Dec 20 2011

## Normal probability models for x-ray measurements

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 *A*_{j} = ∫_{ℒ}a_{j}(r)*d***r** 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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