Richard Kowar

Attenuation Models in Photoacoustics

The aim of this chapter is to review attenuation models in photoacoustic imaging and discuss their causality properties. We also derive integro-differential equations which the attenuated waves are satisfying and highlight the ill–conditionness of the inverse problem for calculating the unattenuated wave from the attenuated one, which has been discussed in Chap. 3.The difficult issue of effects of and corrections for the attenuation of acoustic waves in photoacoustic imaging has been studied [8, 1, 10, 6], although no complete conclusion on the feasibility of this models has been reached. Mathematical models of attenuation are formulated in the frequency domain, since the attenuation is known to be strongly frequency dependent. Let G0, G be the Green functions of the wave equation and the attenuated wave equation, respectively. The common attenuation model reads as follows:

KACZMARZ METHODS FOR REGULARIZING NONLINEAR ILL-POSED EQUATIONS II: APPLICATIONS

In part I we introduced modified Landweber-Kaczmarz methods and have established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.

Causality analysis of frequency‐dependent wave attenuation

The work is inspired by thermo-and photoacoustic imaging, where recent efforts are devoted to take into account attenuation and varying wave speed parameters. In this paper we derive and analyze causal equations describing propagation of attenuated pressure waves. We also review standard models, like frequency power laws, and the thermo-viscous equation and show that they lack causality in the parameter range relevant for biological photoacoustic imaging. To discuss causality in mathematical rigor we use the results and concepts of linear system theory. We present some numerical experiments, which show the physically unmeaningful behavior of standard attenuation models, and the realistic behavior of the novel models. PACS numbers: 43.35.Ud, 43.20.Hq

Integral equation models for thermoacoustic imaging of acoustic dissipative tissue

The main task of thermoacoustic imaging is the estimation of a function, denoted by , which depends on the electromagnetic absorption function and the optical scattering properties of the tissue. In the absence of acoustic dissipation, the parameter function can be estimated from one of the three types of projections (spherical, circular or planar). In the case of acoustic dissipative wave propagation in tissue, it is no longer possible to explicitly calculate the projection of from the respective pressure data (measured by point, planar or line detectors). The goal of this paper is to derive for each of the three types of pressure detectors, an integral equation that allows estimation of the respective projections of . The advantage of this approach is that known reconstruction formulas for using the respective projection data can then be exploited.

On the Landweber iteration for the solution of a parameter identification problem in a hyperbolic partial differential equation of second order

In this paper we study the problem of recovering a onedimensional parameter in a hyperbolic partial differential equation of second order. We consider the integral statement of the inverse problem and investigate the properties of the parameter-to-solution mapping, which allows to give a rigorous analysis of the Landweber iteration.

On Time Reversal in Photoacoustic Tomography for Tissue Similar to Water

This paper is concerned with time reversal in photoacoustic tomography of dissipative media that are similar to water. Under an appropriate condition, it is shown that the time reversal method in [A. Wahab, Modeling and Imaging of Attenuation in Biological Media, Ph.D. dissertation, Ecole Polytechnique Palaiseau, Paris, 2011; H. Ammari et al., in Mathematical and Statistical Methods for Imaging, AMS, Providence, RI, 2011, pp. 151--163] based on the noncausal thermoviscous wave equation can be used if the noncausal data are replaced by a time shifted set of causal data. We investigate a similar imaging functional for time reversal and an operator equation with the time reversal image as the right-hand side. If required, an enhanced image can be obtained by solving this operator equation. Although time reversal (for noise-free data) does not lead to the exact initial pressure function, the theoretical and numerical results of this paper show that regularized time reversal in dissipative media similar to wat...

Estimation of the density, the wave speed and the acoustic impedance function in ultrasound imaging

In this paper, we derive integral equations for estimating the density function ? and the wave speed function c or the acoustic impedance function from boundary measurement data. A nonlinear integral equation for the special case ? = const and c ? const is discussed in detail. We show that the inverse problem of estimating the acoustic impedance function from boundary data measured via a zero-offset transducer configuration can be decomposed into two successively solvable inverse problems. The first inverse problem corresponds to the estimation of the spherical mean function of the acoustic impedance function from a set of one-dimensional integral equations. The second inverse problem corresponds to the estimation of the acoustic impedance function which is closely related to the estimation of the electromagnetic absorption function in transparent media. We conclude the paper with some numerical experiments.

Iterative methods for photoacoustic tomography in attenuating acoustic media

The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation 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 an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. 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. 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.

Time reversal for photoacoustic tomography based on the wave equation of Nachman, Smith and Waag

One goal of photoacoustic tomography (PAT) is to estimate an 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 [J. Acoust. Soc. Am. 88, 1584 (1990)]. This equation is a correction of the thermoviscous wave equation such that its solution has a finite wave front speed and, in contrast, it can model several relaxation processes. In this sense, it is more accurate than the thermoviscous wave equation. For simplicity, we focus on the case of one relaxation process. We derive an exact formula for the time reversal image I, which depends on the relaxation time τ(1) and the compressibility κ(1) of the dissipative medium, and show I(τ(1),κ(1)) → φ for κ(1) → 0. This implies that I = φ holds in the dissipation-free case and that I is similar to φ for sufficiently small compressibility κ(1). Moreover, we show for tissue similar to water that the small wave number approximation I(0) of the time reversal image satisfies I(0) = η(0)*(x)φ with η[over ̂](0)(|k|) ≈ const. for |k| ≪ 1/c(0)τ(1), where φ denotes the initial pressure function. For such tissue, our theoretical analysis and numerical simulations show that the time reversal image I is very similar to the initial pressure function φ and that a resolution of σ≈0.036mm is feasible (for exact measurement data).

On Causality of Thermoacoustic Tomography of Dissipative Tissue

Since all attenuation models for dissipative media that come into question for thermoacoustic tomography (TAT) violate causality, a causal attenuation model for TAT is proposed. A goal of this article is to discuss causality in the context of dissipative wave propagation and TAT. In the process we shortly discuss the frequency power law, a causal attenuation model (with a constant wave front speed which can be adjusted via an additional parameter) and the respective wave equation. Afterwards an integral equation model for estimating the unattenuated pressure data of TAT from the attenuated pressure data of TAT is presented and discussed. Our numerical results show a fast decrease of resolution of TAT for increasing distance of the object of interest from the pressure detector.