Peter Elbau

Reconstruction formulas for photoacoustic sectional imaging

The literature on reconstruction formulas for photoacoustic tomography is vast. The various reconstruction formulas differ in the measurement devices used and geometry on which the data are sampled. In standard photoacoustic imaging (PAI), the object under investigation is illuminated uniformly. Recently, sectional PAI 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 PAI.

Generalized Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed, and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting (see Schuster et al. [13]). We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer [11], and generalize existing convergence rates results. The key tool in our analysis is a homogeneous source condition, which we put into relation to the other existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces (see Groetsch [7]).

Non-local functionals for imaging

Non-local functionals have been successfully applied in a variety of applications, such as spectroscopy or in general filtering of time-dependent data. We mention the patch-based denoising of image sequences [Boulanger et al. IEEE Transactions on Medical Imaging (2010)]. Another family of non-local functionals considered in these notes approximates total variation denoising. Thereby we rely on fundamental characteristics of Sobolev spaces and the space of functions of finite total variation (see [Bourgain et al. Journal d’Analyse Mathematique 87, 77–101 (2002)] and several follow up papers). Standard results of the calculus of variations, like for instance the relation between lower semi-continuity of the functional and convexity of the integrand, do not apply, in general, for the non-local functionals. In this paper we address the questions of the calculus of variations for non-local functionals and derive relations between lower semi-continuity of the functionals and separate convexity of the integrand. Moreover, we use the new characteristics of Sobolev spaces to derive novel approximations of the total variation energy regularisation. All the functionals are well-posed and reveal a unique minimising point. Even more, existing numerical schemes can be recovered in this general framework.

Inverse problems of combined photoacoustic and optical coherence tomography

Optical coherence tomography (OCT) and photoacoustic tomography are emerging non‐invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform photoacoustic tomography and OCT imaging at once. In this paper, we present a mathematical model describing the dual experiment. Because OCT is mathematically modelled by Maxwells 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 Maxwells equations. As a by‐product, we therefore derive a new mathematical model of photoacoustic tomography based on Maxwells 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 with a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible.

Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

ABSTRACT In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

Mathematical Methods of Optical Coherence Tomography.

In this chapter a general mathematical model of Optical Coherence Tomography (OCT) is presented on the basis of the electromagnetic theory. OCT produces high resolution images of the inner structure of biological tissues. Images are obtained by measuring the time delay and the intensity of the backscattered light from the sample considering also the coherence properties of light. The scattering problem is considered for a weakly scattering medium located far enough from the detector. The inverse problem is to reconstruct the susceptibility of the medium given the measurements for dierent positions of the mirror. Dierent approaches are addressed depending on the dierent assumptions made about the optical properties of the sample. This procedure is applied to a full eld OCT system and an extension to standard (time and frequency domain) OCT is briey presented.

Modelling the Effect of Focusing Detectors in Photoacoustic Sectional Imaging

To effectively use photoacoustic tomography to obtain cross-sectional images, a combination of a focused laser illumination and focusing acoustic detectors is used. In this work, we discuss how to incorporate cylindrically shaped focusing detectors in the mathematical modelling of photoacoustics and derive approximative reconstruction formulas. Moreover, we show how such focusing detectors combined with a focused illumination can yield a quantitative reconstruction of the material properties (the transport coefficient and the product of the Gruneisen parameter and the absorption coefficient) of a weakly scattering medium.

Quantitative Thermoacoustic Tomography with microwaves sources

Abstract We investigate a quantitative thermoacoustic tomography process. We aim to recover the electric susceptibility and the conductivity of a medium when the sources are in the microwaves range. We focus on the case where the source signal has a slow time-varying envelope. We present the direct problem coupling equations for the electric field, the temperature variation and the pressure (to be measured via sensors). Then we give a variational formulation of the inverse problem which takes into account the entire electromagnetic, thermal and acoustic coupled system, and perform the formal computation of the optimality system.

The inverse scattering problem for orthotropic media in polarization-sensitive optical coherence tomography

In this paper we provide for a first time, to our knowledge, a mathematical model for imaging an anisotropic, orthotropic medium with polarization-sensitive optical coherence tomography. The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell’s equations. Our reconstruction method is based on the second-order Born-approximation of the electric field.

Quantitative photoacoustic imaging in the acoustic regime using SPIM

While in standard photoacoustic imaging the propagation of sound waves is modeled by the standard wave equation, our approach is based on a generalized wave equation with variable sound speed and material density, respectively. In this paper we present an approach for photoacoustic imaging, which in addition to recovering of the absorption density parameter, the imaging parameter of standard photoacoustics, also allows to reconstruct the spatially varying sound speed and density, respectively, of the medium. We provide analytical reconstruction formulas for all three parameters based in a linearized model based on single plane illumination microscopy (SPIM) techniques.