Simulation of a functional solution to the acoustic tomography problem for data from quasi-point transducers

Two variants of a functional-analytical algorithm intended for solving inverse tomography problems are discussed and numerically carried out. The acoustic fields that are transmitted and received by transducers, which are equivalent to point ones, serve as experimental data. These data are used to calculate the classical or generalized scattering amplitude, and the scatterer characteristics are then reconstructed. The algorithm requires neither model linearization, no iterations for refining the estimates of scatterers, thus making it attractive for solving acoustic-tomography problems in different applications. The results of the numerical reconstruction of inhomogeneities in the sound velocity and absorption in a medium are presented.     

Separate reconstruction of the speed of sound,density of the medium,and coefficient of absorption in tomographic problems

A procedure is proposed for separating contributions to the scatterer function by inhomogeneities of the speed of sound, density of a medium, and coefficient of absorption. The scatterer function is reconstructed by solving the inverse problem. The power index of the absorption coefficient’s frequency dependence is determined simultaneously. The resistance to interference of this procedure is investigated by simulations in the multifrequency mode.     

Reconstruction of spatial distributions of sound velocity and absorption in soft biological tissues using model ultrasonic tomographic data

A two-step algorithm is used to reconstruct the spatial distributions of the acoustic characteristics of soft biological tissues-the sound velocity and absorption coefficient. Knowing these distributions is urgent for early detection of benign and malignant neoplasms in biological tissues, primarily in the breast. At the first stage, large-scale distributions are estimated; at the second step, they are refined with a high resolution. Results of reconstruction on the base of model initial data are presented. The principal necessity of preliminary reconstruction of large-scale distributions followed by their being taken into account at the second step is illustrated. The use of CUDA technology for processing makes it possible to obtain final images of 1024 × 1024 samples in only a few minutes.     

Solution of the three-dimensional acoustic inverse scattering problem. The modified Novikov algorithm

For the first time, three-dimensional model scatterers of various strengths and size are numerically reconstructed on the basis of the monochromatic functional-analytical Novikov algorithm. The algorithm allows for the multiple scattering processes and does not impose stringent constraints on the scatterer strength. The resulting scatterer estimate approaches the true value after the width of the scatterer’s spatial spectrum is restricted to a region with a radius of about 2k ₀. The noise robustness of the algorithm, i.e., the robustness to random errors in experimental data, is sufficiently high for diagnostic applications. However, the amount of numerical operations required by the algorithm is great.     

Computation of the far-field time domain solution of the scattering problem.

This paper presents a new technique for calculating the time domain (transient) far-field scattered pressure. The scattering problem is divided in two steps; the first step evaluates the field distribution inside the scatterer, and the second step generates the far-field scattered pressure by 3-D Radon transform of these data for each time step and summing over time. The algorithm results in considerable saving in CPU time and memory by simplifying the calculation along the path from scatterer to receiver. This technique can also be used in two dimensions.     

Reconstruction of Optical Energy Deposition for Backward Optoacoustic Imaging

Visualizing optical properties, such as the optical absorption coefficient, helps us to obtain structural information of biological tissues. In this paper, we present an efficient reconstruction algorithm for optical energy deposition in backward optoacoustic imaging. Note that econstruction of optical energy deposition is the first step to imaging the optical absorption coefficient distribution. This algorithm is derived from the optoacoustic wave equations with line focusing, in which the focusing techniques were utilized to reduce the reconstruction problem from three dimensions (3-D) to one dimension (1-D). Simulations and experiments were conducted to verify efficacy of this algorithm. In the simulations, optoacoustic signals were generated based on the solution of the optoacoustic wave equations. In the experiments, a 3-D backward mode optoacoustic imaging system was built. The system consisted of a Nd YAG laser for optical irradiation and an acoustic detection system with a broadband hydrophone. A phantom was used to illustrate validity of the proposed algorithm. The results show that optical energy deposition can be efficiently reconstructed in both simulations and experiments.     

