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Jan 30 2015

## The constant covariance approximation to the CRLB with pileup

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[2], 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[1], 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

The method to compute SNR in my paper, “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”[1], 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

This post continues the discussion of my paper “Signal to noise ratio of energy selective x-ray photon counting systems with pileup”[1], 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.

Dec 29 2014

## SNR with pileup-2: Overall plan and NQ detector statistics with pileup

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.

Dec 15 2014

## SNR with pileup-1

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.

Nov 09 2014

## Improve noise by throwing away photons?

Photon counting systems with pulse height analysis (PHA) count the number of photons whose energy falls within a set of energy ranges, which I will call bins. Usually the bins are contiguous, non-overlapping, and span the incident energy spectrum so each photon falls within one bin. A paper[6] by Wang and Pelc showed that the A-vector noise variance can be decreased by using bins that are not contiguous. That is, if we use bins that only cover the low and high energy regions and do not include intermediate energies, we can lower the noise variance. Photons with energies in these intermediate regions are not counted i.e. they are thrown away. Improving noise by throwing away photons is an interesting concept and I will discuss it in this post. It turns out to be an example where the choice of the quality measure fundamentally changes the hardware design, which happens often, so it is important to study it.

Sep 15 2014

## Correlated noise reduction

The noise of the components of the A-vector is highly correlated and the previous post showed a way to produce low noise images analogous to conventional, non-energy selective images by “whitening” the A-vector data. That is good but is there a way to use the correlation to produce lower noise material selective images such as bone or soft tissue canceled? It turns out there are many methods that are seemingly different but are all based on the correlation. Al Macovski introduced the idea and his group at Stanford published several papers on it. It has been used in commercial systems. For example, the Fuji Corporation used an elaborate iterative method to reduce the noise in their “sandwich” photostimulable screen detector system[4]. Other companies like GE are more secretive but I think that they used a similar method with their voltage switching flat panel system.

In this post, I will describe a linear least mean squares method, which is a simplified version of the approach introduced by Cao et. al.[2], who also did her work at Stanford. This approach has straight-forward theory, is easy to implement and is effective at reducing noise. One problem with the approach is that it may change the quantitative values of the data in CT and Kalendar et al.[5] published an enhancement that may retain the quantitative information. However, if quantitative data are important, then the software can extract data from the underlying images guided by an operator using the noise-reduced image.

Sep 04 2014

## Dimensionality and noise in energy selective x-ray imaging-Part 3 low noise conventional images

I have been discussing my recently published paper, Dimensionality and noise in energy selective x-ray imaging, available for free download here. In this post, I will show how to create low noise images with properties analogous to conventional images from the energy spectrum data used in the previous two posts of this series to compute the A-vector images. The results verify that the noise in the ’conventional’ images computed from energy spectrum information is lower than images computed from the total number of photons only.

Aug 22 2014

## Dimensionality and noise in energy selective x-ray imaging-Part 2

The previous post in this series discussed the mathematics behind the increase in noise with the dimensionality, the number of basis functions used to approximate the attenuation coefficient. The series of posts is based on my recently published paper, Dimensionality and noise in energy selective x-ray imaging, available for free download here. This post describes simulations of the increase in noise with an object composed of body materials and an x-ray tube spectrum. The next post will show how to make low-noise images with the same properties as conventional x-ray images from the energy spectrum data. The main purpose of these last two posts is providing and explaining the code to reproduce the images in the paper.

Jul 23 2014