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Coefficients of a finite impulse response linear filter model for ultrasonic propagation are found using a statistical estimation. The model employs a Green’s function to describe scattering received by a two-dimensional array from a random-medium volume illuminated by a transmit beam. The frequency response of this Green’s function is factored into a homogeneous-transmission term and a path-dependent aberration term. Relative amplitude and phase of the aberration response are estimated over the frequency band of the transmit–receive system by using scattering from closely situated volumes and assuming that the aberration is the same for each volume. The amplitude and phase are obtained from the power and cross-power spectra of signals at positions throughout the receive aperture. The homogeneous response is estimated by averaging and is removed to isolate the aberration response. Propagation path aberration parameters calculated from pulse-echo measurements of random scattering through a tissue-mimicking aberration phantom are similar to corresponding parameters calculated for the same aberrator and array position by using echoes from a point-like reflector. The results indicate the approach is capable of describing, in addition to time-shifts, changes in waveform amplitude and shape produced by propagation through a distributed aberration.

Correction of aberration in ultrasound imaging uses the response of a point reflector or its equivalent to characterize the aberration. Because a point reflector is usually unavailable, its equivalent is obtained using statistical methods, such as processing reflections from multiple focal regions in a random medium. However, the validity of methods that use reflections from multiple points is limited to isoplanatic patches for which the aberration is essentially the same. In this study, aberration is modeled by an offset phase screen to relax the isoplanatic restriction. Methods are developed to determine the depth and phase of the screen and to use the model for compensation of aberration as the beam is steered. Use of the model to enhance the performance of the noted statistical estimation procedure is also described. Experimental results obtained with tissue-mimicking phantoms that implement different models and produce different amounts of aberration are presented to show the efficacy of these methods. The improvement in b-scan resolution realized with the model is illustrated. The results show that the isoplanatic patch assumption for estimation of aberration can be relaxed and that propagation-path characteristics and aberration estimation are closely related.

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape. Calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable. Numerical results are presented to demonstrate the accuracy and efficiency of the method.

Relations between the power spectrum of scattering by a plane and by a volume of a statistically isotropic random medium are developed from two basic expressions. One gives the power spectrum of scattering by an n-dimensional isotropic medium as a one-dimensional transform of a rotationally symmetric correlation function of medium variations. The other describes the power spectrum of scattering by an (n-k)-dimensional cross section as a projection of the power spectrum in n-dimensional space. The results are used to determine the power spectrum of scattering by a volume from the power spectrum of scattering by a plane within the volume and also to relate the moments of power scattered in spaces of various dimensions. To illustrate the relations, calculations of power spectra and selected moments for volume scattering are made from power spectra having analytic forms as well as form power spectra computed in two dimensions from images of closely packed spheres and of pig liver sections. Numerical results obtained by processing power spectra having closed form expressions for volume scattering are in close agreement with theoretical values. Calculations for packed spheres show general concentrations of power explained by theory but required substantial smoothing of data in two dimensions to obtain consistent results because the images were not true cross sections. Results from sections of pig liver agree wth expected relationships. The theoretical expressions and numerical data indicate how quantitative information about the three-dimensional structure can be obtained from two-dimensional sections when the basic isotropy conditions are satisfied.

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Pressure scattered by cylindrical and spherical objects with elevation-focused illumination and reception has been analytically calculated, and corresponding cross sections have been reconstructed with a two-dimensional algorithm. Elevation focusing was used to elucidate constraints on quantitative imaging of three-dimensional objects with two-dimensional algorithms. Focused illumination and reception are represented by angular spectra of plane waves that were efficiently computed using a Fourier interpolation method to maintain the same angles for all temporal frequencies. Reconstructions were formed using an eigenfunction method with multiple frequencies, phase compensation, and iteration. The results show that the scattered pressure reduces to a two-dimensional expression, and two-dimensional algorithms are applicable when the region of a three-dimensional object within an elevation-focused beam is approximately constant in elevation. The results also show that energy scattered out of the reception aperture by objects contained within the focused beam can result in the reconstructed values of attenuation slope being greater than true values at the boundary of the object. Reconstructed sound speed images, however, appear to be relatively unaffected by the loss in scattered energy. The broad conclusion that can be drawn from these results is that two-dimensional reconstructions require compensation to account for uncaptured three-dimensional scattering.