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.
more –>;
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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In this post, I continue the discussion of my paper “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”
[2], which is available for free download
here. The computation of the SNR is based on the approach described in my previous paper, “
Near optimal energy selective x-ray imaging system performance with simple detectors[1]”, which is available for free download
here. The approach is extended to data with pileup.
The “Near optimal …” paper shows that regardless whether there is pileup or not, if the noise has a multivariate normal distribution and if the feature is sufficiently thin so the covariance in the background and feature regions is approximately the same, the performance is determined by the signal to noise ratio. So the first thing that has to be done is to show that the data with pileup satisfy these conditions. My plan for the discussion of the SNR paper is therefore as follows.
- First, I will use the idealized model and the Matlab function to generate random recorded counts described in the previous post to develop code to compute random samples of data from NQ and PHA detectors with pileup. I will use these functions to derive and validate formulas for the expected values and covariance of the data. These are required to compute the SNR.
- I will then use these models and software to determine the conditions so that the probability distribution of the data can be approximated as multivariate normal.
- Next, I will show how to use the Cramèr-Rao lower bound (CRLB) to compute the A-vector covariance with pileup data. I will use this to show that we can, under some conditions, use the constant covariance approximation to the CRLB with pileup data just as we can with non-pileup data as shown in this post.
- Finally, I will apply these results to compute the reduction of SNR as pileup increases.
In this post, I will use the Matlab function discussed in the previous post to compute random samples of recorded photon counts (N) and total energy (Q) data. These are data from an NQ detector with pileup. I will use these data to validate the formula for the covariance derived in Appendix C of the SNR paper. I will present Matlab code to reproduce Fig. 9 of the paper, which shows the covariance and the correlation of the data as a function of the dead time. I will also use the same data to validate the formulas for the expected value and variance of the recorded counts and the total energy as a function of dead time. These formulas are described in Section 2.E of the paper.
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In the next posts, I will discuss my paper “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”, which is available for free download
here. The paper uses an idealized model to derive limits on the effects of pileup on the SNR of A-vector data. There have been many papers (see, for example Overdick et al.
[4] Taguchi et al.
[3], and Taguchi and Iwanczyk
[6]) that use more or less realistic models of photon counting detectors to predict the quality of images computed from their data. These models are necessarily complex since state of the art is relatively primitive compared with the extreme count rate requirements in diagnostic imaging. The complexity of detailed models makes it hard to generalize from the results. Moreover, as research continues, the properties of the detectors will improve and their response will approach an idealized limit. This is the case with the energy integrating detectors used in state of the art medical imaging systems whose noise levels have been reduced so that the principal source of noise is the fundamental quantum noise that is present in all measurements with x-ray photons.
In this post, I will describe the rationale for an idealized model of photon counting detectors with pulse height analysis with pileup and illustrate it with the random data it generates. The following posts will show how the model can be applied to compute the SNR of systems with pileup and to compare the SNR to the full spectrum optimal value. The model will be used to determine the allowable response time so that the reduction in SNR due to pileup is small.
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You may ask, what is the fundamental advantage of the new estimator? Yes, it is faster than the iterative method but so what? With Moore’s law, we can just throw silicon at the problem by doing the processing in parallel. I have two responses. The first is that not only is the iterative estimator slow but it also takes a random time to complete the calculation. This is a substantial problem since CT scanners are real-time systems. The calculations have to be done in a fixed time or the data are lost. The gantry cannot be stopped to wait for a long iteration to complete!
The second problem is that, as it has been implemented in the research literature, the iterative estimator requires measurement of the x-ray tube spectrum and the detector energy response to compute the likelihood for a given measurement. These are difficult measurements that cannot be done at most medical institutions. Because of drift of the system components, the measurements have to be done periodically to assure accurate results. There may be a way to implement an iterative estimator with simpler measurements but I am not aware of it.
In this post, I will show how the parameters required for the new estimator can be determined from measurements on a phantom placed in the system. This could be done easily by personnel at medical institutions and is similar to quality assurance measurements now done routinely on other medical systems.
more → Continue reading “Parameters for the estimator”
The past two posts have discussed estimators for A-vector data. I showed that with the same number of measurement spectra as the A-vector dimension, any estimator that solves the deterministic equations is the maximum likelihood estimator (MLE) and it will achieve the Cramèr-Rao lower bound (CRLB). If there are more measurement spectra than the dimension, then the polynomial estimator, which works well for the equal case, has very poor performance giving a variance that can be several hundred times larger than the CRLB. I showed by simulations that with more measurements than dimension the iterative MLE does give a variance close to the CRLB but it has substantial problems. Common to all iterative algorithms, the computation time is long and random. It may fail to converge at all if the initial estimate is too far from the actual value. As it was implemented by Schlomka et al.
[2], it also requires measurements of the x-ray source spectrum and the detector spectral response. These are difficult, time consuming and require laboratory equipment that is not usually available in medical institutions.
In this post, I will give an intuitive explanation for the operation of a new estimator that I introduced in my paper
[1] “Estimator for photon counting energy selective x-ray imaging with multi-bin pulse height analysis,” which is available for free download
here. The estimator is efficient and can be implemented with data that can be measured at medical institutions. The details of the estimator are described in the paper. Here, I will discuss the background and give a rationale on how it works.
more → Continue reading “Rationale for the new estimator”
