I will discuss three practical problems in tomographic imaging all of
which are non-linear but for quite different reasons. The
monochromatic attenuation tomography problem ("CT") but with sources
and detectors that are not points. Electrical Impedance Tomography
(EIT). And a non-Abelian problem in neutron spin tomography.
In CT the problem arises as data from a number of lines integrals is
combined, but it is their exponentials that are added hence the
nonlinearity. In the neutron spin case we have a matrix ODE solved
along rays than cannot be solved simply using an exponential as matrix
multiplication is non commutative. In EIT we cannot even simplify to
ODEs along rays - everything effects every things else. But we can
still separate lack of superposition and saturation as aspects of the
non-linearity.
Overall my aim is to draw some general lessons about how to approach a
new inverse problem and understand its non-linearity.