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(a) Form of the sounding pulse; (b) signals recorded at one of the transducers. Model 1 (solid line), model 2 (dotted line) and model 3(a) (dashed line).
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We develop efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We treat the inverse problem as a coefficient inverse problem for unknown coordinate-dependent functions that characterize both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed regio...
Citations
... When the degree of model-mismatch is low, i.e. when the numerical model accurately represents the measurement scenario, these approaches have the potential to reconstruct accurate, high resolution images. This approach, for which there is a considerable literature in the seismic community [1], has begun to be explored in earnest for medical applications [2,18,19,47,[50][51][52][53][54]56]. Full-wave approaches are discussed further in section 8, but the biggest challenge with such schemes is the computational cost. ...
An efficient and accurate image reconstruction algorithm for ultrasound tomography in soft tissue is described and demonstrated, which can recover accurate sound speed distribution from acoustic time series measurements. The approach is based on a second-order iterative minimisation of the difference between the measurements and a model based on a ray-approximation to the heterogeneous Green's function. It overcomes the computational burden of full-wave solvers while avoiding the drawbacks of time-of-flight methods. Through the use of a second-order iterative minimisation scheme, applied stepwise from low to high frequencies, the effects of scattering are incorporated into the inversion.
... when the numerical model accurately represents the measurement scenario, these approaches have the potential to reconstruct accurate, high resolution images. This approach, for which there is a considerable literature in the seismic community [1], has begun to be explored in earnest for medical applications [19,54,51,55,56,20,57,60,58,2]. Full-wave 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t ...
... In other words, in (48), g(x r ; x) acts as a forward propagator from x to x r , whereas in Eqs. (51) and (53), it acts as a backward propagator from x r to x. Note that the Green's function obeys the reciprocity relation g(x; x ) = g(x ; x). ...
... We emphasise here again the reason for using the Hessian in the inversion. The residual, δp res , which appears in the expression for the gradient, (51), depends only on the points along a ray linking e and r, and therefore it includes only refractive effects. However, the pressure perturbation δp which appears in the expression for the Hessian, (53), can be affected by any point in the medium, because it is obtained from the action of the Frećhet derivative, on the sound speed perturbation, δc, which can vary anywhere. ...
... The first term in (41) is the functional gradient ∇F, and is calculated using (54). The second term is an action of the Hessian matrix on a perturbation to the sound speed, and is calculated using (56) and (46). These formulae are functions of the Green's functions g(c (n) ; ω, x; x e ) and g(c (n) ; ω, x r ; x). ...
An efficient and accurate image reconstruction algorithm for ultrasound tomography (UST) is described and demonstrated, which can recover accurate sound speed distribution from acoustic time series measurements made in soft tissue. The approach is based on a second-order iterative minimisation of the difference between the measurements and a model based on a ray-approximation to the heterogeneous Green's function. It overcomes the computational burden of full-wave solvers while avoiding the drawbacks of time-of-flight methods. Through the use of a second-order iterative minimisation scheme, applied stepwise from low to high frequencies, the effects of scattering are incorporated into the inversion.
... Iterative algorithms for solving the inverse problem are based on representing the gradient of the residual functional between the experimentally measured and the theoretically computed wave fields, which depends on the longitudinal-wave propagation velocity. The principle of operation of the numerical algorithm is described in detail in [5,12,14,16,18,27]. A distinctive feature of the numerical realization of the algorithms employed in this study is the use of high-frequency sources of ultrasonic radiation (above 1 MHz). ...
... The computations were carried out on 384 CPU computing cores (1 core carried out computations for 1 source). In such a configuration, for one iteration on 1 core, it is necessary to perform 3 times the computation of wave propagation in time in the scalar wave model (1) on a grid of 700 Â 700 points of spatial coordinates and 1500 time steps [5,12,14,16,27]. The computation time for 40 iterations on 384 CPU cores was about 30 min. ...
This paper is concerned with the use of a supercomputer to solve tomographic inverse problems of ultrasonic nondestructive testing in the framework of a scalar wave model. The problem of recovering the velocity of a longitudinal wave in a solid is formulated as a coefficient inverse problem, which in this formulation is nonlinear. First, the algorithms were tested on real data obtained in experiments on a test bench for ultrasound tomography examinations. Ultrasound in the 2–8 MHz band was used for sounding. The experiment employed a rotating transducer system. A rotating transducer system substantially increases the number of emitters and detectors in a tomographic scheme and makes it possible to neutralize the image artifacts. An important result of this study is an experimental confirmation of the adequacy of the underlying mathematical model. The proposed scalable numerical algorithms can be efficiently parallelized on CPU– supercomputers. The computations were performed on 384 computing CPU cores of the “Lomonosov–2” supercomputer at Lomonosov Moscow State University.
... This parameter plays a significant role in the quality of the reconstructed image in the bone [198]. Goncharsky et al. [264] calculated proper adjoint operators for different attenuation models added to the acoustic wave equation. In future work, the attenuation should be considered as a property of the medium and object. ...
... We write the residual functional of argument between experimental data and computed field for a given at boundary S as [28,29] (2) ...
The article is devoted to developing methods of ultrasonic tomography for nondestructive testing. The inverse problem of reconstructing velocity sections is treated as a nonlinear coefficient inverse problem for a scalar wave equation. Effective iterative methods for solving this problem using supercomputers have been developed. These methods use direct formulas for computing the gradient of a residual functional between the computed wave field and the field experimentally measured on detectors. The effectiveness of the algorithms was tested on real experimental data. The first experiments were carried out on dedicated solid-state samples in the simplest arrangement, with the signals recorded by standard ultrasonic antenna arrays at a frequency of 5 MHz. It is shown that using the developed tomographic methods with reflection and transmission schemes in a real experiment, it is possible not only to detect the boundaries of reflectors, but also to determine the velocity sections inside the reflectors.
