Inverse Transport Theory and Tomography (10w5063)

Objectives

The purpose of this workshop is to bring together specialists in the general area of inverse transport theory. Inverse transport encompasses many theoretical and practical areas. In case of no scattering, inverse transport includes e.g. classical tomography, with its applications in medical and geophysical imaging. In the presence of particle scattering, applications include optical tomography (propagation of photons through human tissues) and radiation through the atmosphere (which includes the important problem in global warming of radiation through clouds). In the presence of highly scattering media, inverse transport includes the inverse theory of diffusion equations, as it is used in optical tomography and in electrical impedance tomography. We plan to invite pure and applied mathematicians and scientists in other fields working on those problems.

Inverse transport theory studies the following problem. Consider particles, such as photons for concreteness, sent through a medium that can absorb and scatter them. The mathematical model is to study a density function of particles of the type $u(x,theta)$, where $xin mathbf{R}^n$, $thetain {S}^{n-1}$, that solves the transport equation begin{equation} label{1} thetacdotnabla +sigma(x) u - int_{S^{n-1}} k(x,theta,theta')u(x,theta'), dtheta'=f end{equation} in some domain $Omega$ with appropriate boundary conditions. Here $sigma(x,theta)$ models the absorption of the medium, and the scattering (collision) kernel $k(x,theta,theta')$ is the rate at which articles at $x$ change their velocity from $theta'$ to $theta$. The possibly vanishing source term is denoted by $f(x,v)$. Inverse transport then consists in reconstructing the optical parameters $sigma(x)$ and $k(x,theta,theta')$ from measurements of $u(x,theta)$, typically at the domain's boundary. The theory of reconstructions of $sigma(x)$ and $k(x,theta,theta')$ from full measurements of solutions $u$ on $partialOmegatimes S^{n-1}$ is well developed cite{S-IO-03}. In most applications, however, full measurements are hardly available and many inverse problems related to (ref{1}) remain open.

In the non-scattering regime, the classical X-ray tomography problem is to reconstruct a function, say with compact support, from its line integrals [(Xf)(x,omega) = int_{-infty}^infty f(x+somega), ds, quad xin mathbf{R}^n, ; omegain S^{n-1}.
]
While this problem has an explicit solution, many practical questions still remain, like the most effective numerical inversions, limited angle reconstructions, etc. A more realistic model is to account for some absorption of the medium. This leads to the attenuated X-ray transform [(X_sigma f)(x,omega) = int_{-infty}^infty w(x+somega ,omega)f(x+somega), ds, quad xin mathbf{R}^n, ; omegain S^{n-1},]where $w(x,omega) = nt_{-infty}^0sigma(x+tomega), dt$ is the total absorption along the ray ${x+tomega; ; tle0}$, and $sigma$ is the absorption coefficient. It has been proved recently by Bugkheim and Novikov that $X_sigma$ is injective with an explicit formula for the inverse.

In fact, the inversion of $X_sigma$ is a partial case of the inverse problem for the transport equation (ref{1}) when $k=0$. Despite that, the inversion of the attenuated X-ray transform has been treated with different methods including complex analysis, and microlocal methods. The results for that partial case are much more complete. We refer to cite{BT-SIMA-07,FSU-08} for some recent work on the subject. We feel that the relationship between complex analysis and microlocal methods could be pushed further and we plan to invite speakers to talk about recent work in this area.

