Linh V. Nguyen
We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both time-domain and frequency-domain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.
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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 prove a uniqueness theorem for compactly supported initial data for the variable speed wave equation arising in models of thermoacoustic tomography, given measurements on a part of the boundary. The proof is based on domain of dependence arguments and D. Tataru's unique continuation theorem.
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…
The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic and Photoacoustic Tomography.
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 th…
The time reversal method in thermoacoustic tomography is used for approximating the initial pressure inside a biological object using measurements of the pressure wave made on a surface surrounding the object. This article presents error estimates for the time reversal method in the cases of variable, non-trapping sound speeds.
Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. Solution of this problem is needed (both in 2-D and 3-D) because frequently the region of interest cannot be completely surrounded by the detectors, as it happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2-D (similar methods are applicable in the 3-D case). Our method is based on the numerical approximation of plane waves by certain single layer pot…
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.
Remarks on the general Funk-Radon transform and thermoacoustic tomography
研究光滑流形中一族超曲面定义的一般积分变换的性质，包括Sobolev范数估计、值域条件和积分变换核的逼近定理，并讨论在热/光/光声断层成像中出现的球面平均变换的应用。
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…
On reconstruction formulas and algorithms for the thermoacoustic tomography
本文综述了热声断层成像中唯一性建立、反演公式与算法的最新进展，数学上归结为波动方程的特殊逆问题或常声速下的球面平均变换问题。
Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
本文提出一种简单方法，适用于任意点探测器几何布局和满足非捕获条件的变声速，实现热声和光声断层扫描的重建，并证明了重建的唯一性。
Mathematical Challenges Arising in Thermoacoustic Tomography with Line Detectors
本文针对线探测器热声CT建立数学框架，将三维问题简化为二维波动方程初始数据重建，并推导出快速数值实现的解析重建公式。
The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. The article discusses known and provides new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.
Mathematics of Photoacoustic and Thermoacoustic Tomography
本文综述了热声和光声层析成像的数学模型、问题及算法，重点介绍了这种混合成像方法在分辨率和对比度上的优势。
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.
This work considers the time-domain fluorescence diffuse optical tomography (FDOT). We recover the distribution of fluorophores in biological tissue by the boundary measurements. With the Laplace transform and the knowledge of complex analysis, we build the uniqueness theorem of this inverse problem. After that, the numerical reconstructions are considered. We introduce a non-iterative inversion strategy by peak detection and an iterative inversion algorithm under the framework of regularizing scheme, then give several numerical examples in three-dimensional space illustrating the performance …
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…
Ultrasound modulated optical tomography, also called acousto-optics tomography, is a hybrid imaging modality that aims to combine the high contrast of optical waves with the high resolution of ultrasound. We follow the model of the influence of ultrasound modulation on the light intensity measurements developed in [Bal Schotland PRL 2010]. We present sufficient conditions ensuring that the absorption and diffusion coefficients modeling light propagation can locally be uniquely and stably reconstructed from the corresponding available information. We present an iterative procedure to solve such…
Magneto-Acousto-Electric Tomography (MAET), also known as the Lorentz force or Hall effect tomography, is a novel hybrid modality designed to be a high-resolution alternative to the unstable Electrical Impedance Tomography. In the present paper we analyze existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with MAET. It consists in applying to the data one of the algorithms of Thermo-Acoustic tomography, followed by solving the Neumann problem for the Laplace equation and the Poisson equation. For the particular case when the …
We consider the problem of elastic diffraction tomography, which consists in reconstructing elastic properties (i.e. mass density and elastic Lamé parameters) of a weakly scattering medium from full-field data of scattered waves outside the medium. Elastic diffraction tomography refers to the elastic inverse scattering problem after linearization using a firstorder Born approximation. In this paper, we prove the Fourier diffraction theorem, which relates the 2D Fourier transform of scattered waves with the Fourier transform of the scatterer in the 3D spatial Fourier domain. Elastic wave mode s…
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 propose and analyze a mathematical model of Modulated Luminescent Tomography. We show that when single X-rays or focused X-rays are used as an excitation, the problem is similar to the inversion of weighted X-ray transforms. In particular, we give an explicit inversion in the case of Dual Cone X-ray excitation.
Magneto-Acousto-Electric Tomography with Magnetic Field Measurements: Modeling, Inversion and Stability
本文提出了一种结合超声与静态磁场的磁声电断层成像方法，通过建立准静态数学模型和引入伴随问题，实现了电导率的两步反演并证明了其稳定性。
This paper is devoted to a Radon-type transform arising in a version of Photoacoustic Tomography that uses integrating circular detectors. We show that the transform can be decomposed into the spherical Radon transform and the Funk-Minkowski transform. An inversion formula and a range description are obtained by combining existing inversion for- mulas and range descriptions for the spherical Radon transform and the Funk-Minkowski transform. Numerical simulations are performed to demonstrate our proposed algorithm.
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct a function $f$ supported in an $n$-dimensional ball $B$, if the spherical means of $f$ are known over all geodesic spheres centered on the boundary of $B$.} We propose a new unified approach based on the idea of analytic continuation. This approach gives explicit inversion formulas not only for the Euclidean space $\bbr^n$ (as in the original set-up) but al…
Photoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT, in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed…
Direct inversion scheme of time-domain fluorescence diffuse optical tomography by asymptotic analysis of peak time
本文提出一种基于峰值时间渐近分析的FDOT直接反演方案，通过构造球体与四面体并推导半径与峰值时间的渐近关系，实现点目标位置的精确重建，数值测试验证了其准确性与性能。
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo- acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in Rn surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.
Diffraction Tomography, Fourier Reconstruction, and Full Waveform Inversion
本文综述衍射层析成像，比较低/高对比度物体重建，并实现全波形反演方法，该方法利用总场，对多次散射更稳健。
Full Field Inversion of the Attenuated Wave Equation: Theory and Numerical Inversion
本文研究空间变化声速与阻尼下的全场光声断层成像，证明单时刻全场波动反演问题的唯一性与稳定性，并提出基于迭代和变分正则化的数值反演算法，通过全角度和有限角度数据模拟验证了方法有效性。
V-Line Tensor Tomography in a Disk: Theoretical and Numerical Reconstruction
研究圆盘内对称m-张量场的V线变换，给出核的显式刻画并推导新的反演公式，通过数值实验验证了向量场和对称2-张量场反演算法的有效性与鲁棒性。
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