Spectral integral representations of monostatic backscattering from three-dimensional distributions of sediment volume inhomogeneities

A theory is developed for generating short time, monostatic reverberation realizations caused by three-dimensionally distributed volume inhomogeneities in stratified media. A wave number integral approach to treating the propagation to and from the scatterers, combined with a two-dimensional spectral representation of the azimuthally averaged scatterer realizations and a novel numerical implementation, combine to yield an efficient, high fidelity reverberation simulator for predicting monostatic backscatter from horizontally stratified sediments.     

Modulation-spectral method for analyzing the amplitude-phase structure of optical inhomogeneities of objects

An original solution to the phase problem in optics is considered as applied to the problems of recording and analysis of the amplitude-phase structure of optical fields used for studying fine structures and inhomogeneities in steady-state objects producing effects to fractions of the wavelength period. The problem is solved by probing objects using radiation with a known structure. Intensity distributions of the probing field are detected at the exit from the object by using the modulation-spectral method directly for the spatial frequency spectrum and for the spatial frequency spectrum subjected to additional modulation formed in a special way, which is realized in the plane under study and provides visualization of the phase information contained in the light field in some form. The intensity distributions obtained make it possible to calculate the two-dimensional amplitude-phase structure of the field analyzed and, hence, the fine structure of the optical inhomogeneities of the object analyzed for the chosen probing direction. For steady-state objects, probing in a number of directions is possible. Information on the bulk structure of the inhomogeneities under study can be obtained by using the information available on the symmetry of the object. Two variants of action of the medium on probing radiation are considered. In the first one, the action is related to spatial field modulation (described by the multiplication operation); in the second one, the action leads to redistribution of radiation in the plane studied (described by the convolution operation).     

Imaging with acoustic antenna array through a thick layer of large-scale inhomogeneities

The problem of determining the positions and levels of signals by an array in a thick layer of largescale inhomogeneities is considered. The problem is solved in two steps. At the first step, the method of wave front inversion is used to find the fractions of the spatial spectra at which one of the signals predominates. At the second step, within each spectral fraction, the signal is processed by using both the method [5] of taking the logarithms of complex functions with a subsequent filtering and the method of parameter estimation. The results of numerical modeling are presented.     

On fractal structure of large-scale electron-density inhomogeneities of traveling disturbances in the midlatitude ionosphere

We present the results of studies of the multifractal structure of slow (of duration τ ≈ 10 s) fluctuations of the received-signal amplitudes in special experiments on radio-raying of the midlatitude ionosphere by signals from orbital satellites in 2004–2006. It is shown, in particular, that the method of multifractal analysis of amplitude records of the received signals yields information on the spectrum of large-scale ionospheric inhomogeneities, which is inaccessible for the classical method of radio scintillations. From the results of measurements with the use of multifractal processing of experimental data, we found that large-scale (tens of kilometers) quasiregular electron-density inhomogeneities of traveling ionospheric disturbances (TIDs) have a power-law spectrum. It is exactly the power-law form of the spatial spectrum of large-scale inhomogeneities of TIDs that can be the reason for the observed multifractal structure of the intermittency of slow fluctuations of the received-signal amplitudes. However, under conditions of a developed small-scale turbulence of TIDs, the observed multifractal structure of the received signals is, as a rule, stipulated by the spatial inhomogeneity of the variance of the integral electron-density fluctuations of small-scale inhomogeneities on scales comparable with the sizes of large-scale inhomogeneities of TIDs. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 3, pp. 191–198, March 2008.     

