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Oct 30 2013

Rationale for the new estimator

Tag: Implementation,Noise,Physicsadmin @ 10:20 am
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.

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Oct 18 2013

Why is polynomial estimator variance so large?

Tag: Math,Noiseadmin @ 9:11 am
Anyone with experience in energy selective imaging is struck by the terrible performance of polynomial estimators discussed in my last post. This is most likely due to the fact that in the past the number of spectra was almost always equal to the dimension of the A-vector. In this case, as I showed in the last post, any estimator that solves the deterministic, noise free equations is the maximum likelihood estimator (MLE). With equal number of spectra and dimension, the polynomial estimator is accurate for low-noise data so it provides an ’efficient’ estimator. That is its covariance is equal to the Cramèr-Rao lower bound (CRLB). In this post, I examine the reason for the poor performance with more measurements than the A-vector dimension.

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Oct 01 2013

Estimators for Energy-selective imaging—Part 1

Tag: Implementation,Math,Noise,Physicsadmin @ 5:46 pm

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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