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Characteristic functionals are one of the main analytical tools used to quantify the statistical properties of random fields and generalized random fields. The viewpoint taken here is that a random field is the correct model for the ensemble of objects being imaged by a given imaging system. In modern digital imaging systems, random fields are not used to model the reconstructed images themselves since these are necessarily finite dimensional. After a brief introduction to the general theory of characteristic functionals, many examples relevant to imaging applications are presented. The propagation of characteristic functionals through both a binned and list-mode imaging system is also discussed. Methods for using characteristic functionals and image data to estimate population parameters and classify populations of objects are given. These methods are based on maximum likelihood and maximum a posteriori techniques in spaces generated by sampling the relevant characteristic functionals through the imaging operator. It is also shown how to calculate a Fisher information matrix in this space. These estimators and classifiers, and the Fisher information matrix, can then be used for image quality assessment of imaging systems.

Characteristic functionals in imaging and image-quality assessment: tutorial