Errors in ultrasonic scatterer size estimates due to phase and amplitude aberration

Current ultrasonic scatterer size estimation methods assume that acoustic propagation is free of distortion due to large-scale variations in medium attenuation and sound speed. However, it has been demonstrated that under certain conditions in medical applications, medium inhomogeneities can cause significant field aberrations that lead to B-mode image artifacts. These same aberrations may be responsible for errors in size estimates and parametric images of scatterer size. This work derives theoretical expressions for the error in backscatter coefficient and size estimates as a function of statistical parameters that quantify phase and amplitude aberration, assuming a Gaussian spatial autocorrelation function. Results exhibit agreement with simulations for the limited region of parameter space considered. For large values of aberration decorrelation lengths relative to aberration standard deviations, phase aberration errors appear to be minimal, while amplitude aberration errors remain significant. Implications of the results for accurate backscatter and size estimation are discussed. In particular, backscatter filters are suggested as a method for error correction. Limitations of the theory are also addressed. The approach, approximations, and assumptions used in the derivation are most appropriate when the aberrating structures are relatively large, and the region containing the inhomogeneities is offset from the insonifying transducer.     

Monte Carlo simulation in the theory of wave propagation in random media: II. Scintillation index, second and fourth moments

The purpose of this article (comprising parts I and II) is to develop and test the approach of combining a path-integral technique and a complex-valued Monte Carlo method to calculate the highest moments of the Green function of the stochastic wave equation for media with random small-scale inhomogeneities against the background of large-scale inhomogeneities. In part II calculations of the second and fourth moments of the Green function and the scintillation index have been performed for 1D and 2D cases in the framework of three models: a model of the stochastic wave equation and models of parabolic and Markov approximations. The finiteness of the correlation radius of inhomogeneities has been shown to be the reason for the significant difference between the Markov approximation and the other two. The results obtained prove that the applicability of the parabolic approximation (without the Markov approximation) is much wider than might be expected. A comparison has been made showing good agreement with reliable results for 1D media. The Monte Carlo results have exhibited the singularities existing at the localization centres and forming exponential decay of the second moment from distances of about one wavelength. The unexpected sharp oscillations interrupting the exponential decay of the Green function moments have been obtained at distances from the localization centre of several tens of times the average distance between scatterers. The effect of weak large-scale inhomogeneities on the behaviour of the second moment has also been investigated.     

Off-resonance correction using an estimated linear time map

Images acquired in the presence of magnetic field deviations and reconstructed without taking into account the off-resonance, are distorted and corrupted with artifacts. Several post-processing algorithms have been developed for correcting the distortion when it is not possible to fix the field inhomogeneities. These off-resonance correction methods are, in general, slow and computing intensive. To make them faster they are usually adapted to a particular situation or approximated. One of these approximations is to assume that the field map is linear. Although this assumption makes the algorithm fast and robust it is not well suited for arbitrary field maps. On the other hand, there are k-space trajectories with an almost linear time map (time at which each k-space value is acquired), such as 2DFT and EPI. This paper presents an algorithm for off-resonance correction based on a linear time map approximation. This approximation allows a fast algorithm that takes advantage of the almost linearity of the time map and uses the whole field map to correct the images. The proposed correction algorithm reduces the off-resonance induced artifacts while being fast. The linear approximation of the time map needs to be done only once for each trajectory because it does not depend on the acquired image or field map data. The method can also be extended to a multi-plane approximation for sequences with more complex time maps.     

Causes of the reconstructed cross appearing in lensless Fourier transform digital holography

Causes of the bright cross appearing in the reconstructed field of lensless Fourier transform digital hologram (LFDH) are presented. Firstly, as LFDH's reconstruction was directly performed by Fourier transform algorithm, the intensity distribution of reconstructed image plane was just LFDH's spectrum, and three parts of the reconstructed field are imaged in the spectral plane. Meanwhile their intensities were almost the same. Hence, the ratio of signal to noise of the reconstructed image was obviously lower than that of the conventional digital hologram. Further, non-uniformity of the background intensity and random noises usually existed in a practical LFDH. Meanwhile their spectra spread through the central and the edge areas; this also led to decrease in the ratio of signal to noise of the reconstructed image. Specially, as the two-dimensional Fourier transform algorithm was performed row by row, and then column by column, the low-frequency spectrum induced by the non-uniformity of the background and random noises along the directions of rows and columns would concentrate on the central column and row, respectively, and what is gotten looked like a cross. Therefore, the cross appearing in LFDH's reconstructed field should be attributed to the combination of the background and the Fourier transform algorithm itself.     

