Image Reconstruction in Oil Refinery

Abstract

One can recover a function F (x, y) denned in a domain T from the knowledge of its integrals along all lines. In fact, parametrizing the segments σ_θ,t as shown in Figure 7.1, where σθ,_t = T ∩ Σ_θ,t, the function

$$\tilde f\left( {\theta ,t} \right) = \int\limits_{{\sigma _{\theta ,t}}} {f\left( {x,y} \right)ds}$$

is called the Radon transform of f and its inverse is given by an explicit formula (see, for instance I.M. Gelfand and G.E. Shilov, Generalized Functions, Vol. 1, Academic Press, New York, 1964).