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Jan 30 2012

A-space covariance from x-ray detector noise

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.

Jan 24 2012

Detection theory with A-space data

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.

Jan 14 2012