  3D Compton scattering imaging is an upcoming concept exploiting the
scattering of photons induced by the electronic structure of the object under
study. The so-called Compton scattering rules the collision of particles with
electrons and describes their energy loss after scattering. Although physically
relevant, multiple-order scattering was so far not considered and therefore,
only first-order scattering is generally assumed in the literature. The purpose
of this work is to argument why and how a contour reconstruction of the
electron density map from scattered measurement composed of first- and
second-order scattering is possible (scattering of higher orders is here
neglected). After the development of integral representations for the first-
and second-order scattering, this is achieved by the study of the smoothness
properties of associated Fourier integral operators (FIO). The second-order
scattered radiation reveals itself to be structurally smoother than the
radiation of first-order indicating that the contours of the electron density
are essentially encoded within the first-order part. This opens the way to
contour-based reconstruction techniques when using multiple scattered data. Our
main results, modeling and reconstruction scheme, are successfully implemented
on synthetic and Monte-Carlo data.
