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.
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In the next few posts I will discuss my paper,
Dimensionality and noise in energy selective x-ray imaging, available for free download
here. I will elaborate on the physical and mathematical background and explain how to reproduce the figures.
With my approach to energy selective imaging, the x-ray attenuation coefficient is approximated as a linear combination of functions of energy multiplied by constants that are independent of energy. The number of functions required is the dimensionality. The basic premise of the paper is that the dimensionality is really a pragmatic tradeoff between more information, which requires larger dimensionality, and the increase in noise, which requires higher dose and more expensive equipment to reduce it to a level where the resultant images are clinically useful. The bottom line of the paper is that with biological materials such as soft tissue, bone, and fat, only two dimensions are practical but if an externally administered contrast agent with a high atomic number element such as iodine is included then three and maybe more dimensions are possible.
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The past two posts have discussed estimators for A-vector data. I showed that with the same number of measurement spectra as the A-vector dimension, any estimator that solves the deterministic equations is the maximum likelihood estimator (MLE) and it will achieve the Cramèr-Rao lower bound (CRLB). If there are more measurement spectra than the dimension, then the polynomial estimator, which works well for the equal case, has very poor performance giving a variance that can be several hundred times larger than the CRLB. I showed by simulations that with more measurements than dimension the iterative MLE does give a variance close to the CRLB but it has substantial problems. Common to all iterative algorithms, the computation time is long and random. It may fail to converge at all if the initial estimate is too far from the actual value. As it was implemented by Schlomka et al.
[2], it also requires measurements of the x-ray source spectrum and the detector spectral response. These are difficult, time consuming and require laboratory equipment that is not usually available in medical institutions.
In this post, I will give an intuitive explanation for the operation of a new estimator that I introduced in my paper
[1] “Estimator for photon counting energy selective x-ray imaging with multi-bin pulse height analysis,” which is available for free download
here. The estimator is efficient and can be implemented with data that can be measured at medical institutions. The details of the estimator are described in the paper. Here, I will discuss the background and give a rationale on how it works.
more → Continue reading “Rationale for the new estimator”
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.
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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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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 will summarize where I stand in the discussion of my paper, “Near optimal energy selective x-ray imaging system performance with simple detectors.” You will find it useful to download a free copy of the paper to follow along with this discussion.
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The NKQ detector introduced in my paper[Alvarez2010] is a generalization of the NQ where instead of just counting the photons we use pulse height analysis (PHA) to separate the photons into K bins depending on their energies and simultaneously measure their total energy. This gives performance similar to PHA with the number of bins plus one so it may be easier to implement. As with the NQ detector, the integrated energy Q can be used in regions where high count rate leads to excessive pulse pileup.
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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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