The diagonalized contrast source inversion approach for elastic wave inversion

This study focuses on the inverse scattering of objects embedded in a homogeneous elastic background. The medium is probed by ultrasonic sources, and the scattered fields are observed along a receiver array. The goal is to retrieve the shape, location, and constitutive parameters of the objects through an inversion procedure. The problem is formulated using a vector integral equation. As is well-known, this inverse scattering problem is nonlinear and ill-posed. In a realistic configuration, this nonlinear inverse scattering problem involves a large number of unknowns, hence the application of full nonlinear inversion approaches such as Gauss-Newton or nonlinear gradient methods might not be feasible, even with present-day computer power. Hence, in this study we use the so-called diagonalized contrast source inversion (DCSI) method in which the nonlinear problem is approximately transformed into a number of linear problems. We will show that, by using a three-step procedure, the nonlinear inverse problem can be handled at the cost of solving three constrained linear inverse problems. The robustness and efficiency of this approach is illustrated using a number of synthetic examples.     

Imaging strongly scattering media using a multiple frequency distorted Born iterative method

The distorted Born iterative (DBI) method is a powerful approach for solving the inverse scattering problem for ultrasound tomographic imaging. This method alternates between solving the inverse scattering problem for the scattering function and the forward scattering problem for the total field and the inhomogeneous Green's function. The algorithm is initialized using the basic Born inverse solution. One fundamental problem is the algorithm diverges for strongly scattering media. This is caused by the limitation of the Born assumption in estimating the initial step of the algorithm. We present a multiple frequency DBI approach to alleviate this problem, thus extending the applicability of the DBI method to the level of dealing with biological tissue. In this multiple frequency approach, a low frequency DBI-based solution, is used to initialize the algorithm at higher frequencies. The low frequency allows convergence of the algorithm to a contrast level that is close to the true level, however, with a poor spatial resolution. The high frequency improves the spatial resolution while preserving convergence because the difference between the true contrast and the initial contrast is relatively small. We present numerical simulations that demonstrate the ability of this method to reconstruct strongly scattering regions.     

超声层析成像的迭代重建

利用变形Born迭代方法,建立了超声衍射重建算法。在迭代过程中,为了解决超声逆散射问题中的非线性性,需要反复地求解前向散射方程和逆散射方程,以达到全场和未知函数的近似,较好地重建物体内部的断层图象。由于逆散射方程是一个不适定性的方程组,要用正则化方法处理方程的不适定性问题,使迭代方法收敛于问题的真实解,才能成功地应用于较高对比度物体的图象重建问题。用Picard准则对不适定问题进行了分析,给出了通过简单图形．确定模型受噪声污染情况以及正则化方法适用范围的方法。在重建实验中。对建立的图像重建算法进行了实验仿真。达到了较好的效果。     

Image reconstruction by backprojection from frequency-domain optical measurements in highly scattering media

The reconstruction of the location and optical properties of objects in turbid media requires the solution of the inverse problem. Iterative solutions to this problem can require large amounts of computing time and may not converge to a unique solution. Instead, we propose a fast, simple method for approximately solving this problem in which calculated effective absorption and reduced scattering coefficients are backprojected to create an image of the objects. We reconstructed images of objects with centimeter dimensions embedded in a diffusive medium with optical characteristics similar to those of human tissue. Data were collected by a frequency-domain spectrometer operating at 120 MHz with a laser diode light source emitting at 793 nm. Intensity and phase of the incident photon density wave were collected from linear scans at different projection angles. Although the positions of the objects are correctly identified by the reconstructed images, the optical parameters of the objects are recovered only qualitatively.     

