Mar 17 2016

## Estimator for contrast agents-3 Monte Carlo simulation

admin @ 10:51 am
In this post I continue the discussion of the paper, “Efficient, non-iterative estimator for imaging contrast agents with spectral x-ray detectors,” which is 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. Here I describe the Matlab code to reproduce the figures that summarize the Monte Carlo simulation of the estimators’ performance. The Monte Carlo simulation verifies that the new estimator achieves the Cramèr-Rao lower bound (CRLB) and compares it to an iterative estimator. The simulation code is included with the package for this post.

Sep 10 2015

## Beam hardening 4: consistency

admin @ 2:58 pm
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.

Jul 10 2015

## Beam hardening 3: nonlinear reconstruction

admin @ 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.

Jun 19 2015

## Beam Hardening 2-the no-linearize theorem

admin @ 1:45 pm
The last post showed that beam hardening causes a nonlinearity between the log of the measurements and the A-vector. It is natural to think that we can eliminate the beam hardening artifacts by measuring the nonlinearity and then “linearizing” it with an inverse transformation. In this post, I will show that this is not possible in general. Although there are some special cases when we can linearize and a linearizing transformation may reduce the artifacts, we cannot do this for every object. I will show that this is due to the fact that we need at least a two dimension basis set to represent the attenuation coefficient.

Jun 12 2015

## Beam hardening 1

admin @ 2:50 pm
Beam hardening artifacts were seen soon after the introduction of CT. Radiologists noticed a ring of increased Hounsfield numbers against the inside of the skull. At first they thought the increase was due to the difference between the white matter in the interior and the gray matter in the cortex of the brain but images of skulls filled with only water also showed the ring so it was obvious the increased values were an artifact.
The EMI corporation, which produced the first CT scanners, must have known about the artifact but they were notoriously close mouthed about the scanner design. In their first scanner the patient stuck his head into a plastic bladder filled with water and the x-ray system measured through the head surrounded by the water. This reduced the dynamic range requirements for the electronics but it also reduced the beam hardening nonlinearity as well as other artifacts as I will show.
In Al Macovski’s group at Stanford, we quickly figured out that the change in average energy of the transmitted photons as the object thickness increases, spectral shift as we called it, would produce a nonlinear relationship between the logarithm of the measurements and the line integral of the attenuation coefficient. We also showed that this nonlinearity could produce the artifact. We were quite interested in it because it was an effect of x-ray energy on the image and we wanted to extract energy dependent information.
Fig. 1↓ shows that the change in average energy and the effective attenuation coefficient as object thickness increases are both quite large. In this post, I will show how this change leads to a nonlinearity between the log of the measurements and the line integral of the object. I will derive expressions for the magnitude of the nonlinearity. These will lead to ways to reduce the nonlinearity and therefore the artifacts. In later posts I will show that the nonlinearity cannot in general be corrected using a lookup table, the no-linearize theorem. I will then describe a general way to understand the effect of the nonlinearity on the reconstructed CT image. Finally, I will examine whether iterative reconstruction methods can be used to correct the artifacts by making the projections and the image consistent.

Feb 09 2015

## SNR with pileup

Tag: Math,Noise,Physicsadmin @ 2:32 pm
In this post I conclude my discussion of my “SNR with pileup …” paper. I will present and explain the code to reproduce the “bottom line” figures 3 to 5 of the paper that show the decrease of the SNR as the pileup parameter η increases. The decrease is rapid and when the value of η reaches 1, all the counting and PHA detectors have SNR smaller than the energy integrating detector. The NQ detector SNR decreases rapidly and approaches but marginally stays above the Q SNR since it uses that signal.

Feb 09 2015

## Monte Carlo validation of variance formulas

admin @ 11:26 am
My “SNR with pileup …” paper presented a set of theoretical formulas for the noise of NQ and PHA detectors in Tables I and II. I have discussed the individual formulas and Monte Carlo simulations of their validity in past posts in this series. In Section 2.K of the paper, I presented an overall test of the formulas that compared the A-vector component variances with a Monte Carlo simulation of the random detector data processed with a maximum likelihood estimator (MLE). In this post, I expand the discussion in the paper and present code to reproduce Fig. 2.

Jan 30 2015

## The constant covariance approximation to the CRLB with pileup

Tag: Math,Noise,Physicsadmin @ 12:06 pm
In my last post, I showed that the probability distribution of photon counting detector data with pileup is multivariate normal for the counts typically used in material selective imaging. With the normal distribution and a linear model, the Cramèr-Rao lower bound (CRLB) for the covariance of the A-vector data includes a term that depends on the change in the measurement data covariance with A. Without pileup I show in this post and in the Appendix of my “Dimensionality and noise …” paper, available for free download here, that the change in covariance term is negligible for large enough counts. In Appendix B of my “SNR with pileup …” paper, I show that the term is also negligible with pileup. In this post, I will present and explain the code to reproduce the figures in that section.

Jan 26 2015

## Probability distribution with pileup

Tag: Math,Noise,Physicsadmin @ 5:13 pm
The method to compute SNR in my paper, “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”, assumes that the noisy data have a multivariate normal distribution. Appendix A of the paper describes a Monte Carlo simulation to study the conditions under which the normal distribution assumption is valid. In this post, I will expand on the discussion in the paper and present Matlab code to reproduce the figures.

Jan 05 2015

## SNR with pileup-3 PHA detector statistics with pileup

Tag: Math,Physics,softwareadmin @ 11:31 am
This post continues the discussion of my paper “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”, which is available for free download here. Following the road map described in my last post, I am deriving and validating formulas for the statistics of photon counting detectors with pileup. In this post, I describe formulas for the expected value and covariance of pulse height analysis data with pileup and present software to verify the formulas with Monte Carlo simulations.

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