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A Reconstruction Method in Transmission Tomography
Transmission tomography seeks to reconstruct the density of an object
from images made by X-rays passed through the object. In the simplest
continuous 2D model of the actual discrete 3D problem, it reduces to the
classical (Radon) problem to find the function from its parallel
projections at all angles and has analytical solution. Passing to the
discrete case leads to ill-conditioned equations which require
I will present an alternative reconstruction method for the discrete 3D
case. It supposes that the object is contained within a known
rectangular box and limits the reconstruction goal to the estimation of
a finite number of amplitudes of the expansion of the density in this
box into a discrete Fourier series. As input, it uses the projections of
the density on planes passing through the center of the box and parallel
to its sides.
The method is based on simple relations between the amplitudes of the
density expansion and the amplitudes of the Fourier expansion of the
'folded' projections - the periodic functions obtained from the usual
projections by adding their periodically shifted copies so that one
period of the 'folded' projection fills the part of the projection plane
contained within the box.
If the projection is made orthogonal to the wave vector corresponding to
the element of the Fourier expansion of the density, the relation
between the respective amplitudes becomes trivial and the amplitude of
this element can be found from the projection data immediately.
Generally, the amplitudes have to be estimated from projection data
interpolated between available projection directions closest to
directions orthogonal to the wave vectors for corresponding harmonics.
This efficient reconstruction method needs neither regularization nor
matrix inversion. It is illustrated by simulated examples.