-International audienceThis paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field normalΦ:M→frakturso()n from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of CΦ increases, the statistical error in the recovery of Φ converges to 0 in L2(M)‐distance at a rate that is algebraic in 1/N and approaches 1/N for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map CΦ → Φ. Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC

We consider the statistical inverse problem of recovering a function f : M → R, where M is a smooth compact Riemannian manifold with boundary, from measurements of general X-ray transforms Ia(f ) of f , corrupted by additive Gaussian noise. For M equal to the unit disk with 'flat' geometry and a = 0 this reduces to the standard Radon transform, but our general setting allows for anisotropic media M and can further model local 'attenuation' effects -both highly relevant in practical imaging problems such as SPECT tomography. We study a nonparametric Bayesian inference method based on standard Gaussian process priors for f . The posterior reconstruction of f corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty that does not require the calculation of the singular value decomposition of the forward operator Ia. We prove Bernstein-von Mises theorems for a large family of one-dimensional linear functionals of f , and they entail that posterior-based inferences such as credible sets are valid and optimal from a frequentist point of view. In particular we derive the asymptotic distribution of smooth linear functionals of the Tikhonov regulariser, which attains the semi-parametric information lower bound. The proofs rely on an invertibility result for the 'Fisher information' operator I * a Ia between suitable function spaces, a result of independent interest that relies on techniques from microlocal analysis. We illustrate the performance of the proposed method via simulations in various settings.1. Introduction. The Radon transform and its variants play a key role in image reconstruction problems, with important applications in physics, engineering and other areas of scientific imaging. The classical case is where a function f in R 2 is reconstructed from integrals over straight lines:where ω ⊥ is the rotation of ω by 90 degrees counterclockwise. Often it is natural to confine the function f to a bounded subset M of Euclidean space such MSC 2010 subject classifications: Primary 62G20; secondary 58J40, 65R10, 62F15

Abstract. We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but still ill-posed, if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.