This is the last post in my series discussing my paper, “Near optimal energy selective x-ray imaging system performance with simple detectors”. In the last post I showed plots of the signal to noise ratio (SNR) of images with different types of energy-selective detectors. In this post, I show images illustrating these differences. These images were not included in the paper but they are based on its approach. The images are calculated from a random sample of the energy spectrum at each point in a projection image. These data are then used to make images with (a) the total energy, which are comparable to the detectors now used in commercial systems, (b) the total number of photons, (c) an N2Q detector, and (d) an optimal full spectrum by weighting the spectrum data before summing, as described in Tapiovaara and Wagner (TW). I use the theory developed in my paper, to make images from A-space data using data from the N2Q detector. In order to do this, I need an estimator that achieves the Cramèr-Rao lower bound (CRLB). For this I use the A-table estimator I introduced in my paper “Estimator for photon counting energy selective x-ray imaging with multibin pulse height analysis” available for download here.
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In this post, I continue the discussion of my paper, “Near optimal energy selective x-ray imaging system performance with simple detectors.” I summarized many of the theoretical formulas in my last post. Here, I will discuss the formulas for the SNR with x-ray tube spectra with different voltages and different object thicknesses. I will present code to reproduce Figs. 7-9 of the paper. The results show that the energy-selective detectors have SNR that is approximately 4 times larger than the SNR with energy-integrating detectors, the detectors used in almost all conventional medical x-ray systems.
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The standard way to handle optional arguments in a Matlab function is to put them at the end of the call list and only include them if you want to change them. This has a lot of problems. First, what do you do if you want to change an argument that is somewhere in the middle of the list but leave the rest unchanged? Some would suggest just putting commas for the unchanged arguments but as far as I know the behavior is undefined. Does this substitute an empty variable ‘[]‘ for the other arguments? Also, this makes it hard to read the code. You have to count up the number of commas to figure out which argument is being changed.
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I have prepared an ebook that compiles and organizes the posts in this blog to today’s date. You can access it by sending me email:
Energy-selective x-ray imaging and other topics.
I plan to update the book regularly and I will post an entry to the blog when an updated version is available.
In this post, I continue to discuss the results in my paper “Near optimal energy selective x-ray imaging system performance with simple detectors[Alvarez2010].” 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. The paper uses statistical detection theory to show that the performance depends on the signal to noise ratio (SNR) and derives a formula (see this post) to compute the SNR as a function of the detector spectral response and noise properties. In this post, I use the formulas for the NQ (simultaneous photon counts and integrated energy) detector data statistics from my last post to compute the SNR. We can use the formulas to show that the NQ signal (almost) always has a larger SNR than the N and Q individual signals. The SNRs are equal if the spectrum has zero-width.
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In this post, I will discuss the signal to noise ratio (SNR) of a photon counting detector with pulse height analysis (PHA). I will show that it approaches the ideal full-spectrum SNR as the number of bins gets large. I will also show that we can get quite close to the ideal value even with a small number of bins, which was one of the main points in my paper.
I will begin by re-visiting a result from my last post that the Cramèr-Rao lower bound (CRLB) with multivariate normal log of photon count data assuming a constant covariance is nearly equal to the accurate matrix that includes the variation of the covariance. The derivation becomes problematic with a large number of bins since the mean value in each bin may be small enough so the normal approximation to the counts is not valid. I will show an alternate derivation that uses the Poisson model of the count data, which is valid for small counts. The result is formally the same as with the normal approximation so the CRLB formula can be applied for any number of bins. This result is new and was not included in my paper.
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My last post showed that detection performance is determined by the signal to noise ratio (SNR). I derived a formula for the SNR with multispectral measurements, which depends on the covariance of the A-space data. This post shows how to compute the A-space covariance from the x-ray data noise and the effective attenuation coefficient matrix. In general, this depends on the type of estimator used so instead I will use the Cramèr-Rao lower bound (CRLB), which is the minimum covariance for any unbiased estimator. This gives a general result independent of the specific estimator implementation. I will show that the constant covariance CRLB is sufficiently accurate for our purposes. These results will allow us to compute signal to noise ratios for limited energy resolution measurements that are directly comparable to the Tapiovaara-Wagner[Tapiovaara1985] optimal SNR with complete energy information.
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In my last posts I discussed the background for applying statistical detection theory to x-ray imaging. In this post, I will show how to incorporate the A-space description into the model. This will lead me to discuss the effect of the basis set functions on the approximation or representation error of the attenuation coefficients of body materials. I will show that there are optimal functions that minimize the error but that other basis functions, such as the attenuation coefficient functions of different materials, do not lead to substantially larger errors. Once the A-space description is in the model, we can derive a signal to noise ratio that is directly comparable to the Tapiovaara-Wagner SNR[Tapiovaara1985] and we can compare the SNR with limited energy resolution to the ideal with complete spectral information.
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In my previous post, I showed that multivariate normal is a good model for x-ray measurements and in my last post I described the general properties of this distribution. In this post, I will discuss statistical detection theory with the normal model. I will show that the performance is characterized by a suitably defined signal to noise ratio. This will enable me to close the loop to the main topic of this series of posts, which is to explain the results in my paper, “Near optimal energy selective x-ray imaging system performance with simple detectors[Alvarez2010]”, which is available for free download here.
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