Biomedical Imaging

$\iff \underset{\_}{\widehat{f}}=A^{-1}X.(7.47)$

Hence, the ‘simple’ inversion of the noisy data is equal to the ML estimator.

In the more general case that A is not invertible, the ML estimator for Poisson distributed random data must be obtained by solving the optimization problem

$\underset{\_}{\widehat{f}}=\text{argmin}\left[{\displaystyle \sum _i}\left({\displaystyle \sum _j{A}_{ij}{f}_j-X_i\log }{\displaystyle \sum _j{A}_{ij}{f}_j}\right)\right].(7.48)$

Apart from Poisson distributions, the ML approach can be applied to all kinds of probability functions P(X|μ), including the simple case of Gaussian noise. Of course, the functionals to be minimized will look different from those in the Poisson case.

7.1.3Bayesian inference techniques

In the preceding section, we assumed the object to be deterministic and we did not impose any constraints on the object or ...