Algorithm for near-field reconstruction based on radial-shearing interferometry

A new iterative algorithm to be used to precisely reconstruct near-field distribution from an interferogram of a laser output generated by a cyclic radial-shearing interferometer is proposed. First, by use of a window function around the zero-frequency part of the Fourier transform of the interferogram and calculation of the inverse Fourier transform of the zero-frequency part, we obtain the background intensity distribution of the interferogram. Then, according to the iterative algorithm, the near-field distribution of the laser output is precisely reconstructed from the background intensity distribution obtained in the first step. A numerical simulation and an actual experiment of the near-field reconstruction of the laser output with arbitrary amplitude distribution are implemented successfully.     

Detection of inhomogeneities in biological tissues by the methods of confocal microscopy

The model of detection of local inhomogeneities of scattering and absorbing types in biological tissues by the methods of reflection and transmission confocal microscopy has been developed on the basis of the theory of vision in scattering media. General equations for calculation of the image contrast of an inhomogeneity against the background of a scattering medium are derived. The influence of the object characteristics and observation-system parameters on the maximum detection depth of inhomogeneities is analyzed. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 5, pp. 625–639, May, 1998.     

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.     

Monte Carlo simulation in the theory of wave propagation in random media: I. Modified and Feynman path integrals

The purpose of this article (comprising parts I and II) is to develop and test the approach of combining a path-integral technique and a complex-valued Monte Carlo method to calculate the highest moments of the Green function of the stochastic wave equation for a random medium against the background of large-scale inhomogeneities. In part I, the new modified path-integral representations of the Green function moments of the stochastic wave equation have been developed. The limiting transition of these representations to the Feynman path integrals corresponding to the parabolic approximation is discussed. Path-integral representations for Green function moments are given for three models: a model of the stochastic wave equation and models of parabolic and Markov approximations. The Metropolis algorithm underlying the Monte Carlo method for calculating real and complex-valued path integrals is discussed in brief. Numerical results are presented in part II of the article.     

基于交替隐式有限差分法的快速早期乳腺癌时域微波断层成像

采用时域微波断层成像技术进行早期乳腺癌检测能够准确地获得乳房的电参数分布,具有明确的物理解释和医学诊断价值.临床应用讲究即时性,为了提高检测的速度,本文将交替隐式有限差分法应用到乳腺癌检测中,基于正反演时间步进成像算法进行成像分析,结果显示在保证精度的前提下,采用交替隐式有限差分法的成像时间可缩短为传统时域有限差分法的23%,提高了微波断层成像技术的临床可应用性.     

On the possibility of using the maximum entropy method in ultrasonic nondestructive testing for scatterer visualization from a set of echo signals

We have studied the possibility of solving the inverse scattering problem in the Born approximation, i.e., the reconstruction of scatterer images from the measured set of echo signals. We have considered generalization of the classical combined SAFT (C-SAFT) algorithm to the case of multiple reflections from uneven boundaries of the tested object taking into account the transformation of the wave type for several positions of the antenna grid, which makes it possible to obtain high-quality scatterer images. Representation of the direct problem in matrix form makes it possible to switch to solving the inverse problem, which can be solved using the Tikhonov regularization procedure, because it is an ill-posed. We have considered the possibility of using the entropy of the image estimate as the stabilizing functional that forms the essence of the maximum entropy method (MEM). The advantage of the MEM over the conventionally used linear C-SAFT method has been shown. The ray model taking into account reflections of rays from the boundaries of the tested object with uneven boundaries has been used for constructing the function estimate. We have demonstrated the ability of the MEM to obtain the scatterer images with superresolution and to suppress the “side lobes” of the function of the point scattering on the collapsed set of echo signals. The use of echo signals reflected from the boundaries of the tested object makes it possible to reconstruct the scatterer shape more exactly. Examples of images reconstructed by the MEM on echo signals obtained in the numerical and model experiments have been presented.