR, the size of the power control step and the link gain. The analysis and simulation results show that if the step size is properly set according to the window size, the proposed algorithm can achieve a small convergence region and a fast convergence rate.     

Complexity results for reconstructing binary images with disjoint components from horizontal and vertical projections

The problem of reconstructing a special class of binary images from their horizontal and vertical projections is considered. We present a general framework for analyzing the worst case complexity of this task if the image consists of more than one pairwise disjoint component. Applying the presented technique we analyze the complexity of reconstructing canonical hv-convex binary images. We also present parameterized complexity results on general and so-called glued hv-convex images. Moreover, we study how our results are related to the reconstruction of permutation matrices from four projections.     

Bounds on the quality of reconstructed images in binary tomography

Binary tomography deals with the problem of reconstructing a binary image from its projections. In particular, there is a focus on highly underdetermined reconstruction problems for which many solutions may exist. In such cases, it is important to have a quality measure for the reconstruction with respect to the unknown original image.     

Inverse conductivity Problem with Internal Data

This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameter-dependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.     

Early warning detection of runaway initiation using non-linear approaches

We present a method for early detection of runaway initiation in chemical reactors using only temperature measurements and based on the calculation of the divergence of the system. The method is based on state space reconstruction techniques and is illustrated using simulated as well as experimental datasets. The results show that the method is able to distinguish between runaway and non-runaway situations and it does not produce false alarms during controlled heating/cooling experiments.     

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.     

MR image reconstruction from undersampled data by using the iterative refinement procedure

Magnetic resonance imaging (MRI) reconstruction from sparsely sampled data has been a difficult problem in medical imaging field. We approach this problem by formulating a cost functional that includes a constraint term that is imposed by the raw measurement data in k-space and the L¹ norm of a sparse representation of the reconstructed image. The sparse representation is usually realized by total variational regularization and/or wavelet transform. We have applied the Bregman iteration to minimize this functional to recover finer scales in our recent work. Here we propose nonlinear inverse scale space methods in addition to the iterative refinement procedure. Numerical results from the two methods are presented and it shows that the nonlinear inverse scale space method is a more efficient algorithm than the iterated refinement method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear PDE based numerical methods for cell tracking in zebrafish embryogenesis

The paper presents numerical algorithms leading to an automated cell tracking and reconstruction of the cell lineage tree during the first hours of animal embryogenesis. We present results obtained for large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. Our approach consists of three basic steps – the image filtering, the cell centers detection and the cell trajectories extraction yielding the lineage tree reconstruction. In all three steps we use nonlinear partial differential equations. For the filtering the geodesic mean curvature flow in level set formulation is used, for the cell center detection the motion of level sets by a constant speed regularized by mean curvature flow is used and the solution of the eikonal equation is essential for the cell trajectories extraction. The core of our new tracking method is an original approach to cell trajectories extraction based on finding a continuous centered paths inside the spatio-temporal tree structures representing cell movement and divisions. Such paths are found by using a suitably designed distance function from cell centers detected in all time steps of the 3D+time image sequence and by a backtracking in the steepest descent direction of a potential field based on this distance function. We also present efficient and naturally parallelizable discretizations of the aforementioned nonlinear PDEs and discuss properties and results of our new tracking method on artificial and real 4D data.     

Parallel Block-Iterative Reconstruction Algorithms for Binary Tomography

A convex programming approach to binary tomographic image reconstruction in noisy environments is proposed. Conventional constraints are mixed with new constraints on the sinogram. A convex objective is then minimized over the resulting feasibility set via a parallel block-iterative method. The new constraints involve noise-based confidence regions and a binarity-promoting total variation constraint.     

Generic iterative subset algorithms for discrete tomography

Discrete tomography deals with the reconstruction of images from their projections where the images are assumed to contain only a small number of grey values. In particular, there is a strong focus on the reconstruction of binary images (binary tomography). A variety of binary tomography problems have been considered in the literature, each using different projection models or additional constraints. In this paper, we propose a generic iterative reconstruction algorithm that can be used for many different binary reconstruction problems. In every iteration, a subproblem is solved based on at most two of the available projections. Each of the subproblems can be solved efficiently using network flow methods. We report experimental results for various reconstruction problems. Our results demonstrate that the algorithm is capable of reconstructing complex objects from a small number of projections.     

A Penalization Approach for Tomographic Reconstruction of Binary Axially Symmetric Objects

We propose a variational method for tomographic reconstruction of blurred and noised binary images based on a penalization process of a minimization problem settled in the space of bounded variation functions. We prove existence and/or uniqueness results and derive a penalized optimality system. Numerical simulations are provided to demonstrate the relevance of the approach.     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

On a nonlocal model of image segmentation

We understand an image as binary grey ‘alloy’ of a black and a white component and use a nonlocal phase separation model to describe image segmentation. The model consists in a degenerate nonlinear parabolic equation with a nonlocal drift term additionally to the familiar Perona-Malik model. We formulate conditions for the model parameters to guarantee global existence of a unique solution that tends exponentially in time to a unique steady state. This steady state is solution of a nonlocal nonlinear elliptic boundary value problem and allows a variational characterization. Numerical examples demonstrate the properties of the model.Dedicated to Klaus Kirchgässner on the occasion of his 70th birthdayReceived: November 12, 2002; revised: April 8, 2003     

Detection of Subsurface Bubbles with Discrete Electromagnetic Geotomography

In electromagnetic geotomography, the inversion is very ill-posed due to the limita- tion of an angular range of projections, and consequently the reconstructed image is different than the true image. Nevertheless, the reconstruction can be much im- proved if we are interested only in a piece of information on the interior volume of the examined object, e.g. detection of some anomalies (bubbles). We assume that the image can be represented by binary values, hence: Discrete Electromagnetic Geotomography (DEG). In this paper, we discuss two discrete image reconstruction algorithms. The first one uses the D.C. programming to find the integer solution to the penalized least-squares problem. The penalty term is based on the Gibbs distribution. In the second one, the mean field annealing is applied to maximize the Gibbs-Boltzmann distribution associated with the penalized least-squares function. The numerical results are presented for noise-free and noisy data.     

基于相空间重构的降噪技术研究

探讨了基于相空间重构的局部线性映射算法在非线性时间序列降噪技术中的应用,并给出了算法中主要参数的选取方法.实验结果表明,该算法的降噪效果明显优于传统的线性信号滤波技术.并且针对多数实测数据的原始动态模型未知的特点,提出通过计算降噪前后时序信号的关联维数作为评判降噪效果的工具,克服了已有方法中无法计算该类时序信号降噪水平的缺点.     

Control reconstruction under changes of part of the coordinates of a nonlinear delay dynamical system

We consider the problem of dynamic reconstruction of a variable input of a nonlinear delay system on the basis of an inexact measurement of part of the phase vector. We present a solution algorithm on the basis of the method of auxiliary control models.