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## 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:
 (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