In the high scattering regime, i.e., when $sigma$ and $k$ are large and reasonable assumptions are satisfied, the transport equation (ref{1}) is well-approximated by a diffusion equation, which takes the form begin{equation}
label{eq:3} -nablacdot D(x) nabla U + sigma_a U =0,end{equation} where $D$ and $sigma_a$ are typically scalar-valued functions. Mathematically, this is a diffusion equation, which in the case of vanishing absorption, is an example of the well-studied elliptic problem begin{equation} label{2} sum_{i,j}partial_i sigma_{ij}partial_j u=0 quad mbox{in $Omega$},end{equation} where $sigma_{ij}(x)$ is positive definite. Then we are lead to the inverse problem of recovering $sigma$ from the knowledge of the Dirichlet-to-Neumann map [Lambda :f to sigmafrac{partial u}{partial f}bigg|_{partialOmega},]where $u$ is the solution of (ref{2}) with boundary data $u=f$. That inverse problem is known as Electric Impedance Tomography, where $sigma$ is a unknown conductivity (anisotropic in general), $u$ is the voltage, and $Lambda$ is the voltage to current map. It was first proposed by Calder'on in cite{calderon80}, and has seen a tremendous progress since then. Uniqueness of recovery and constructive algorithms has been established by Sylvester and Uhlmann, and Nachmann in dimensions $nge3$, and by Astala and Paivarinta for $n=2$. The first of those works started the Complex Geometric Optics, and led to progress in solving many other problems. Several teams used those theoretical advances to develop numerical reconstructions and even devices that do the reconstruction in real time. EIT has promising applications in medical imaging for early detection of cancer. Some of them, obtained by the group at RPI led by David Isaacson, will be presented at the workshop. We also plan other talks by both pure math and applied specialists in EIT.

As we have mentioned, the theory for (ref{2}) and (ref{eq:3}) is well advanced. The inverse transport theory for (ref{1}) from angularly measurements is much less advanced. By angular measurement, we mean e.g. measurements of the particle current at the boundary: begin{equation} label{4} J(x) = int_{S^{n-1}} u(x,v) vcdot n(x) dmu(v), end{equation} where $n(x)$ is the outward unit normal to $Omega$ at $xinpartialOmega$. In the diffusion approximation of transport, $J(x)$ takes the form $sigmafrac{partial u}{partial
f}big|_{partialOmega}$. We thus see that angularly averaged measurements in inverse transport theory are of the same kind as the reconstruction of $sigma$ in (ref{2}) from the Dirichlet-to-Neumann map. Moreover, the recent results obtained in cite{BLM-IPI-08} show that the linearized inverse problem for (ref{1}) from angularly averaged measurements is very similar to the Calder'on problem. The same complex geometrical optics solutions are being used. This surprising analogy should be explored further and we expect that the interactions of specialists from inverse diffusion and from inverse transport theory will provide further advances in this area.

In media with variable, and possibly, anisotropic speed, the mathematical model is the transport equation on a Riemannian manifold. This links inverse transport with integral geometry on Riemannian manifolds. That link has been exploited by Sharafutdinov cite{Sh} and McDowall cite{Mc}, who will be among the speakers. In particular, energy methods, widely used on Riemannian manifolds, have been used in inverse transport.

-BLM-IPI-08
G.~Bal, I.~Langmore, and F.~Monard. Inverse transport with isotropic sources and angularly averaged measurements. Inverse Probl. Imaging, (1), pp. 23-42, 2008.

-BT-SIMA-07
G.~Bal and A.~Tamasan. Inverse source problems in transport equations. SIAM J. Math. Anal., 39(1):57--76, 2007.

-calderon80
A.P. Calderon. On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, pages 65--73, 1980.

- FSU-08
B.~Frigyik, P.~Stefanov, and G.~Uhlmann. The x-ray transform for a generic family of curves and weights. J. Geom. Anal., 18 , pp. 81-97, 2008.

-Mc
S. McDowall. An inverse problem for the transport equation in the presence of a Riemannian metric. Pac. J. Math., 216 (2004), no.1, 107--129.

-Sh
V.~Sharafutdinov. The inverse problem of determining the source in the stationary transport equation on a Riemannian manifold. J. Math. Sci. (New York)}, 96(4)(1999), 3430--3433.

-S-IO-03
P.~Stefanov. Inverse Problems in Transport Theory; in: {em Inside Out: Inverse problems and applications}, volume~47 of MSRI publications, Ed. G. Uhlmann.
Cambridge University Press, Cambridge, UK, 2003.