Optical coherence tomography (OCT) and photoacoustic tomography (PAT) are emerging non-invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform PAT and OCT imaging at once. In this paper we present a mathematical model describing the dual experiment.
Since OCT is mathematically modelled by Maxwell's equations or some simplifications of it, whereas the light propagation in quantitative photoacoustics is modelled by (simplifications of) the radiative transfer equation, the first step in the derivation of a mathematical model of the dual experiment is to obtain a unified mathematical description, which in our case are Maxwell's equations. As a by-product we therefore derive a new mathematical model of photoacoustic tomography based on Maxwell's equations.
It is well known by now, that without additional assumptions on the medium, it is not possible to uniquely reconstruct all optical parameters from either one of these modalities alone. We show that in the combined approach one has additional information, compared to a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible.
In this paper we perform quantitative reconstruction of the electric susceptibility and the Grüneisen parameter of a non-magnetic linear dielectric medium using measurement of a multi-modal photoacoustic and optical coherence tomography system. We consider the mathematical model presented in [11], where a Fredholm integral equation of the first kind for the Grüneisen parameter was derived. For the numerical solution of the integral equation we consider a Galerkin type method.
Photoacoustic tomography (PAT) is a hybrid imaging modality that intends to construct high-resolution images of optical properties of heterogeneous media from measured acoustic data generated by the photoacoustic effect. To date, most of the model-based quantitative image reconstructions in PAT are performed with either the radiative transport equation or its classical diffusion approximation as the model of light propagation. In this work, we study quantitative image reconstructions in PAT using the simplified $P_2$ equations as the light propagation model. We provide numerical evidences on t…
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.
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.
We analyze the mathematical model of multiwave tomography with a variable speed with integrating measurements on planes tangent to a sphere surrounding the source. We prove sharp uniqueness and stability estimates with full and partial data and propose a time reversal algorithm which recovers the visible singularities.
We treat the stability issue for the three dimensional inverse imaging modality called Quantitative Photoacoustic Tomography. We provide universal choices of the illuminations which enable to recover, in a Hölder stable fashion, the diffusion and absorption coefficients from the interior pressure data. With such choices of illuminations we do not need the nondegeneracy conditions commonly used in previous studies, which are difficult to be verified a-priori.
In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.
In the mathematical analysis of photoacoustic imaging, it is usually assumed that the acoustic pressure (Dirichlet data) is measured on a detection surface. However, actual ultrasound detectors gather data of a different type. In this paper, we propose a more realistic mathematical model of ultrasound measurements acquired by the Fabry--Perot sensor. This modeling incorporates directional response of such sensors. We study the solvability of the resulting photoacoustic tomography problem, concluding that the problem is well-posed under certain assumptions. Numerical reconstructions are impleme…
We study an inverse initial source problem that models Photoacoustic tomography measurements with array detectors, and introduce a method that can be viewed as a modification of the so called back and forth nudging method. We show that the method converges at an exponential rate under a natural visibility condition, with data given only on a part of the boundary of the domain of wave propagation. In this paper we consider the case of noiseless measurements.
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 development of efficient and accurate image reconstruction algorithms is one of the cornerstones of computed tomography. Existing algorithms for quantitative photoacoustic tomography currently operate in a two-stage procedure: First an inverse source problem for the acoustic wave propagation is solved, whereas in a second step the optical parameters are estimated from the result of the first step. Such an approach has several drawbacks. In this paper we therefore propose the use of single-stage reconstruction algorithms for quantitative photoacoustic tomography, where the optical parameter…
This paper is concerned with robust preconditioning of wave equations constrained linear inverse problems from boundary observation data. The main result of this paper is a concept for regularization parameter robust preconditioning. Analogous concepts have been developed for control problems based on elliptic partial equations before.
The goal of quantitative photoacoustic tomography is to determine optical and acoustical material properties from initial pressure maps as obtained, for instance, from photoacoustic imaging. The most relevant parameters are absorption, diffusion and Grueneisen coefficients, all of which can be heterogeneous. Recent work by Bal and Ren shows that in general, unique reconstruction of all three parameters is impossible, even if multiple measurements of the initial pressure (corresponding to different laser excitation directions at a single wavelength) are available. Here, we propose a restriction…
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…
This paper is concerned with the stability issue in determining absorption and diffusion coefficients in quantitative photoacoustic imaging. Assuming that the optical wave is generated by point sources in a region where the optical coefficients are known, we derive pointwise H{ö}lder stability estimate of the inversion. This result shows that the reconstruction of the optical coefficients is stable in the region close to the optical illumination sources and deteriorate exponentially far away. Our stability estimate is therefore in accordance with known experimental observations. Mathematics su…
In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based ba…
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 …
Quantitative image reconstruction in photoacoustic tomography requires the solution of a coupled physics inverse problem involvier light transport and acoustic wave propagation. In this paper we address this issue employing the radiative transfer equation as accurate model for light transport. As main theoretical results, we derive several stability and uniqueness results for the linearized inverse problem. We consider the case of single illumination as well as the case of multiple illuminations assuming full or partial data. The numerical solution of the linearized problem is much less costly…
The literature on reconstruction formulas for photoacoustic tomography (PAT) is vast. The various reconstruction formulas differ by used measurement devices and geometry on which the data are sampled. In standard photoacoustic imaging (PAI), the object under investigation is illuminated uniformly. Recently, sectional photoacoustic imaging techniques, using focusing techniques for initializing and measuring the pressure along a plane, appeared in the literature. This paper surveys existing and provides novel exact reconstruction formulas for sectional photoacoustic imaging.
Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT measurement setup, the used data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full…
We consider the reconstruction of the diffusion and absorption coefficients of the diffusion equation from the internal information of the solution obtained from the first step of the inverse photoacoustic tomography (PAT). In practice, the internal information is only partially provided near the boundary due to the high absorption property of the medium and the limitation of the equipment. Our main contribution is to prove a Hölder stability of the inverse problem in a subregion where the internal information is reliably provided based on the stability estimation of a Cauchy problem satisfied…
Almost all known image reconstruction algorithms for photoacoustic and thermoacoustic tomography assume that the acoustic waves leave the region of interest after a finite time. This assumption is reasonable if the reflections from the detectors and surrounding surfaces can be neglected or filtered out (for example, by time-gating). However, when the object is surrounded by acoustically hard detector arrays, and/or by additional acoustic mirrors, the acoustic waves will undergo multiple reflections. (In the absence of absorption they would bounce around in such a reverberant cavity forever). T…
We present a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in photoacoustic tomography (PAT). This framework comprises of minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in PAT. Further, the objective functional consists of an $L^1$ regularization term that promotes s…
Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging modality that has great potential for a wide range of biomedical imaging applications. In this Note, we derive a hybrid reconstruction formula that is mathematically exact and operates on a data function that is expressed in the temporal frequency and spatial domains. This formula explicitly reveals new insights into how the spatial frequency components of the sought-after object function are determined by the temporal frequency components of the data function measured with a circular or sph…
The goal of \emph{photoacoustic tomography} (PAT) is to estimate an \emph{initial pressure function} $φ$ from pressure data measured at a boundary surrounding the object of interest. This paper is concerned with a time reversal method for PAT that is based on the dissipative wave equation of Nachman, Smith and Waag\cite{NaSmWa90}. This equation has the advantage that it is more accurate than the \emph{thermo-viscous} wave equation. For simplicity, we focus on the case of one \emph{relaxation process}. We derive an exact formula for the \emph{time reversal image} $\I$, which depends on the \emp…
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…
In this article, we revisit iterative methods for solving the inverse problem of photoacoustic tomography in free space. Recently, there have been interesting developments on explicit formulations of the adjoint operator, demonstrating that iterative methods are an attractive choice for photoacoustic image reconstruction. In this work, we propose several modifications of current formulations of the adjoint operator which help speed up the convergence and yield improved error estimates. We establish a stability analysis and show that, with our choices of the adjoint operator, the iterative meth…
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.
Fast algorithms enabling optimization and deep learning for photoacoustic tomography in a circular detection geometry
Research Unit of Mathematical Sciences, University of Oulu（奥卢大学数学科学研究所）；Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Department of Mathematics, University of Arizona（亚利桑那大学数学系）；Department of Technical Physics, University of Eastern Finland（东部芬兰大学技术物理系）
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.
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.
Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography (PAT), we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first- and second-order linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purpose, we also establish the stability of the reconstruction of the absorption coefficients with respect t…
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 study the inverse source problem in photoacoustic tomography (PAT) for mixed data, which denote a weighted linear combination of the acoustic pressure and its normal derivative on an observation surface. We consider in particular the case where the data are only available on finite time intervals, which accounts for real-world usage of PAT where data are only feasible within a certain time interval. Extending our previous work, we derive explicit formulas up to a smoothing integral on convex domains with a smooth boundary, yielding exact reconstruction for circular or elliptical domains. We…
In this paper, we study the photoacoustic tomography problem for which we seek to recover both the initial state of the pressure field and the wave speed of the medium from the knowledge of a single boundary measurement. The goal is to propose practical assumptions to define a set of initial conditions and wave speeds over which uniqueness for this inverse problem is guaranteed. The main result of the paper is that given two sets of wave speeds and pressure profiles, they cannot produce the same acoustic measurements if the relative difference between the wave speeds is much smaller than the r…
