Statistics of ultrasonic scatterer size estimation with a reference phantom

A theoretical expression for the variance of scatterer size estimates is derived for a modified least squares size estimator used in conjunction with a reference phantom method for backscatter coefficient measurement. A Gaussian spatial autocorrelation function is assumed. Simulations and phantom experiments were performed to verify the results for backscatter and size variances. The dependence of size estimate errors upon free experimental parameters is explored. Implications of the findings for the optimization of scatterer size estimation are discussed. The utility of scatterer size parametric imaging is examined through the signal to noise ratio comparison with standard ultrasonic B-mode imaging.     

Spectral estimation for characterization of acoustic aberration

Spectral estimation based on acoustic backscatter from a motionless stochastic medium is described for characterization of aberration in ultrasonic imaging. The underlying assumptions for the estimation are: The correlation length of the medium is short compared to the length of the transmitted acoustic pulse, an isoplanatic region of sufficient size exists around the focal point, and the backscatter can be modeled as an ergodic stochastic process. The motivation for this work is ultrasonic imaging with aberration correction. Measurements were performed using a two-dimensional array system with 80 x 80 transducer elements and an element pitch of 0.6 mm. The f number for the measurements was 1.2 and the center frequency was 3.0 MHz with a 53% bandwidth. Relative phase of aberration was extracted from estimated cross spectra using a robust least-mean-square-error method based on an orthogonal expansion of the phase differences of neighboring wave forms as a function of frequency. Estimates of cross-spectrum phase from measurements of random scattering through a tissue-mimicking aberrator have confidence bands approximately +/- 5 degrees wide. Both phase and magnitude are in good agreement with a reference characterization obtained from a point scatterer.     

Correlation of ultrasonic scatterer size estimates for the statistical analysis and optimization of angular compounding

Ultrasonic scatterer size estimates generally have large variances due to the inherent noise of spectral estimates used to calculate size. Compounding partially correlated size estimates associated with the same tissue, but produced with data acquired from different angles of incidence, is an effective way to reduce the variance without making dramatic sacrifices in spatial resolution. This work derives theoretical approximations for the correlation between these size estimates, and the coherence between their associated spectral estimates, as functions of ultrasonic system parameters. A Gaussian spatial autocorrelation function is assumed to adequately model scatterer shape. Both approximations compare favorably with simulation results, which consider validation near the focus. Utilization of the correlation/coherence expressions for statistical analysis and optimization is discussed. Approximations, such as the invariance of phase and amplitude terms with angle, are made to obtain closed-form solutions to the derived spectral coherence near the focus and permit analytical optimization analysis. Results indicate that recommended parameter adjustments for performance improvement generally depend upon whether, for the system under consideration, the primary source of change in total coherence with rotation is phase term variation due to the change in the relative position of scattering sites, or field amplitude term variation due to beam movement.     

Scatterer size estimation in pulse-echo ultrasound using focused sources: theoretical approximations and simulation analysis

The speckle in ultrasound images has long been thought to contain information related to the tissue microstructure. Many different investigators have analyzed the frequency characteristics of the backscattered signals to estimate the scatterer acoustic concentration and size. Previous work has been mostly restricted to unfocused or weakly focused ultrasound sources, thus limiting its implementation with diagnostically relevant fields. Herein, we derive equations capable of estimating the size of a scatterer for any reasonably focused source provided that the velocity potential field in the focal region can be approximated as a three-dimensional Gaussian beam, scatterers are a sufficient distance from the source, and the field is approximately constant across the scatterer. The calculations show that, when estimating the scatterer size, correcting for focusing requires a generalized attenuation-compensation function that includes both attenuation and focusing along the beam axis. The Gaussian approximation is validated by comparing the ideal velocity potential field for three spherically focused sources with f-numbers of 1, 2, and 4 to the Gaussian approximation for frequencies from 2 to 14 MHz. The theoretical derivations are evaluated by simulating the backscatter by using spherically focused sources (f-numbers of 1, 2, and 4) adjacent to attenuating media (0.05 to 1 dB/cm/MHz) that contain scatterers with Gaussian impedance distributions. The generalized attenuation-compensation function yielded results accurate to 7.2% while the traditional attenuation-compensation functions that neglected focusing had errors as high as 103%.     

