High spatial resolution is a central aspect of tomographic imaging. In photoacoustic tomography, this requires accurate models for acoustic wave propagation and corresponding efficient image reconstruction methods. In this article we consider such models accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. Our approach comes with a clear convergence analysis that is not shared by existing approaches for the considered model class. In contrast to time reversal the employed adjoint wave equation is again damping and thus has a stable solution. This stability property can be clearly seen in our numerical results. Indeed, if the attenuation is not too strong then the reconstruction results are even better than in the absence of attenuation. Moreover, the presented numerical results clearly demonstrate the efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case, where alternative methods cannot be directly applied.
A different class of algorithms extends the time reversal technique to the attenuated case (see @cite_48 @cite_24 @cite_38 @cite_3 @cite_37 @cite_69 @cite_4 @cite_0 ). Note that the time reversal of the attenuated wave equation yields a noise amplifying equation. Therefore regularization methods have to be incorporated in its numerical solution. Opposed to the time reversal, the adjoint wave equation used in our approach is again damping and no regularization is required for its stable solution. This yields a clear convergence analysis for our method by using standard regularization theory @cite_58 @cite_14 @cite_57 . We are not aware of similar existing results for PAT in attenuating acoustic media. The approaches which are closest to our work seem @cite_74 @cite_67 . In @cite_74 discrete iterative methods are considered, where the problem is first discretized and the adjoint is computed from the discretized problem. Further, both works @cite_74 @cite_67 use attenuation models based on the fractional Laplacian (see @cite_29 @cite_61 ) which yields an equation that is non-local in space. It is not obvious how to extend these approaches to model which can also include memory.
@cite_61 @cite_38 @cite_37 @cite_69 @cite_4 @cite_14 @cite_67 @cite_48 @cite_29 @cite_3 @cite_24 @cite_0 @cite_57 @cite_74 @cite_58
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The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simula... Photoacoustic imaging is based on the generation of acoustic waves in a semitransparent sample after illumination with short pulses of light or radio waves. The goal is to recover the spatial distribution of absorbed energy density inside the sample from acoustic pressure signals measured outside the sample (photoacoustic inverse problem). We have proposed a numerical method to calculate directly the time reversed field by retransmitting the measured pressure on the detection surface in reversed temporal order. This model-based time reversal method can solve the photoacoustic inverse problem exactly for an arbitrary closed detection surface. Recently we presented a set up which requires a single rotation axis and line detectors perpendicular to the rotation axis. Using a two-dimensional reconstruction method, such as time reversal in two dimensions, and applying the inverse two-dimensional radon transform afterwards gives an exact reconstruction of a three-dimensional sample with this set up. The resolution in photoacoustic imaging is limited by the acoustic bandwidth and therefore by acoustic attenuation, which can be substantial for high frequencies. This effect is usually ignored in reconstruction algorithms but has a strong impact on the resolution of small structures. It is demonstrated that the model based time reversal method allows to partly compensate this effect. In this paper, we derive time reversal imaging functionals for two strongly causal acoustic attenuation models, which have been proposed recently. The time reversal techniques are based on recently proposed ideas of for the thermo-viscous wave equation. Here and there, an asymptotic analysis provides reconstruction functionals from first order corrections for the attenuating effect. In addition, we present a novel approach for higher order corrections. Copyright © 2013 John Wiley & Sons, Ltd. In this article we study the reconstruction problem in TAT PAT on an attenuating media. Namely, we prove a reconstruction procedure of the initial condition for the damped wave equation via Neumann series that works for arbitrary large smooth attenuation coefficients extending the result of Homan in (2013 Inverse Problems Imaging 7 1235–50). We also illustrate the theoretical result by including some numerical experiments at the end of the paper. Inspired by the recent advances on minimizing nonsmooth or bound-constrained convex functions on models using varying degrees of fidelity, we propose a line search multi-grid (MG) method for full-wave iterative image reconstruction in photoacoustic tomography (PAT) in heterogeneous media. To compute the search direction at each iteration, we decide between the gradient at the target level, or alternatively an approximate error correction at a coarser level, relying on some predefined criteria. To incorporate absorption and dispersion, we derive the analytical adjoint directly from the first-order acoustic wave system. The effectiveness of the proposed method is tested on a total-variation penalized Iterative Shrinkage Thresholding algorithm (ISTA) and its accelerated variant (FISTA), which have been used in many studies of image reconstruction in PAT. The results show the great potential of the proposed method in improving speed of iterative image reconstruction. In this article we study the inverse problem of thermoacoustic tomography (TAT) on a medium with attenuation represented by a time- convolution (or memory) term, and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters. Under the assumption of being able to measure data on the whole boundary, we prove uniqueness and stability, and propose a convergent reconstruction method for a class of smooth variable sound speeds. By a suitable modification of the time reversal technique, we obtain a Neumann series reconstruction formula. Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation. In this study, a linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation. A new definition of the fractional Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional Laplacian. Under the Szabo’s smallness approximation, where attenuation is assumed to be much smaller than the wave number, the linear model is found consi... We consider a mathematical model of thermoacoustic tomography and other multi-wave imaging techniques with variable sound speed and attenuation. We find that a Neumann series reconstruction algorithm, previously studied under the assumption of zero attenuation, still converges if attenuation is sufficiently small. With complete boundary data, we show the inverse problem has a unique solution, and modified time reversal provides a stable reconstruction. We also consider partial boundary data, and in this case study those singularities that can be stably recovered. The reconstruction of photoacoustic images typically neglects the effect of acoustic absorption on the measured time domain signals. Here, a method to compensate for acoustic absorption in photoacoustic tomography is described. The approach is based on time-reversal image reconstruction and an absorbing equation of state which separately accounts for acoustic absorption and dispersion following a frequency power law. Absorption compensation in the inverse problem is achieved by reversing the absorption proportionality coefficient in sign but leaving the equivalent dispersion parameter unchanged. The reconstruction is regularized by filtering the absorption and dispersion terms in the spatial frequency domain using a Tukey window. This maintains the correct frequency dependence of these parameters within the filter pass band. The method is valid in one, two and three dimensions, and for arbitrary power law absorption parameters. The approach is verified through several numerical experiments. The reconstruction of a carbon fibre phantom and the vasculature in the abdomen of a mouse are also presented. When absorption compensation is included, a general improvement in the image magnitude and resolution is seen, particularly for deeper features. Existing approaches to image reconstruction in photoacoustic computed tomography (PACT) with acoustically heterogeneous media are limited to weakly varying media, are computationally burdensome, and or cannot effectively mitigate the effects of measurement data incompleteness and noise. In this work, we develop and investigate a discrete imaging model for PACT that is based on the exact photoacoustic (PA) wave equation and facilitates the circumvention of these limitations. A key contribution of the work is the establishment of a procedure to implement a matched forward and backprojection operator pair associated with the discrete imaging model, which permits application of a wide-range of modern image reconstruction algorithms that can mitigate the effects of data incompleteness and noise. The forward and backprojection operators are based on the k-space pseudospectral method for computing numerical solutions to the PA wave equation in the time domain. The developed reconstruction methodology is investigated by use of both computer-simulated and experimental PACT measurement data.