Jul 10

## Beam hardening 3: nonlinear reconstruction

*L*,

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 *c*_{1} and *c*_{2} are constants and *P*_{1} and *P*_{2} are sets of line integrals, which I will also refer to as projections. Letting *c*_{1} = (∂*L*)/(∂*x*)(0) and *c*_{2} = (∂^{2}*L*)/(∂*x*^{2})(0) in the Taylor’s series in Eq. 1↑, the reconstruction of the logarithm of the x-ray measurement is

*L*] =

*c*

_{1}ℛ[

*x*] +

*c*

_{2}ℛ[

*x*

^{2}].

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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