Jul 10
Beam hardening 3: nonlinear reconstruction
where x is the object thickness. Since line integrals are linear operators, the inverse operator, that is the image reconstruction operator ℛ, is also linear so that
where c1 and c2 are constants and P1 and P2 are sets of line integrals, which I will also refer to as projections. Letting c1 = (∂L)/(∂x)(0) and c2 = (∂2L)/(∂x2)(0) in the Taylor’s series in Eq. 1↑, the reconstruction of the logarithm of the x-ray measurement is
The first term is the reconstruction of the projections, which is what we want, while the second term, the reconstruction of the squares of the projections, leads to artifacts. There are also be higher order terms but I will assume they are negligible although they can be analyzed similarly to the discussion here.
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