The last two articles discussed the use of energy information to increase the SNR of x-ray imaging systems. They assumed that the attenuation coefficient is a continuous function of energy and that the energy spectrum is measured with perfect resolution. But we know from my posts here, here, here, and here that the attenuation coefficient can be expressed as a linear combination of two functions of energy. In addition, as I discussed in my posts about deadtime, the extremely high count rates required for medical x-ray systems severely limit the energy resolution and the complexity of the signal processing.
My paper “Near optimal energy selective x-ray imaging system performance with simple detectors”, which is available for free download here, discusses the use of the two-function decomposition in the signal processing. By transforming the problem from infinite to finite dimensions, the decomposition allows us to get near ideal SNR using low energy-resolution measurements, which may be possible with high speed photon counting detectors.
more –>;
In this post, I continue the discussion of my paper “Near optimal energy selective x-ray imaging system performance with simple detectors”, which is available for free download here. I will describe a Monte Carlo simulation of the SNR with energy information discussed in my last post. The simulation traces the paths of individual photons. This is, perhaps, more fundamental than my previous simulations that relied on statistical models for the detector data. I develop models for the random path lengths and use them to simulate the imaging task for SNR described in my last post. The model is validated by comparing the energy spectrum of photons transmitted through an object to the theoretical formula. I also provide an estimate of the errors in the Monte Carlo results.
more –>;
In the next posts I will discuss some of the results in my recent paper, which is available for free download here. The paper discusses fundamental limits on the signal to noise ratio of x-ray detectors with energy spectrum information. It also describes how we can design practical systems with low energy resolution detectors whose performance gets close to the optimal limit.
more –>;
The logarithm of the data is often used in x-ray imaging systems because it is (very) approximately proportional to the line integral. The statistics of the log of the counts in photon counting detectors are are different than those of the counts as summarized in my last post and I derive them in this post. I then test the formulas using a Monte Carlo simulation.
more –>;
I used a Monte Carlo simulation to test the formulas derived in my last post for the mean value, variance, and covariance of pulse height analysis (PHA) data as a function of deadtime. In this post, I discuss the simulation software and the results. The formulas were quite accurate except for the covariance where the relatively small value and the difficulty in getting good statistics for variance estimates in general caused a spread around the theoretical values.
more –>;
In previous posts, I discussed the mean and variance and the energy spectrum of photon counting with deadtime. In this post, I will describe the statistics of pulse height analysis (PHA) data as a function of the deadtime of the detector. I will analyze the idealized case with perfect energy bins with zero transition width and no overlap and no added electronic noise. With these assumptions and no deadtime, the number of counts in each bin is Poisson distributed with a mean value equal to the number of incident photons and the data in different bins are independent. With deadtime, the PHA data mean and variance are smaller than those with no deadtime. In addition, the data in different bins become negatively correlated.
In my next post, I will describe a Monte Carlo simulation to validate the formulas derived here.
more –>;
My previous post discussed the mean and variance of photon counts with deadtime. In this post, I describe a model for the energy spectrum that might be measured by a photon counter with perfect pulse height analysis (PHA). Again, my purpose is to gain insight so I will use a highly simplified model. I derive a theoretical formula for the measured spectrum and then a Monte Carlo simulation to validate the model.
more –>;
The main problem with photon counting detectors in medical x-ray systems is the large count rates, which can be greater than 108/sec. The maximum count rate is limited in part by the deadtime and a critical question is the effects of deadtime on the image quality. This post starts the derivation of a simplified model to understand these effects. I do not claim that the model describes a real detector in detail. Instead, I want to get an insight into the magnitude of the effects. For a given mean incident count rate what are small deadtimes that do not affect the results and what values are so large that the images are totally degraded? I start in this post by deriving formulas for the mean value and variance of the recorded counts as a function of deadtime. I found that books and papers that cover this topic, like Parzen and Yu and Fessler, refer to other much more mathematical books for the central limit theorem of renewal processes, which forms the basis for the derivation. Here, I present a derivation, which, although decidedly not mathematically rigorous, gives insight into the assumptions of the formulas. I then validate the formulas with a Monte Carlo simulation. As always, I provide the code and data to reproduce the simulation results.
more –>;
I added a 2D Gaussian object to the projection simulator described in my previous post. I also cleaned up the code.
more –>;
I previously discussed the rationale, the C++ implmentation, and the the Matlab interface for a computed tomography projection simulator. In this post, I discuss a Matlab-only implementation of a simulator. The simulator is limited to ellipses and parallel lines but it is simple and can be (fairly) easily extended to other object types and geometries.
more –>;
