An Iterative Algorithm for Angle-limited Three-dimensional Image Reconstruction

We establish an improved GP iterative algorithm for the extrapolation of band-limited function to fully 3-dimensional image reconstruction by the convolution-backprojection algorithm. Numerical experiments demonstrate that the image resolving power of IGP algorithm is better than that of the original GP algorithm for noisy data.     

A Direct convolution algorithm for image reconstruction

In image reconstruction algorithms, the choices of filter functions and interpolating functions are very important for the computational speed and the quality of the image reconstructed, especially, for fan-beam geometry, the occurrence of the singular integral operator may lead tosome great oscillations compared to the original image. In this paper we will give a direct convolu-tion algorithm which needs not the complex computations occuring in the Fourier transform, then using a circle integral we obtain a stable computational program. Different from all other previouswindow functions used by many pioneer researchers, in our algorithm we choose a window func tion similar to Gabor‘s window function e-x^2/2 , which can be regarded as the approximation to the inverse Fourier transform of a locally integrable frequency function. Also we point out that such reconstruction algorithm procedures can be used to deal with the SPECT projection data with constant attenuation.     

An algorithm for dynamic reconstruction of the right-hand side of a second-order equation with distributed parameters

The dynamic reconstruction of the right-hand side of a second-order differential equation is considered. A solution algorithm is proposed that is robust to information noise and computational errors. The algorithm is constructed using dynamic inversion theory.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

A multi-scale image reconstruction algorithm for electrical capacitance tomography

Electrical capacitance tomography (ECT) is considered as a promising process tomography (PT) technology, and its successful applications depend mainly on the precision and speed of the image reconstruction algorithms. In this paper, based on the wavelet multi-scale analysis method, an efficient image reconstruction algorithm is presented. The original inverse problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the smallest scale. At different scales, the inverse problem is solved by a generalized regularized total least squares (TLS) method, which is developed using a combinational minimax estimation method and an extended stabilizing functional, until the solution of the original inverse problem is found. The homotopy algorithm is employed to solve the objective functional. The proposed algorithm is tested by the noise-free capacitance data and the noise-contaminated capacitance data, and excellent numerical performances and satisfactory results are observed. In the cases considered in this paper, the reconstruction results show remarkable improvement in the accuracy. The spatial resolution of the reconstructed images by the proposed algorithm is enhanced and the artifacts in the reconstructed images can be eliminated effectively. As a result, a promising algorithm is introduced for ECT image reconstruction.     

A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction

In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while the subproblem is solved by the truncated conjugate gradient method, which has been well developed for well-posed problems. The trust region method, due to its robustness in global convergence, seems to be a promising way to deal with ill-posed problems.     

An efficient self-adaptive model for chaotic image encryption algorithm

In this paper, an efficient self-adaptive model for chaotic image encryption algorithm is proposed. With the help of the classical structure of permutation-diffusion and double simple two-dimensional chaotic systems, an efficient and fast encryption algorithm is designed. However, different from most of the existing methods which are found insecure upon chosen-plaintext or known-plaintext attack in the process of permutation or diffusion, the keystream generated in both operations of our method is dependent on the plain-image. Therefore, different plain-images will have different keystreams in both processes even just only a bit is changed in the plain-image. This design can solve the problem of fixed chaotic sequence produced by the same initial conditions but for different images. Moreover, the operation speed is high because complex mathematical methods, such as Runge–Kutta method, of solving the high-dimensional partial differential equations are avoided. Numerical experiments show that the proposed self-adaptive method can well resist against chosen-plaintext and known-plaintext attacks, and has high security and efficiency.     

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)     

Fixing and extending some recent results on the ADMM algorithm

Numerical Algorithms - We investigate the techniques and ideas used in Shefi and Teboulle (SIAM J Optim 24(1), 269–297, 2014) in the convergence analysis of two proximal ADMM algorithms for...     

An algorithm for surface reconstruction from planar contours using smoothing splines

This paper presents a fast algorithm for constructing a smooth three-dimensional surface over a set of cross-sectional contours. We assume that these sections are perpendicular to the z-axis and first consider the case that the surface can be represented in cylindrical coordinates. An approximation is then determined for r(θ, z) by using tensor product splines which satisfy certain boundary constraints. The algorithm is an extension of an existing semi-automatic surface fitting algorithm. The knots of the spline are chosen automatically but a single parameter is expected to control the tradeoff between closeness of fit and smoothness of fit.Both open and closed surfaces can be represented. In particular we demonstrate the use of a non-linear transformation for obtaining smooth closed surfaces.The algorithm can easily be extended to the reconstruction of surfaces which cannot be represented in cylindrical coordinates. A number of applications are also briefly discussed such as the calculation of volumes and the intersection with other surfaces. We have applied the method in practice to obtain a 3-D integrated image of the cerebral blood vessels and CT view of tumor for stereotactic neurosurgery.     

A general framework for ADMM acceleration

Numerical Algorithms - The Alternating Direction Multipliers Method (ADMM) is a very popular algorithm for computing the solution of convex constrained minimization problems. Such problems are...     

Continuous-time image reconstruction for binary tomography

Binary tomography is the process of reconstructing a binary image from a finite number of projections. We present a novel method for solving binary tomographic inverse problems using a continuous-time image reconstruction (CIR) system described by nonlinear differential equations based on the minimization of a double Kullback–Leibler divergence. We prove theoretically that the divergence measure monotonically decreases in time. Moreover, we demonstrate numerically that the quality of the reconstructed images of the nonlinear CIR system is better than those from an iterative reconstruction method.     

基于Arnold变换的图像逆置乱算法

针对Arnold变换的周期依赖于图像的阶数这一特性,提出了一种反变换算法.该算法通过分析加密图像任一点处两坐标分量间关系,得到原图像相应点的坐标,从而实现图像的解密.该反变换也可作为图像置乱的正变换,相应的反变换就是Arnold变换.在此基础上,把二维反变换算法推广到m维的情形.实验结果表明,对于已应用Arnold变换进行预处理的置乱图像,在无须计算原图像变换周期的前提下可快速实现图像的逆置乱,该过程具有确定性,其迭代次数与预处理置乱次数相等.     

An alternative approach to the image reconstruction for parallel data acquisition in MRI

Magnetic resonance imaging with parallel data acquisition requires algorithms for reconstructing the patient's image from a small number of measured k‐space lines. In contrast to well‐known algorithms like SENSE and GRAPPA and its flavours we consider the problem as a non‐linear inverse problem. Fast computation algorithms for the necessary Fréchet derivative and reconstruction algorithms are given. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(?nL){O(\sqrt{n}L)} for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.     

Generalized symmetric ADMM for separable convex optimization

The alternating direction method of multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a generalized symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of p block variables while the other has q block variables, where \(p \ge 1\) and \(q \ge 1\) are two integers. The two grouped variables are updated in a Gauss–Seidel scheme, while the variables within each group are updated in a Jacobi scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that GS-ADMM is globally convergent with a worst-case \({\mathcal {O}}(1/t)\) ergodic convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible and attractive on choosing and using larger stepsizes of the dual variable. Besides, two special cases of GS-ADMM, which allows using zero penalty terms, are also discussed and analyzed. Compared with several state-of-the-art methods, preliminary numerical experiments on solving a sparse matrix minimization problem in the statistical learning show that our proposed method is effective and promising.