... The simplest absorption model is used in this paper [9]. The inverse problem of ultrasound tomography can then be formulated as a coefficient inverse problem of reconstructing the unknown coefficients c(r) and a(r) in the wave equation, given the measurements of the acoustic pressure on the surface S made with different positions q of the sources: ...
This paper is dedicated to developing effective methods of 3D acoustic tomography. The inverse problem of acoustic tomography is formulated as a coefficient inverse problem for a hyperbolic equation where the speed of sound and the absorption factor in three-dimensional space are unknown. Substantial difficulties in solving this inverse problem are due to its nonlinear nature. A method which uses short sounding pulses of two different central frequencies is proposed. The method employs an iterative parallel gradient-based minimization algorithm at the higher frequency with the initial approximation of unknown coefficients obtained by solving the inverse problem at the lower frequency. The efficiency of the proposed method is illustrated via a model problem. In the model problem an easy to implement 3D tomographic scheme is used with the data specified at a cylindrical surface. The developed algorithms can be efficiently parallelized using GPU clusters. Computer simulations show that a GPU cluster capable of performing 3D image reconstruction within reasonable time.
... These results are based on the possibility of constructing iterative gradient algorithms to solve inverse problems in wave tomography. An expression for the gradient of the residual functional has been derived, which is the difference between the experimental wave field measured at the detectors and the theoretical wave field calculated for the given coefficients of a differential equation [5,15,16]. To calculate the residual functional gradient in this formulation, it is necessary to apply an additional boundary condition, such as the Neumann condition, at the boundary of the computational domain. ...
... The use of low sounding frequencies combined with the small wave velocity difference in breast and water allows efficient techniques to be developed for ultrasound tomography diagnosis in the case considered. The proposed methods were tested both on simulated data sets and on experimental data obtained on ultrasound tomography examination benches [16,[28][29][30][31][32]. ...
... □ Remark 1. The representation of the Fréchet derivative of the residual functional derived can be used to construct efficient iterative algorithms for solving both 2D and 3D inverse problems of ultrasound tomography [16,28,29]. In the numerical implementation, an approximate solution is obtained by means of finite difference methods. ...
In this paper, the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain. Efficient methods for solving the inverse problems of wave tomography for the case of transparent boundary conditions are presented. The algorithms are designed for supercomputers. We prove the Fréchet differentiability theorem for the residual functional and derive an exact expression for the Fréchet derivative in the case of a transparent boundary in the direct and conjugate problems. The expression for the Fréchet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary. The effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low-frequency wave tomography. The model problem is tailored to apply to the differential diagnosis of breast cancer.
... The methods developed in the 1970s-1990s for solving inverse problems [8,9] are the most striking mathematical results of the last century. The inverse problem considered in this paper is a coefficient inverse problem [3]. The solution algorithms rely on the processing power of modern GPU clusters. ...
... The images reconstructed from experimental data showed that a spatial resolution of ≈ 1.5 mm is attainable in a low-frequency setup with a central wavelength of ≈ 3 mm, which can be implemented in practice. The algorithms developed by the authors in [3,5] for layer-by-layer ultrasound tomography schemes were used to solve the inverse problems. ...
... We use the iterative gradient method [4] to minimize the functional. Representations of the gradient Φ c (u(c, a)), Φ a (u(c, a)) were obtained in [3,5]. Finite difference time-domain method [6] was used to compute the wavefields. ...
The paper considers the use of supercomputers in design of medical ultrasound tomography devices. The mathematical models describing the wave propagation in ultrasound tomography should take into account such physical phenomena as diffraction, multiple scattering, and so on. The inverse problem of wave tomography is posed as a coefficient inverse problem with respect to the wave propagation velocity and the absorption factor. Numerous simulations made it possible to determine the optimal parameters of an ultrasound tomograph in order to obtain a spatial resolution of 1.5 mm suitable for early-stage breast cancer diagnosis. The developed methods were tested both on model problems and on real data obtained at the experimental test bench for tomographic studies. The computations were performed on GPU devices of Lomonosov-2 supercomputer at Lomonosov Moscow State University.
... Коэффициентная обратная задача волновой томографии является нелинейной и для ее решения необходимо использовать суперЭВМ [4][5][6]. Используемая в ультразвуковой томографии математическая модель хорошо описывает как дифракционные эффекты, так и эффекты поглощения [7,8]. ...
Статья посвящена разработке методов формирования акустических зондирующих импульсов в задачах ультразвуковой томографии. Обратная задача формирования акустических зондирующих импульсов рассматривается в рамках линейной модели. Эта задача является некорректной и требует использования регуляризирующих алгоритмов. Для численного решения использована тихоновская схема регуляризации. Разработанные алгоритмы протестированы на решении модельных задач и с помощью специально поставленного эксперимента, в котором акустический тракт включает в себя цифровой генератор импульсов, усилитель, источник акустического излучения, акустический детектор, предусилитель и аналого-цифровой преобразователь. Экспериментально подтверждены как адекватность линейной модели, так и высокая эффективность предложенных алгоритмов.
This paper is concerned with developing the methods of forming acoustic sounding pulses in ultrasound tomography applications. The inverse problem of forming acoustic sounding pulses is considered in the framework of linear models. This problem is ill-posed and requires the use of regularizing algorithms. Tikhonov's regularization scheme is used to solve the problem numerically. The developed algorithms are tested on model problems as well as on experimental data. In the experimental setup, the acoustic path includes a digital waveform generator, an amplifier, an ultrasound emitter, a hydrophone with a preamplifier, and an analog-digital converter. The applicability of the linear model and the efficiency of the proposed algorithms are substantiated experimentally.
