The next series of posts discuss my recently published paper, “Efficient, non-iterative estimator for imaging contrast agents with spectral x-ray detectors,” available for free download
here. The paper extends the previous A-table estimator,
see this post, to three or more dimension basis sets so it can be used with high atomic number contrast agents. It also compares the A-table estimator to an iterative estimator.
This post describes the software to implement the new estimator. The next posts describe the code for an iterative estimator, compare the performance of the new estimator to the iterative estimator and the CRLB, compare the new estimator with a neural network estimator, and finally discuss an alternate implementation using a neural network as the interpolator.
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In the posts on beam hardening, I have shown that it causes the estimate of the line integral using the single average energy assumption to be nonlinearly related to the actual line integral. The nonlinearity causes any noncircularly symmetric object to look different when you look at it from different angles. This inconsistency results in artifacts in the reconstructed images. That is, the reconstructed image has features in it that are not in the original object. The inconsistency brings up some interesting questions. Is there a way to test the data to determine whether it is inconsistent? If so, is there a way to subtract the inconsistent part and will the result be equal to the original object without artifacts? In this post, I will review some prior research into this subject.
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The
first post in the discussion of beam hardening derived a Taylor’s series for logarithm of the x-ray measurement
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 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
ℛ[L] = c1ℛ[x] + c2ℛ[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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