We consider the inverse source problem of thermo- and photoacoustic tomography, with data registered on an open surface partially surrounding the source of acoustic waves. Under the assumption of constant speed of sound we develop an explicit non-iterative reconstruction procedure that recovers the Radon transform of the sought source, up to an infinitely smooth additive error term. The source then can be found by inverting the Radon transform.
Our analysis is microlocal in nature and does not provide a norm estimate on the error in the so obtained image. However, numerical simulations show that this error is quite small in practical terms. We also present an asymptotically fast implementation of this procedure for the case when the data are given on a circular arc in 2D.
We consider the problem of thermoacoustic tomography (TAT), in which one needs to reconstruct the initial value of a solution of the wave equation from its value on an observation surface. We show that if some geometric rays for the equation do not intersect the observation surface, the reconstruction in TAT is not Hölder stable.
We derive an explicit method for reconstructing singularities of the initial data in a thermoacoustic tomography problem, in the case of variable sound speed and limited boundary data. In order to obtain this explicit formula we assume the metric induced by the sound speed does not have conjugate points inside the region to be observed.
We consider the mathematical model of photoacoustic and thermoacoustic tomography in media with a variable sound speed. When the sound speed is known, the explicit reconstruction formula by P. Stefanov and G. Uhlmann (Inverse Problems, 25(7):075011, 16, 2009) can be used. We study how a modelling error in the sound speed affects the reconstruction formula and quantify the effect in terms of a stability estimate.
The spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. Under a non-trapping hypothesis, we show that sparsity of the interior transmission spectrum implies a range separation condition for the thermoacoustic operator. In odd dimension greater than or equal to three, we prove that the transmission spectrum for a pair of radially symmetric non-trapping sound speeds is countable, and conclude that the ranges of the associated thermoacoustic maps have only trivial intersection.
In this article, we consider the inverse source problem arising in photoacoustic tomography in elastic media. We show that the time reversal method, proposed by Tittelfitz [Inverse Problems 28.5 (2012): 055004], converges with the sharp observation time without any constraint on the speeds of the longitudinal and shear waves.
We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.
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 Neuma…
We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to O(n^3 log n) and O(n^3 log^2 n) for the 3D techniques (…
We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary.
We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval, greater than the diameter of the domain. In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary.
We investigate the problem of recovering the initial displacement f for a solution u of a linear, isotropic, non-homogeneous elastic wave equation, given measurements of u on [0,T] x \partial Ω, where Ω\subset\R^3 is some bounded domain containing the support of f. For the acoustic wave equation, this problem is known as thermoacoustic tomography (TAT), and has been well-studied; for the elastic wave equation, the situation is somewhat more subtle, and we give sufficient conditions on the Lamé parameters to ensure that recovery is possible.
We explore the effect of sampling rates when measuring data given by $Mf$ for special operators $M$ arising in Thermoacoustic Tomography. We start with sampling requirements on $Mf$ given $f$ satisfying certain conditions. After this we discuss the resolution limit on $f$ posed by the sampling rate of $Mf$ without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when $Mf$ is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing.
Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stability estimates are obatined for an arbitrary variable principal part of that operator. Convergence of the Quasi-Reversibility Method to the exact solution is also established for this case. Both complete…
In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
We develop a paradigm using microlocal analysis that allows one to characterize the visible and added singularities in a broad range of incomplete data tomography problems. We give precise characterizations for photo- and thermoacoustic tomography and Sonar, and provide artifact reduction strategies. In particular, our theorems show that it is better to arrange Sonar detectors so that the boundary of the set of detectors does not have corners and is smooth. To illustrate our results, we provide reconstructions from synthetic spherical mean data as well as from experimental photoacoustic data.
In thermoacoustic and photoacoustic tomography an electromagnetic wave is sent through a medium, heating it and therefore generating an elastic expansion that in turns generates an acoustic wave that is measured outside the medium. The general problem is to recover both the inhomogeneous sound speed and the initial pressure from the boundary measurements of the solution of the acoustic wave equation with a single measurement of the pressure. We show that one can recover both the sound speed and the initial pressure assuming a monocitiy condition on the sound speed that includes the physically …
This paper concerns thermoacoustic tomography and photoacoustic tomography, two couple-physics imaging modalities that attempt to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. We give sufficient conditions to recover both the sound speed of the medium being probed and the source.
The attenuation of ultrasound waves in photoacoustic and thermoacoustic imaging presents an important drawback in the applicability of these modalities. This issue has been addressed previously in the applied and theoretical literature, and some advances have been made on the topic. In particular, stability inequalities have been proposed for the inverse problem of initial source recovery with partial observations under the assumption of unique determination of the initial pressure. The main goal of this work is to fill this gap, this is, we prove the uniqueness property for the inverse proble…
We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.
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.
We study the problem of recovery the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not=0$ with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation $u_{tt} -c^2(x)Δu =0$ with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.
We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtaine…
We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if the solutions of the wave equation with the source and sound speed $(f_1,c_1)$ and $(f_2,c_2)$ agree on the boundary of a bounded region $Ω$, then \[ \int_Ω(c_2^{-2}-c_1^{-2})φdy=0,\] for every harmonic function $φ\in C(\barΩ)$, which holds without any knowledge of the source. We also show t…
We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing solution of this problem in terms of the measured data, under the assumption of constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yi…
