  Multi-Energy Computed Tomography (ME-CT) is a medical imaging modality aiming
to reconstruct the spatial density of materials from the attenuation properties
of probing x-rays. For each line in two- or three-dimensional space, ME-CT
measurements may be written as a nonlinear mapping from the integrals of the
unknown densities of a finite number of materials along said line to an equal
or larger number of energy-weighted integrals corresponding to different x-ray
source energy spectra. ME-CT reconstructions may thus be decomposed as a
two-step process: (i) reconstruct line integrals of the material densities from
the available energy measurements; and (ii) reconstruct densities from their
line integrals. Step (ii) is the standard linear x-ray CT problem whose
invertibility is well-known, so this paper focuses on step (i). We show that
ME-CT admits stable, global inversion provided that (a well-chosen linear
transform of) the differential of the transform in step (i) satisfies
appropriate orientation constraints that makes it a P-matrix. We introduce a
notion of quantitative P-function that allows us to derive global stability
results for ME-CT in the determined as well as over-determined (with more
source energy spectra than the number of materials) cases. Numerical
simulations based on standard material properties in imaging applications (of
bone, water, contrast agents) and well accepted models of source energy spectra
show that ME-CT is often (always in our simulations) either (i) non-globally
injective because it is non-injective locally (differential not of full rank),
or (ii) globally injective as soon as it is locally injective (differentials
satisfy our proposed constraints).