Comparison of the born iterative method and tarantola's method for an electromagnetic time-domain inverse problem

Two methods of solving the nonlinear two-dimensional electromagnetic inverse scattering problem in the time domain are considered. These are the Born iterative method and the method originally proposed by Tarantola for the seismic reflection inverse problems. The former is based on Born-type iterations on an integral equation, whereby at each iteration the problem is linearized, and its solution is found via a regularized optimization. The latter also uses an iterative method to solve the nonlinear system of equations. Although it linearizes the problem at each stage as well, no optimization is carried out at each iteration; rather the problem as a whole is posed as a (regularized) optimization. Each method is described briefly and its computational complexity is analyzed. Tarantola's method is shown to have a lower numerical complexity compared to the Born iterative method for each iteration, but in the examples considered, required more iterations to converge. Both methods perform well when inverting a smooth profile; however, the Born iterative method gave better results in resolving localized point scatterers.     

Two-dimensional inverse problem of diffusion tomography: the approach applicable for small inclusions

A method for solving the two-dimensional inverse problems of optical diffusion tomography is proposed. The method is especially designed for the imaging of small inclusions embedded in the backgrounds of strongly scattering media. Numerical simulations show that the results are stable with respect to external noise at the boundary of the sample. The location of an inclusion is obtained with an accuracy of the order of several photon transport mean-free paths in the medium in cases both with and without noise in the scattering data used for the solution of the inverse problem.     

Solution of the Inverse Problems of Scattering of Electromagnetic Fields by Elongated Defects

We formulate the inverse problem of scattering of electromagnetic fields by thin defects and analyze numerical algorithms used for its solution. It is shown that, in the two-dimensional case, the shape of a thin defect is completely determined by the scattered field given on a certain curve for a fixed value of the wave number. For the solution of the inverse scattering problem, we propose to use the procedure of iterative regularization based on the gradient methods. We deduce expressions for the Fréchet derivative of the operator of direct scattering problem with Dirichlet conditions imposed on the surface of a scatterer.     

Ultrasound imaging using variations of the iterative Born technique [biomedical diagnosis

The iterative Born method is an inverse technique that has been used successfully in ultrasound imaging. However, the calculation cost of the standard iterative Born method is high, and parallel computation is limited to the forward problem. In this work, two methods are introduced to increase the rate of convergence of the iterative Born algorithm. These methods are tested on three different objects. The results are promising, with both algorithms giving accurate results at lower computational cost. The first method, referred to as the coarse resolution initial value (CRIV) method, uses the iterative Born algorithm for a coarse grid to quickly estimate the initial value of the object to be reconstructed. From this initial value, the final image is obtained for a finer grid with additional iterations. The cost of this method is 40% less than that of the iterative Born technique. The second method, the quadriphase source (QS) method, simultaneously uses four single sources, and object reconstruction for each is performed in parallel; the reconstruction results for all four sources then are averaged to obtain the final image. The cost of this method is 20% less than that of the standard iterative Born method. When the object to be reconstructed is of low contrast and/or has a small phase shift, the QS method is very promising because parallel computation can be used to solve both the forward and inverse problems. However, the QS method fails for high contrast objects.     

Application of the sinc basis moment method to the reconstruction of infinite circular cylinders

A solution of the ultrasonic scattering and inverse scattering problem has been obtained by solving the inhomogeneous Helmholtz wave equation by the sinc basis moment method. In this numerical study, the algorithm of S.A. Johnson and M.L. Tracy (1983) has been applied to the reconstruction of an infinite circular cylinder that is subject to an incident cylindrical wave of ultrasound and is surrounded by a homogeneous coupling medium. For weak scattering cylinders, successful reconstructions have been obtained using the known exact solution for the scattered field as the input data for the algorithm. A detailed discussion of sampling requirements for this algorithm is presented, and the threshold derived correlates well with results of a numerical study of variation of the sampling density. Effects of varying object contrast, object size, grid size, sampling density, and method of iteration are investigated. Because the algorithm is slow, optimization of computation is described.     

