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Fourier Slice Theorem
The Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform of the original object. It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2D inverse Fourier transform.
Start by defining the 2D Fourier transform of the object function as:
Define the projection at angle ?, P?(t) and its transform by:
For simplicity q=0 which leads to v=0
As the phase factor is no-longer dependent on y, the integral can be split.
The part in brackets is the equation for a projection along lines of constant x
Substituting in
Thus the following relationship between the vertical projection and the 2D transform of the object function:
The Fourier Slice theorem relates the Fourier transform of the object along a radial line.
Collection of projections of an object at a number of angles
For the reconstruction to be made it is common to determine the values onto a square grid by linear interpolation from the radial points. But for high frequencies the points are further apart resulting in image degradation.
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