  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.
