Another way to get energy selective data is to measure the total number of counts and their total energy. This may not seem to provide energy-dependent information but if you look at it from the point of view of energy weighting, there is information. We can look at the total counts as the integral of the spectrum multiplied by a constant function of energy, where the constant is one. On the other hand, the total energy is the integral of the spectrum multiplied by the function f(E)=E. This weights the higher energies more than the lower and therefore has different information than the counts. This is actually enough difference to give SNR comparable to two-bin PHA.
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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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In my last post, I showed that the multivariate normal, abbreviated multinormal, is a good model for the noise w in a linearized x-ray system model. In this post, I will discuss some of the properties of the multinormal distribution. I will show a rationale for its expression using vectors and matrices. This will lead me to discuss matrix calculus. I will describe diagonalizing and whitening transformations and derive the moment generating functions of the uninormal and multinormal to show that linear combinations of multinormals are also multinormal. This post will provide math background for my discussions of detection and maximum likelihood estimation with the linearized x-ray model.
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In my last post, I described a three part model used in statistical signal processing: (1) an information source produces outputs described by a finite dimensional vector, (2) a probabilistic mapping between the source outputs and the measured data, and (3) a receiver or processor that computes an estimate of the source output or makes a decision about the source based on the data. I showed that in x-ray imaging the information is summarized by the A vector whose components are the line integrals of the coefficients in the expansion of the x-ray attenuation coefficient. The basis set coefficients a(r) depend on the material at points r within the object and the line integrals Aj = ∫ℒaj(r)dr are computed along a line ℒ from the x-ray source to the detector. I then showed the rationale for a linearized model of the probabilistic mapping from A to the logarithm of the detector data L
In this post, I will try to convince you that the multivariate normal is a good model for the noise w. This will lead me to discuss tests for normality including probability plots and statistical tests based on them such as the Shapiro-Wilk test[4] (available online) for univariate data and Royston’s test[3] for multivariate data.
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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.
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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”, 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.
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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.
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