Figure 3.

A graphical illustration of the Projection Slice Theorem. $\mathsf{\Phi }(k_x,k_y)$ is the 2D-FT of $\varphi (x,y)$. The projection given by the RT, $p_{\theta }\left(\rho \right)$, is given by multiple line integrals along lines normal to $\theta$ direction. The slice $\mathsf{\Phi }\left(k_{\theta }\right)$ on the 2D-Fourier plain is along the $\theta$ direction. The Projection Slice Theorem states that the slice $\mathsf{\Phi }\left(k_{\theta }\right)$ is the 1D-FT of the projection $p_{\theta }\left(\rho \right)$.