Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

A,0.0 0.2 0.4 0.6 0.8 1.0Position (fraction of pixel width)o w ~ W 1W 1W 1~Compton Scatter Angle (degrees)BFIGURE 10-21. Not all probability density functions (PDFs) are distributed according tothe gaussian distribution. (A) The probability of an x-ray photon landing at some positionalong the width of a pixel demonstrates a uniformly distributed PDF.(B) The probabilitythat a 20-keV x-ray photon will be scattered at a given angle by the Comptoninteraction is also illustrated. In nature, and especially in physics, there are many shapesof PDFsother than the gaussian. Nevertheless, the gaussian distribution is the most commonlyapplicable PDF,and many measured phenomena exhibit gaussian characteristics.x 3 X 4 X 5) in Equation 10-17 makes computation difficult for larger values of x(for example, 69! = 1.7 X 10 98 ).X-ray and y-ray counting statistics obey the Poisson distribution, and this isquite fortunate. Why? We saw that the gaussian distribution has two parameters,X, and 0", which govern its shape. Knowing one parameter does not imply thatyou know the other, because there is no fixed relationship between 0" and X (asthe children's height example was meant to illustrate). However, with the Poissondistribution (actually its gaussian app~ximation), knowing the mean (X) impliesthat 0" is known as well, since 0" = \iX. In other words, 0" can be predicted fromX. Why is this a fortunate situation for x-ray and y-ray imaging? It is fortunatebecause it means that we can adjust the noise (0") in an image by adjusting the(mean) number of photons used to produce the image. This is discussed more inthe next section.GaussiaPoisson40 50 60ParameterFIGURE 10-28. The Poisson distribution hasbut one parameter, which describes its shape,m, unlike the gaussian distrib!!tion, which isdescribed by two parameters (X and cr). However,the gaussian distribution represents anexcellent approximation to the Poisson distributionwhen cr = vX.