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.     

利用相位差法测量望远镜像差

大气湍流引起的动态波前畸变和望远镜的像差是限制望远镜分辨力的主要因素,如何准确地测量望远镜的像差是进一步提高望远镜分辨能力的关键问题。相位差法利用在焦面和离焦面上同时采集到的短曝光图像,恢复出瞬时波前相位分布,然后根据大气湍流的统计特性进行平均,可以实现对望远镜像差的估计。通过计算机模拟实验,对利用相位差法恢复光瞳上的波前相位和测量望远镜像差进行了研究。模拟研究结果表明,利用相位差法能有效地估计出望远镜像差,估计均方根误差约为0.08个波长。     

On the estimation of backscatter coefficients using single-element focused transducers

The ultimate goal of quantitative ultrasound (QUS) imaging methods based on backscatter coefficient (BSC) estimates is to obtain system-independent structural information about samples. In the current study, three BSC estimation methods were compared and evaluated using the same backscattered pressure datasets in order to assess their consistency. BSC estimates were obtained from two phantoms with embedded glass spheres and compared to theoretical BSCs calculated using size distributions estimated using optical microscopy. Effective scatterer diameter and concentration estimates of the glass spheres were also obtained from the estimated BSCs. One estimation method needed to be compensated by more than an order of magnitude in amplitude in order to produce BSCs comparable to the other two methods. All calibration methods introduced different frequency-dependent effects, which could have noticeable effects on the bias of QUS estimates derived from experimental BSCs. Although in most cases the experimental QUS estimates obtained with all three methods were observed to differ by less than 10%, larger differences are expected depending on both the pressure focusing gain of the transducer (proportional to the ratio of the square of the aperture radius to the product of the wavelength and focal length) and ka range used in the estimation.     

Method of improved scatterer size estimation and application to parametric imaging using ultrasound

The frequency dependence of RF signals backscattered from random media (tissues) has been used to describe the microstructure of the media. The frequency dependence of the backscattered RF signal is seen in the power spectrum. Estimates of scatterer properties (average scatterer size) from an interrogated medium are made by minimizing the average squared deviation (MASD) between the measured power spectrum and a theoretical power spectrum over an analysis bandwidth. Estimates of the scatterer properties become increasingly inaccurate as the average signal to noise ratio (SNR) over the analysis bandwidth becomes smaller. Some frequency components in the analysis bandwidth of the measured power spectrum will have smaller SNR than other frequency components. The accuracy of estimates can be improved by weighting the frequency components that have the smallest SNR less than the frequencies with the largest SNR in the MASD. A weighting function is devised that minimizes the noise effects on the estimates of the average scatterer sizes. Simulations and phantom experiments are conducted that show the weighting function gives improved estimates in an attenuating medium. The weighting function is applied to parametric images using scatterer size estimates of a rat that had developed a spontaneous mammary tumor.     

Eigenfunction analysis of stochastic backscatter for characterization of acoustic aberration in medical ultrasound imaging

Presented here is a characterization of aberration in medical ultrasound imaging. The characterization is optimal in the sense of maximizing the expected energy in a modified beamformer output of the received acoustic backscatter. Aberration correction based on this characterization takes the form of an aberration correction filter. The situation considered is frequently found in applications when imaging organs through a body wall: aberration is introduced in a layer close to the transducer, and acoustic backscatter from a scattering region behind the body wall is measured at the transducer surface. The scattering region consists of scatterers randomly distributed with very short correlation length compared to the acoustic wavelength of the transmit pulse. The scatterer distribution is therefore assumed to be delta correlated. This paper shows how maximizing the expected energy in a modified beamformer output signal naturally leads to eigenfunctions of a Fredholm integral operator, where the associated kernel function is a spatial correlation function of the received stochastic signal. Aberration characterization and aberration correction are presented for simulated data constructed to mimic aberration introduced by the abdominal wall. The results compare well with what is obtainable using data from a simulated point source.     

