Interestingly, posing the tomographic problem as a linear equation system can also be done in Fourier space
. Thanks to the Central Slice Theorem, the relationship between the 3D Fourier transform of the macromolecule model and the 2D Fourier transform of the projection can be expressed as 
This relates the scattered field from a single projection to Fourier space
data of the object function being investigated.
To apply this convolution, which has been generated in real space, it must be transformed to Fourier space
Indeed, in Fourier space
we would obtain the ODE i[omega][??] - [[partial derivative].sub.x](D(x) [[partial derivative].sub.x][??]) = 0 for [??].
where, according to Li's line of reasoning [2,3], [[PI].sup.x.sub.i][D.sub.x] is factorizable in Fourier space
, but [[PI].sup.y.sub.j][D.sub.x] is not.
By Shannon's sampling theorem, at a given real space sampling distance [DELTA][xi] the signal can only be correctly reconstructed up to a frequency Q = [(2[DELTA][xi]).sup.-1] in Fourier space
. Thus, the radial weighting by the function [absolute value of q] in Eq.
This effectively provides information on response at one selected point in Fourier space
, or its multiples.
Walter Norfleet of MIT has concluded that a technique based on identification of the part shape's local sensitivity to each individual pin's position is more efficient than an incremental correction method, or a method in which the correction is calculated in Fourier space
. The Fourier approach, in which a series of superimposed sine waves of various frequencies describes tool shape, couples the entire part shape to motions of individual pins.
We use a method to estimate local orientations in the n-dimensional space from the covariance matrix of the gradient, which can be implemented either in the image space or in the Fourier space
. In a second step, two methods allow us to detect sudden changes of orientation in images.
In Fourier space
, equations (1.1) and (1.2) have the form
During evaluations of the Fourier transforms for one, two and three dimensions, a diagonal linear integral operator was found to be implicit in the 3-D Fourier space
of the scattering potential.
Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space