Statistical Reconstruction Methods. We know that x-ray beams are made of x-ray photons, and rather than computing the deterministic solution directly from an equation, we can explore the randomness of x-ray photons. If we assume that the detector is a photon-counting detector, then we can use statistical tools to build the model.
Statistical Reconstruction Methods. We know that x-ray beams are made of x-ray photons, and rather than computing the deterministic solution directly from an equation, we can explore the randomness of x-ray photons. If we assume that the detector is a photon-counting detector, then we can use statistical tools to build the model. ∫ ( )∫ In Equation (2.3), if we discretize the equation with respect to the j-th projection and the i-th detector pixel, then we can obtain yi = S(e) exp − µ (→r (t)) d t d e, (2.23) E t∈li,j where yi represents the number of photons measured at the i-th pixel of detector. If we further discretize the equation over the energy spectrum, then it can be expressed ( as yi = Σ sk exp − ∫ k t∈li,j µ (→r (t)) d t) . (2.24) Repeating the same steps as (2.12), (2.13) and (2.14), we can obtain yi = s¯i exp {− [Au]i} , (2.25) where s¯i is the average photon density of the j-th projection. If we both consider the errors introduced in the photon diffusion and counting, we can add a noise term yi = s¯i exp {− [Au]i} + ηi. (2.26) One realistic assumption is that the projected data are Poisson distributed as yi ∼ Poisson(s¯i exp {− [Au]i}). (2.27) With this assumption, we can build a maximum likelihood function fy (y; u) = . (2.28) Y s¯yi exp {−yi [Au] } exp (−s¯i exp {− [Au] }) yi! If we ignore the constant terms, the corresponding likelihood function is Y L (u; y) = exp {−yi [Au]i} exp (−s¯i exp {− [Au]i}) . (2.29) i The log-likelihood function can be expressed as Σ l (u; y) = (−yi [Au]i − s¯i exp {− [Au]i}) . (2.30) To maximize the log-likelihood function, it is equivalent to minimizing the negative log-likelihood function. So the objective function is Σ
