Shannon information and receiver operating characteristic analysis for multiclass classification in imaging

We show how Shannon information is mathematically related to receiver operating characteristic (ROC) analysis for multiclass classification problems in imaging. In particular, the minimum probability of error for the ideal observer, as a function of the prior probabilities for each class, determines the Shannon Information for the classification task, also considered as a function of the prior probabilities on the classes. In the process, we show how an ROC hypersurface that has been studied by other researchers is mathematically related to a Shannon information ROC (SIROC) hypersurface. In fact, the ROC hypersurface completely determines the SIROC hypersurface via a non-local integral transform on the ROC hypersurface. We also show that both hypersurfaces are convex and satisfy other geometrical relationships via the Legendre transform.