  This work develops new numerical methods for the solution of the tomography
problem in domains with reflecting obstacles. We compare the solution's
performance for Lambertian reflection, for classical tomography with unbroken
rays and for specular reflection. Our numerical method using Lambertian
reflection improves the solution's accuracy by an order of magnitude compared
to classical tomography with unbroken rays and for tomography in the presence
of a specularly reflecting obstacle the numerical method improves the
solution's accuracy approximately by a factor of three times. We present
efficient new algorithms for the solution's software implementation and analyze
the solution's performance and effectiveness. The new method from this work for
reducing the number of equations in tomography linear systems is applicable to
improving the performance of a wide class of algebraic tomographic
reconstruction methods.
