Inverse Problems and Imaging Web site: http://www.aimSciences.org Volume 2, No. 1, 2008, 23–42 INVERSE TRANSPORT WITH ISOTROPIC SOURCES AND ANGULARLY AVERAGED MEASUREMENTS Guillaume Bal Department of Applied Physics and Applied Mathematics Columbia University, New York NY, 10027, USA Ian Langmore Department of Mathematics University of Washington, Seattle WA, 98195, USA Franc ¸ois Monard Department of Applied Physics and Applied Mathematics Columbia University, New York NY, 10027, USA, and SUPAERO, 31400 Toulouse, France (Communicated by Gunther Uhlmann) Abstract. We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for recon- struction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this er- ror so that it is small when the high frequency content of the scattering kernel is small. 1. Introduction. Linear transport equations are used in many applications such as the propagation of the energy density of waves in heterogeneous media [8, 12, 22, 29], neutrons in nuclear reactors [20], and more recently, near-infra-red photons in tissues and its application in optical tomography, a medical imaging modality [5, 11, 21]. Inverse transport theory consists of reconstructing the constitutive parameters in the transport equation from various measurements. The typical optical parameters one wants to reconstruct are the total absorption coefficient σ and the scattering coefficient k. Several theories have been developed on the reconstruction of such parameters in various settings; see [3, 4, 7, 14, 18, 19, 23, 24, 26]. All rigorous theoretical results of uniqueness and stability [23] are based on full phase-space measurements. What we mean by this is the following. Particle densities depend on their position x and their direction v, which in this paper we assume is normal- ized to |v| = 1. Phase space measurements mean that u(x, v) can be arbitrarily chosen and measured at the domain’s boundary as a function of its phase-space variables (x, v) in n +(n − 1) dimensions for n−dimensional problems. This means having 4(n − 1) dimensions of available data to reconstruct the optical parameters. 2000 Mathematics Subject Classification. 65R32, 92C55, 82C70. Key words and phrases. Inverse Transport Theory, Optical Tomography, Averaged Measure- ments, Complex Geometrical Optics Solutions. 23 c 2008 American Institute of Mathematical Sciences