Temperature-modulated fluorescence tomography in a turbid media

High scattering in biological tissues makes fluorescence tomography inverse problem very challenging in thick medium. We describe an approach termed "temperature-modulated fluorescence tomography" that can acquire fluorescence images at focused ultrasound resolution. By utilizing recently emerged temperature sensitive fluorescence contrast agents, this technique provides fluorescence images with high resolution prior to any reconstruction process. We demonstrate that this technique is well suited to resolve small fluorescence targets located several centimeters deep in tissue.     

Time Domain Inverse Scattering of Buried Inhomogeneous Dielectric Cylinders

This paper presents the studies of time domain inverse scattering for a two-dimensional inhomogeneous dielectric cylinder buried in a half-space by the finite difference time domain (FDTD) method and evolutionary algorithms (EAs). For forward scattering, the FDTD method is employed to calculate the scattered E fields, while for the inverse scattering the evolutionary algorithms are utilized to determine the permittivity of the buried cylindrical scatterer with arbitrary cross section. The results obtained for different examples show that the dynamic differential evolution (DDE) algorithms outperform the non-uniform steady state genetic algorithm (NU-SSGA) variants in terms of finding best optima. The suitability and efficiency of applying these two methods for microwave imaging of 2-D inhomogeneous dielectric cylinders are examined. Moreover, when the measured scattered fields are contaminated with Gaussian noise, DDE is able to yield good reconstructed quality.     

Nonlinear diffraction tomography: The use of inverse scattering for imaging

Our recent inverse scattering work has been to derive inverse scattering theory and algorithms that can be used to process practical experimental data. The theory makes use of computation of the forward scattering solution. Therefore, an efficient forward solver is instrumental to the rapid solution of the inverse scattering problem. The advantage of the more sophisticated theory over a linear theory is that it accounts for multiple scattering effects within the scatterers which often give rise to distortions in an image. A new method to invert strong scatterers, such as metallic scatterers, is presented. © 1996 John Wiley & Sons, Inc.     

An eigenfunction method for reconstruction of large-scale and high-contrast objects

A multiple-frequency inverse scattering method that uses eigenfunctions of a scattering operator is extended to image large-scale and high-contrast objects. The extension uses an estimate of the scattering object to form the difference between the scattering by the object and the scattering by the estimate of the object. The scattering potential defined by this difference is expanded in a basis of products of acoustic fields. These fields are defined by eigenfunctions of the scattering operator associated with the estimate. In the case of scattering objects for which the estimate is radial, symmetries in the expressions used to reconstruct the scattering potential greatly reduce the amount of computation. The range of parameters over which the reconstruction method works well is illustrated using calculated scattering by different objects. The method is applied to experimental data from a 48-mm diameter scattering object with tissue-like properties. The image reconstructed from measurements has, relative to a conventional B-scan formed using a low f-number at the same center frequency, significantly higher resolution and less speckle, implying that small, high-contrast structures can be demonstrated clearly using the extended method.     

Inverse scattering and diffraction tomography in cylindrical background media

A recently developed inverse scattering method based on the distorted-wave Born approximation (DWBA) that applies to objects embedded in known background media [Inverse Probl.19, 855 (2003);20, 1307 (2004)] is implemented for the special case of circularly symmetric scatterers embedded in circularly symmetric backgrounds. The newly developed scheme is applied in a computer-simulation study of optical diffraction tomography (ODT), and the results are compared and contrasted with reconstructions obtained using the filtered backpropagation algorithm (FBP algorithm). Unlike the DWBA-based inversion algorithm, the FBP algorithm does not take into account multiple scattering within the known background, and it is found that the newly implemented scheme yields reconstructions much superior to those obtained using the FBP algorithm. The research reported applies to a number of important applications that include ultrasound nondestructive evaluation testing of cylinders for defects as well as ODT.     

