Inverse slant stack

Tomography in medical imaging is based on the same mathematics as inverse slant stack. Simply stated, (two-dimensional) tomography or inverse slant stacking is the reconstruction of a function given line integrals through it. The inverse slant-stack formula will follow from the definition of two-dimensional Fourier integration:  

$$u(x,t) \eq \int\ e^{{-} i \omega t} \ [\ \int \ e^{{i} k x } \ U(k, \omega ) \ dk\ ]\ d \omega$$ (20)

Substitute $k = \omega p$ and $dk=\omega\,dp$ into (20). Notice that when $\omega$ is negative the integration with dp runs from positive to negative instead of the reverse. To keep the integration in the conventional sense of negative to positive, introduce the absolute value $\vert \omega \vert$.(More generally, a change of variable of volume integrals introduces the Jacobian of the transformation). Thus,

$$\begin{eqnarray} u(x,t)\ \ \ &=&\ \ \ \int\ e^{{-} i \omega t} \ [\ \int \ e^{... ...i} \omega p x } \ \vert \omega \vert \ ] \ d \omega \ \ \ dp\\ end{eqnarray}$$ (21) (22)

Observe that the { } in (22) contain an inverse Fourier transform of a product of three functions of frequency. The product of three functions in the $\omega$-domain is a convolution in the time domain. The three functions are first $U( \omega p, \omega )$,which by (19) is the FT of the slant stack. Second is a delay operator $e^{{i} \omega p x }$, i.e an impulse function of time at time px. Third is an $\vert \omega \vert$ filter. The $\vert \omega \vert$ filter is called a rho filter. The rho filter does not depend on p so we may separate it from the integration over p. Let ``*'' denote convolution. Introduce the delay px as an argument shift. Finally we have the inverse slant-stack equation we have been seeking:  

$$\begin{tabular} {\vert c\vert} \hline \\ $u(x, t)\ =\ rho (... ...t\overline{u}(p, t-px) \ dp$\space \\ \\ \hline\end{tabular}$$ (23)

It is curious that the inverse to the slant-stack operation (14) is basically another slant-stacking operation (23) with a sign change.