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Jul 10

Beam hardening 3: nonlinear reconstruction

Tag: Implementation,Math,Physicsadmin @ 10:28 am
The first post in the discussion of beam hardening derived a Taylor’s series for logarithm of the x-ray measurement L

(1) L(x) = (L)/(x)(0)x + (2L)/(x2)(0)x2

where x is the object thickness. Since line integrals are linear operators, the inverse operator, that is the image reconstruction operator , is also linear so that

(2) [c1P1 + c2P2] = c1[P1] + c2[P2]

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

[L] = c1[x] + c2[x2].

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.

In this post, I will discuss some of the properties of the reconstruction of the nonlinear term and apply it to models of common beam hardening artifacts. This gives us some insight into the types of artifacts that we can expect from beam hardening.

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