Aberrations in fractional-Talbot images of periodic binary gratings

We discuss the ray-optical aberrations which appear in the fractional-Talbot image of a periodic binary grating with coherent illumination. First we examine the complex amplitude of an aberration-free imaging field at a fractional-Talbot plane. We then trace the path of a diffracted ray of specific order which contributes to the fractional-Talbot imaging. Next we formulate the focus-shift and third-order aberrations which arise from a focusing error and a fourth-order approximation of the path length, respectively. We then evaluate the amplitude and phase of an aberrated imaging field that are represented in terms of aberration functions. When the grating period decreases to approach the optical wavelength, the aberrations of lower-order rays are shown to be more influential on the fractional-Talbot imaging field than those of higher-order rays. The theory of aberration discussed here could be very useful in evaluating the fractional-Talbot image of a periodic binary grating.     

Identifying ultrasonic scattering sites from three-dimensional impedance maps

Ultrasonic backscattered signals contain frequency-dependent information that is usually discarded to produce conventional B-mode images. It is hypothesized that parametrization of the quantitative ultrasound frequency-dependent information (i.e., estimating scatterer size and acoustic concentration) may be related to discrete scattering anatomic structures in tissues. Thus, an estimation technique is proposed to extract scatterer size and acoustic concentration from the power spectrum derived from a three-dimensional impedance map (3DZM) of a tissue volume. The 3DZM can be viewed as a computational phantom and is produced from a 3D histologic data set. The 3D histologic data set is constructed from tissue sections that have been appropriately stained to highlight specific tissue features. These tissue features are assigned acoustic impedance values to yield a 3DZM. From the power spectrum, scatterer size and acoustic concentration estimates were obtained by optimization. The 3DZM technique was validated by simulations that showed relative errors of less than 3% for all estimated parameters. Estimates using the 3DZM technique were obtained and compared against published ultrasonically derived estimates for two mammary tumors, a rat fibroadenoma and a 4T1 mouse mammary carcinoma. For both tumors, the relative difference between ultrasonic and 3DZM estimates was less than 10% for the average scatterer size.     

Aberrations in materials with random inhomogeneities

Materials that consist of a random microstructure can affect ultrasonic measurements--reducing signal strength, increasing noise, and reducing measurement accuracy--through scattering and aberration of the acoustic field. To account for these adverse effects a phase screen model, alongside the stochastic wave equation, has been developed. This approach allows the field and study aberrations to be modeled from a statistical point of view. Experimental evidence of aberration and statistical properties of the measured acoustic field are shown. A measured correlation function of the acoustic field is interlinked to mean crystallite size by using a theoretical coherence function that can be mainly described by the correlation length and wave velocity variation of microstructure. The estimation of the mean crystallite size using this technique would provide some insight into material characterization.     

Improved scatterer size estimation using backscatter coefficient measurements with coded excitation and pulse compression

Scatterer size estimates from ultrasonic backscatter coefficient measurements have been used to differentiate diseased tissue from normal. A low echo signal-to-noise ratio (eSNR) leads to increased bias and variance in scatterer size estimates. One way to improve the eSNR is to use coded excitation (CE). The normalized backscatter coefficient was measured from three tissue-mimicking phantoms by using CE and conventional pulsing (CP) techniques. The three phantoms contained randomly spaced glass beads with median diameters of 30, 45, and 82 mum, respectively. Measurements were made with two weakly focused, single-element transducers (f(0)=5 MHz and f(0)=10 MHz). For CE, a linear frequency modulated chirp with a time bandwidth product of 40 was used and pulse compression was accomplished by the use of a Wiener filter. Preliminary results indicated that improved estimation bias versus penetration depth was obtained by using CE compared to CP. The depth of penetration, where the accuracy of scatterer diameter estimates (absolute divergence <25%) were obtained with the 10 MHz transducer, was increased up to 50% by using CE versus CP techniques. In addition, for a majority of the phantoms, the increase in eSNR from CE resulted in a modest reduction in estimate variance versus depth of penetration.     

Localized charge inhomogeneities and phase separation near a second-order phase transition

Localized charge inhomogeneities and phase separation are described in the framework of the phenomenological theory of phase transitions. It is shown that Coulomb interaction determines the charge distribution and the characteristic size of the emerging inhomogeneities. Phase separation associated with charge segregation becomes possible because of a high dielectric constant and a low excess charge density in the localization region. The phase diagram of the system is calculated, and estimates are obtained for the gain in energy associated with the emerging state. The role of Coulomb interaction is exposed, and corresponding estimates are given.     