Comparison of reconstruction algorithms for optical diffraction tomography

A recently developed inverse scattering algorithm [A. J. Devaney and M. Dennison, Inverse Probl., 19, 855 (2003) and M. Dennison and A. J. Devaney, Inverse Probl., 20, 1307 (2004)] is described and applied in a computer simulation study of optical diffraction tomography (ODT). The new algorithm is superior to standard ODT reconstruction algorithms, such as the filtered backpropagation algorithm, in applications employing a limited number of scattering experiments (the so-called limited-view case) and also in cases where multiple scattering occurs between the object being interrogated and the (known) background in which the object is embedded. The new algorithm is compared and contrasted with the filtered backpropagation algorithm in a computer simulation of ODT of weakly inhomogeneous cylindrical objects being interrogated in a limited number of scattering experiments employing incident plane waves. Our study has potential applications in biomedical imaging and tomographic microscopy.     

A singular-value method for reconstruction of nonradial and lossy objects

Efficient inverse scattering algorithms for nonradial lossy objects are presented using singular-value decomposition to form reduced-rank representations of the scattering operator. These algorithms extend eigenfunction methods that are not applicable to nonradial lossy scattering objects because the scattering operators for these objects do not have orthonormal eigenfunction decompositions. A method of local reconstruction by segregation of scattering contributions from different local regions is also presented. Scattering from each region is isolated by forming a reduced-rank representation of the scattering operator that has domain and range spaces comprised of far-field patterns with retransmitted fields that focus on the local region. Methods for the estimation of the boundary, average sound speed, and average attenuation slope of the scattering object are also given. These methods yielded approximations of scattering objects that were sufficiently accurate to allow residual variations to be reconstructed in a single iteration. Calculated scattering from a lossy elliptical object with a random background, internal features, and white noise is used to evaluate the proposed methods. Local reconstruction yielded images with spatial resolution that is finer than a half wavelength of the center frequency and reproduces sound speed and attenuation slope with relative root-mean-square errors of 1.09% and 11.45%, respectively.     

超声逆散射成像问题中正则化参数研究

研究了二维理想情况下,基于精确场描述的超声逆散射成像问题,先用矩量法将波动方程化为离散形式,分别用BI和DBI算法进行迭代重建。影响整个算法的一个关键因素是散射场方程的正则化求解,具有明显的不适定性。文章基于L曲线法,提出以解的范数和残差变化量的加权形式作为确定正则化参数的依据,在迭代过程根据问题不适定性程度,自适应地调整搜索范围。仿真结果表明,该算法可快速地找到最优正则化参数。     

结构逆可靠度最可能失效点的改进搜索算法

确定逆可靠度最可能失效点(MPPIR)是结构逆可靠度分析的核心问题,以改进均值法(AMV)及其改进方法应用最广泛。但当功能函数非线性程度较高或为非凸非凹函数时,AMV易出现周期振荡等不收敛问题。以现有的AMV改进方法为基础,通过迭代过程中控制搜索方向和步长,提出一种MPPIR的改进搜索算法,并结合不精确一维搜索方法给出了具体的计算流程。数值算例分析表明:提出的算法与AMV相比具有更好的收敛性,与弧长搜索法相比不需要采用优化方法确定最优步长,且对于非凸非凹功能函数以及高度非线性功能函数都具有良好的收敛性。     

Theory of total-internal-reflection tomography

A method is presented to reconstruct three-dimensional tomographic images of weakly scattering objects with subwavelength resolution. The method may be applied to data available in phase-sensitive, total-internal-reflection microscopy. The results follow from an analysis of the near-field inverse scattering problem with evanescent waves.     

Using independent component analysis for material estimation in hyperspectral images

We develop a method for automated material estimation in hyperspectral images. The method models a hyperspectral pixel as a linear mixture of unknown materials. The method is particularly useful for applications in which material regions in a scene are smaller than one pixel. In contrast to many material estimation methods, the new method uses the statistics of large numbers of pixels rather than attempting to identify a small number of the purest pixels. The method is based on maximizing the independence of material abundances at each pixel. We show how independent component analysis algorithms can be adapted for use with this problem. We demonstrate properties of the method by application to airborne hyperspectral data.