Images reconstruction. Radon transform

Starring effect and residual signal (bleeding out)

Esempio : densità di spin “puntiforme”

Back projected image R(x,y) i = different angles

Projection or central slice theorem

To remove bleeding out and star effect : filtered back-projection Example : M-filter, see text

Limited to 1D but easily generalized to 2D and 3D Continuous case Real obtained density

Before going ahead some properties of FT

If there’s a linear shift in k-space it can be demonstrasted that :

Sampling function

Relevant FT pairs for next discussion

Discrete FT

The final expression :

Finally :