In a previous post I described the application of statistical estimator theory to energy selective x-ray imaging. I introduced a linearized model for the signal and noise and in a subsequent post I described a linear maximum likelihood estimator (MLE) that achieved the Cramèr-Rao lower bound (CRLB). In many applications, such as CT, the linear model is not sufficiently accurate. In this post, I will start the discussion of my paper[3] “Estimator for photon counting energy selective x-ray imaging with multi-bin pulse height analysis.” The paper describes an estimator that is accurate for a wide dynamic range that also achieves the CRLB and has other desirable properties such as fast and predictable computation time and being implementable in a clinical institution as opposed to a physics lab. This post frames the discussion by describing general aspects of computing the A-vector from energy selective measurements and several estimators that are widely used and their properties.
In this post, I will describe the methods that I use to create the posts on this blog. My approach is to automate the process as much as is comfortable. This reduces the repetitive work required to put a post online, reduces errors and produces a standard package of files with standard format. The standard format makes it easier for readers to follow the post and allows me to re-use the posts; for example, to create an ebook compendium of the posts. My blog discusses technical topics with a lot of mathematics so the methods that I describe will be geared to my interests but I think the general approaches of automating and creating re-useable content are useful for other subjects. The shell scripts and other code I have written were “hacked” together to do a job and designed only for my use so they lack many niceties. I will present them here as an example you can use to create code for your blog.
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.
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.
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.
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.
Computational geometry is an interesting and important topic for imaging in general and x-ray imaging in particular. In this post, I describe the basic formulas for perhaps the most fundamental geometric object—a straight line in three dimensions. The object can also be used in 2D to represent a line in an image.
One of my Matlab programming styles, some would say quirks, is wide use of complex variables. I not only use them in the standard mathematical places but I use them to represent two dimensional spatial vectors and to represent two dimensional quantities in general.
