Topics to be covered:

i) The transport model through an attenuating and scattering medium

- The Forward Problem

- The Albedo operator and its kernel

- An inverse boundary value problem.

ii) The non- attenuated and non-scattering case

- The X-ray Transform and the Radon Transform

- Inversion formulae

- The Helgason-Ludwig Range Conditions

- Stability questions

- Finitely many measurements: non-uniqueness examples

- The filtered Back Projection Algorithm.

iii) The attenuated X-Ray transform

- Novikov’s approach: the Riemann-Hilbert Problem

- Other Inversion Formulae (Natterer, Oleg-Stromberg)

- Bukhgeim’s approach: the theory of A-analytic maps

- Other applications of Bukhgeim’s approach: the Doppler Transform.

iv) The attenuating and scattering case

- Isotropic attenuation and weak scattering: an iterative reconstruction method.

- Anisotropic attenuation. Non-uniqueness and gauge equivalence.

v) Open problems