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*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:

| |
(20) |

Substitute and into (20).
Notice that when 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 .(More generally, a change of variable of volume integrals introduces
the *Jacobian * of the transformation).
Thus,
| |
(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 -domain is
a convolution in the time domain.
The three functions are first ,which by (19) is the FT of the slant stack.
Second is a delay operator , i.e
an impulse function of time at time *px*.
Third is an filter.
The 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:
| |
(23) |

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

** Next:** Plane-wave superposition
** Up:** SLANT STACK
** Previous:** Slant stack and Fourier
Stanford Exploration Project

10/31/1997