Guillaume Bal

Frequency Domain Optical Tomography Based on the Equation of Radiative Transfer

Optical tomography consists of reconstructing the spatial distribution of absorption and scattering properties of a medium from surface measurements of transmitted light intensities. Mathematically, this problem amounts to parameter identification for the equation of radiative transfer (ERT) with diffusion-type boundary measurements. Because they are posed in the phase-space, radiative transfer equations are quite challenging to solve computationally. Most past works have considered the steady-state ERT or the diffusion approximation of the ERT. In both cases, substantial cross-talk has been observed in the reconstruction of the absorption and scattering properties of inclusions. In this paper, we present an optical tomographic reconstruction algorithm based on the frequency-domain ERT. The inverse problem is formulated as a regularized least-squares minimization problem, in which the mismatch between forward model predictions and measurements is minimized. The ERT is discretized by using a discrete ordinates method for the directional variables and a finite volume method for the spatial variables. A limited-memory quasi-Newton algorithm is used to minimize the least-squares functional. Numerical simulations with synthetic data show that the cross-talk between the two optical parameters is significantly reduced in reconstructions based on frequency-domain data as compared to those based on steady-state data.

Inverse transport theory and applications

Inverse transport consists of reconstructing the optical properties of a domain from measurements performed at the domains boundary. This review concerns several types of measurements: time-dependent, time-independent, angularly resolved and angularly averaged measurements. We review recent results on the reconstruction of the optical parameters from such measurements and the stability of such reconstructions. Inverse transport finds applications e.g. in medical imaging (optical tomography, optical molecular imaging) and in geophysical imaging (remote sensing in the Earths atmosphere).

Algorithm for solving the equation of radiative transfer in the frequency domain

We present an algorithm that provides a frequency-domain solution of the equation of radiative transfer (ERT) for heterogeneous media of arbitrary shape. Although an ERT is more accurate than a diffusion equation, no ERT code for the widely employed frequency-domain case has been developed to date. In this work the ERT is discretized by a combination of discrete-ordinate and finite-volume methods. Two numerical simulations are presented.

Inverse diffusion theory of photoacoustics

This paper analyzes the reconstruction of diffusion and absorption parameters in an elliptic equation from knowledge of internal data. In the application of photoacoustics, the internal data are the amount of thermal energy deposited by high frequency radiation propagating inside a domain of interest. These data are obtained by solving an inverse wave equation, which is well studied in the literature. We show that knowledge of two internal data based on well-chosen boundary conditions uniquely determines two constitutive parameters in diffusion and Schrodinger equations. Stability of the reconstruction is guaranteed under additional geometric constraints of strict convexity. No geometric constraints are necessary when 2n internal data for well-chosen boundary conditions are available, where n is spatial dimension. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.

SELF-AVERAGING IN TIME REVERSAL FOR THE PARABOLIC WAVE EQUATION

We analyze the self-averaging properties of time-reversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. This allows us to construct an approximate martingale for the phase space Wigner transform of two wave fields. Using a prioriL2-bounds available in the time-reversal setting, we prove that the Wigner transform in the high frequency limit converges in probability to its deterministic limit, which is the solution of a transport equation.

A “Parareal” Time Discretization for Non-Linear PDE’s with Application to the Pricing of an American Put

In this paper, we introduce a new implementation of the “parareal” time discretization aimed at solving unsteady nonlinear problems more efficiently, in particular those involving non-differentiable partial differential equations. As in the former implementation [3], the main goal of this scheme is to parallelize the time discretization to obtain an important speed up. As an application in financial mathematics, we consider the Black-Scholes equations for an American put. Numerical evidence of the important savings in computational time is also presented.

On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations

After stating an abstract convergence result for the parareal algorithm used in the parallelization in time of general partial differential equations, we analyze the stability and convergence properties of the algorithm for equations with constant coefficients. We show that suitably damping coarse schemes ensure unconditional stability of the parareal algorithm and analyze how the regularity of the initial condition influences convergence in the absence of sufficient damping.

Multi-source quantitative photoacoustic tomography in a diffusive regime

Photoacoustic tomography (PAT) is a novel hybrid medical imaging technique that aims to combine the large contrast of optical coefficients with the high-resolution capabilities of ultrasound. We assume that the first step of PAT, namely the reconstruction of a map of absorbed radiation from ultrasound boundary measurement, has been done. We focus on quantitative photoacoustic tomography, which aims at quantitatively reconstructing the optical coefficients from knowledge of the absorbed radiation map. We present a non-iterative procedure to reconstruct such optical coefficients, namely the diffusion and absorption coefficients, and the Gruneisen coefficient when the propagation of radiation is modeled by a second-order elliptic equation. We show that PAT measurements allow us to uniquely reconstruct only two out of the above three coefficients, even when data are collected using an arbitrary number of radiation illuminations. We present uniqueness and stability results for the reconstructions of two such parameters and demonstrate the accuracy of the reconstruction algorithm with numerical reconstructions from two-dimensional synthetic data.

Time Reversal and Refocusing in Random Media

In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers, time reversed, and finally reemitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the reemitted signals at the location of the initial source is improved when the medium is heterogeneous. Contrary to intuition, multiple scattering enhances the spatial resolution of the refocused signal and allows one to beat the diffraction limit obtained in homogeneous media. This paper presents a quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media. The theory is based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit. Numerical experiments complement the theory.

Electromagnetic Time-Reversal Source Localization in Changing Media: Experiment and Analysis

An experimental study is performed on electromagnetic time reversal in highly scattering environments, with a particular focus on performance when environmental conditions change. In particular, we consider the case for which there is a mismatch between the Greens function used on the forward measurement and that used for time-reversal inversion. We examine the degradation in the time-reversal image with increasing media mismatch, and consider techniques that mitigate such degradation. The experimental results are also compared with theoretical predictions for time reversal in changing media, with good agreement observed