Alexandre Jollivet

Inverse transport theory of photoacoustics

We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high-frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multi-measurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient g(x) in a Henyey–Greenstein phase function model. Finally, we derive stability estimates for the reconstructions.

STABILITY FOR TIME-DEPENDENT INVERSE TRANSPORT

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from full knowledge of the albedo operator at the boundary of a bounded domain of interest. We present optimal stability results on the reconstruction of the absorption and scattering parameters for a given error in the measured albedo operator.

Time-dependent angularly averaged inverse transport

This paper concerns the reconstruction of the absorption and scattering parameters in a time-dependent linear transport equation from knowledge of angularly averaged measurements performed at the boundary of a domain of interest. Such measurement settings find applications in medical and geophysical imaging. We show that the absorption coefficient and the spatial component of the scattering coefficient are uniquely determined by such measurements. We obtain stability results on the reconstruction of the absorption and scattering parameters with respect to the measured albedo operator. The stability results are obtained by a precise decomposition of the measurements into components with different singular behavior in the time domain.

On inverse scattering for the multidimensional relativistic Newton equation at high energies

Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) p=F(x), F(x)=−∇V(x), p=x∕1−(∣x∣2∕c2), p=dp∕dt, x=dx∕dt, x∊C1(R,Rd), where V∊C2(Rd,R), ∣∂xjV(x)∣⩽β∣j∣(1+∣x∣)−(α+∣j∣) for ∣j∣⩽2 and some α>1. We give estimates and asymptotics for scattering solutions and scattering data for the equation for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the x-ray transform PF. Applying results on inversion of the x-ray transform P we obtain that for d⩾2 the velocity valued component of the scattering operator at high energies uniquely determines F. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator does not uniquely determine F. The results of the present work were obtained in the process of generalizing some results of Novikov to the relativistic case.

On inverse problems in electromagnetic field in classical mechanics at fixed energy

In this article, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic and nonrelativistic Newton equations in a static external electromagnetic field (V, B), V∈C2, B∈C1 in classical mechanics. Developing the approach going back to Gerver-Nadinashvili 1983s work on an inverse problem of mechanics, we obtain, in particular, theorems of uniqueness.

Inverse scattering at high energies for the multidimensional Newton equation in a long range potential

We dene scattering data for the Newton equation in a potential V 2 C 2 (R n ;R), n 2, that decays at innity like r for some 2 (0; 1]. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at xed energy of the scattering data is also considered. We then change the denition of the scattering data to study inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of [Novikov, 1999].

Angular Average of Time-Harmonic Transport Solutions

We consider the angular averaging of solutions to time-harmonic transport equations. Such quantities model measurements obtained for instance in optical tomography, a medical imaging technique, with frequency-modulated sources. Frequency modulated sources are useful to separate ballistic photons from photons that undergo scattering with the underlying medium. This paper presents a precise asymptotic description of the angularly averaged transport solutions as the modulation frequency ω tends to ∞. Provided that scattering vanishes in the vicinity of measurements, we show that the ballistic contribution is asymptotically larger than the contribution corresponding to single scattering. Similarly, we show that singly scattered photons also have a much larger contribution to the measurements than multiply scattered photons. This decomposition is a necessary step toward the reconstruction of the optical coefficients from available measurements.

On inverse problems for the multidimensional relativistic Newton equation at fixed energy

In this paper, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic Newton equation with an external potential V, V ∈ C 2 . Using known results, we obtain, in particular, theorems of uniqueness.

On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field

Abstract We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field where V ∈ C 2(, ℝ), B(x) is the n × n real antisymmetric matrix with elements Bi,k (x), Bi,k ∈ C 1(, ℝ) (and B satisfies the closure condition), and ≤ β |j 1|(1 + |x|)–(α + |j 1|) for x ∈ , 1 ≤ |j 1| ≤ 2, 0 ≤ |j 2| ≤ 1, |j 2| = |j 1| – 1, i, k = 1, . . . , n and some α > 1. We give estimates and asymptotics for scattering solutions and scattering data for the equation (∗) for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms P∇V and PBi,k (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform P we obtain that for n ≥ 2 the velocity valued component of the scattering operator at high energies uniquely determines (∇V, B). We also consider the problem of recovering (∇V, B) from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of Novikov [Ark. Mat. 37: 141–169, 1999] for (∗) with B ≡ 0 and of Jollivet [J. Math. Phys. 47: 062902, 2006] for the relativistic version of (∗). We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for (∗) on the one hand and for its relativistic version on the other.