Image reconstruction for hard field tomography

Computed tomography requires the solution of an inverse problem, that is, the reconstruction of an object distribution from measurement data. In hard field tomography this problem can be more specifically referred to as the reconstruction of an object distribution from its line integrals. A first solution to the mathematical problem was given by Johann Radon in 1917 long before anyone thought about computed tomography. The transformation of an object distribution into the space of its line integrals is hence today called the Radon transformation. With the development of computed tomography technology later quite powerful algorithms basing on analytic and algebraic inversion schemes for the Radon transform were developed. This chapter shall introduce the mathematical fundamentals of the forward and inverse problem of hard field computed tomography, the discretization of the problem and further discuss some distinct features and specialties of image reconstruction, such as 3D inversion approaches and concepts for limited and local tomography.