François Monard

Inverse Anisotropic Conductivity from Power Densities in Dimension n ≥ 3

We investigate the problem of reconstructing a fully anisotropic conductivity tensor γ from internal functionals of the form ∇u·γ∇u where u solves ∇·(γ∇u) = 0 over a given bounded domain X with prescribed Dirichlet boundary condition. This work motivated by hybrid medical imaging methods covers the case n ≥ 3, following the previously published case n = 2 [22]. Under knowledge of enough such functionals, and writing ( ) with τ a positive scalar function, we show that all of γ can be explicitly and locally reconstructed, with no loss of scales for τ and loss of one derivative for the anisotropic structure . The reconstruction algorithms presented require rank maximality conditions that must be satisfied by the functionals or their corresponding solutions, and we discuss different possible ways of ensuring these conditions for 𝒞1, α-smooth tensors (0 < α <1).

Inverse anisotropic diffusion from power density measurements in two dimensions

This paper concerns the reconstruction of an anisotropic diffusion tensor γ = (γij)1 ⩽ i, j ⩽ 2 from knowledge of internal functionals of the form γ∇ui · ∇uj with ui for 1 ⩽ i ⩽ I solutions of the elliptic equation ∇ · γ∇ui = 0 on a two-dimensional bounded domain with appropriate boundary conditions. We show that for I = 4 and appropriately chosen boundary conditions, γ may uniquely and stably be reconstructed from such internal functionals, which appear in coupled-physics inverse problems involving the ultrasound modulation of electrical or optical coefficients. Explicit reconstruction procedures for the diffusion tensor are presented and implemented numerically.

Inverse anisotropic conductivity from internal current densities

This paper concerns the reconstruction of a fully anisotropic conductivity tensor γ from internal current densities of the form J = γ∇u, where u solves a second-order elliptic equation ∇ (γ∇u) = 0 on a bounded domain X with prescribed boundary conditions. A minimum number of n + 2 such functionals known on Y⊂X, where n is the spatial dimension, is sufficient to guarantee a unique and explicit reconstruction of γ locally on Y. Moreover, we show that γ is reconstructed with a loss of one derivative compared to errors in the measurement of J in the general case and no loss of derivatives in the special case where γ is scalar. We also describe linear combinations of mixed partial derivatives of γ that exhibit better stability properties and hence can be reconstructed with better resolution in practice.

Numerical implementation of geodesic X-ray transforms and their inversion

We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived in \cite{Pestov2004,Krishnan2010} are implemented in the case of simple and some non-simple metrics. These numerical tools are also used to better understand and gain intuition about non-simple manifolds, for which injectivity and stability of the corresponding integral geometric problems are still under active study.

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

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 it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach.

Reconstruction of a Fully Anisotropic Elasticity Tensor from Knowledge of Displacement Fields

We present explicit reconstruction algorithms for fully anisotropic unknown elasticity tensors from knowledge of a finite number of internal displacement fields, with applications to transient elastography. Under certain rank-maximality assumptions satisfied by the strain fields, explicit algebraic reconstruction formulas are provided. A discussion ensues on how to fulfill these assumptions, describing the range of validity of the approach. We also show how the general method can be applied to more specific cases such as the transversely isotropic one.

Imaging of Anisotropic Conductivities from Current Densities in Two Dimensions

We consider the imaging of anisotropic conductivity tensors

An accurate solver for forward and inverse transport

\gamma=(\gamma_{ij})_{1\leq i,j\leq 2}

Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces

from knowledge of several internal current densities