厚介质情况下激光“热像”的演化规律分析

以“热像”形成的衍射理论模型和分步傅里叶算法为基础，模拟研究了厚介质情况下“热像”的形成特点.重点分析了散射点调制系数(包括振幅调制和相位调制)、散射点与介质前表面之间的距离、散射点大小、入射光强等参数的变化对“热像”点位置以及强度的影响.数值模拟结果与解析理论结果进行了对比，在散射点调制系数和入射光强的变化对“热像”的影响两种情况下，得出了与解析理论基本一致的结论；在散射点大小的变化对“热像”的影响情况下，得出了与解析理论预言不同的结论.此外，还发现散射点与介质前表面距离为零时，在介质后表面也可能出现“热像”效应. 关键词： 热像 小尺度自聚焦 分步傅里叶算法 散射点     

Iteration of transmit-beam aberration correction in medical ultrasound imaging

Simulations of iterative transmit-beam aberration correction using a time-delay and amplitude filter have been performed to study the convergence of such a process. Aberration in medical ultrasonic imaging is usually modeled by arrival-time and amplitude fluctuations concentrated on the transducer array. This is an approximation of the physical aberration process, and may be applied to correct the transmitted signal using a time-delay and amplitude filter. Estimation of such a filter has proven difficult in the presence of severe aberration. Presented here is an iterative approach, whereby a filter estimate is applied to correct the transmit-beam. This beam induces acoustic backscatter better suited for arrival-time and amplitude estimation, thus facilitating an improved filter estimate. Two correlation-based methods for estimating arrival-time and amplitude fluctuations in received echoes from random scatterers were employed. Aberration was introduced using eight models emulating aberration produced by the human abdominal wall. Results show that only a few iterations are needed to obtain corrected transmit-beam profiles comparable to those of an ideal aberration correction filter. Furthermore, a previously developed focusing criterion is found to quantify the convergence accurately.     

小尺寸平面源的误差分析

基于Helmholtz方程的严格远场解，讨论了大发散角光辐射远场振幅和相位对于源包络扰动的依赖关系，并推导了远场振幅、相位误差公式．结果表明，远场振幅和相位误差必须用源的“误差面积”来描述；在远场主要辐射区（即傍轴区），振幅和相位对于源包络扰动不敏感，而在远轴区较敏感． 关键词：     

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.     

Wavefront sensorless adaptive optics for large aberrations

In some adaptive optics systems the aberration is determined not by using a wavefront sensor but by sequential optimization of the adaptive correction element. Efficient schemes for the control of such systems are essential if they are to be effective. A scheme is introduced that permits the efficient measurement of large amplitude wavefront aberrations that are represented by an appropriate series of modes. This scheme uses an optimization metric based on the root-mean-square spot radius (or focal spot second moment) and an aberration expansion using polynomials suited to the representation of lateral aberrations. Experimental correction of N aberration modes is demonstrated with a minimum of N+1 photodetector measurements. The geometrical optics basis means that the scheme can be extended to arbitrarily large aberrations.     

Reconstruction of the fine structure of an acoustic scatterer against the distorting influence of its large-scale inhomogeneities

In the ultrasonic diagnostics of small-size neoplasms of biological tissues at the earliest stage of their development, an efficient way to eliminate the distorting influence of high-contrast or large inhomogeneities of the biological medium is to apply the iterative technique. A simple approach is proposed, which makes it possible with only two iteration steps to achieve an efficient focusing of the tomograph array. At the first step, the unknown distribution of the large-scale inhomogeneities of sound velocity and absorption over the scatterer is reconstructed, where the large-scale inhomogeneities are those whose size exceeds several wavelengths. At the second step, the fine structure of the scatterer is reconstructed against the large-scale background, which can be performed with a high accuracy owing to the evaluation of the background at the first step. The possibility of simultaneous reconstruction of the large-scale and fine structures by the noniterative Grinevich-Novikov algorithm is considered as an alternative. This algorithm reconstructs in an explicit form two-dimensional refractive-absorbing acoustic scatterers of almost arbitrary shape and strength. Taking into account the effects of multiple scattering, this algorithm provides resolution of the fine structure almost as good as that achieved in reconstructing the same structure against an undistorting homogeneous background. The results of numerical simulations of both algorithms are presented.