On the Relation Between ROC and CMC

We formulate a compact relation between the probabilistic Receiver Operating Characteristic (ROC) and the probabilistic Cumulative Match Characteristic (CMC) that predicts every entry of the probabilistic CMC as a functional on the probabilistic ROC. This result is shown to be valid for individual probabilistic ROCs and CMCs of single identities, based on the assumption that each identity has individual mated and nonmated Probabilitic Density Functions (PDF). Furthermore, it is shown that the relation still holds between the global probabilistic CMC of a gallery of identities and the average probabilistic ROC obtained by averaging the individual probabilistic ROCs of these identities involved over constant False Match Rates (FMR). We illustrate that the difference between individual probabilistic ROCs and the difference between global and average probabilistic ROCs provide an explanation for the discrepancies observed in the literature. The new formulation of the relation between probabilistic ROCs and CMCs allows us to prove that the probabilistic CMC plotted as a function of fractional rank, i.e. linearly compressed to a domain ranging from 0 to 1, will converge to the average probabilistic ROC when the gallery size increases. We illustrate our findings by experiments on synthetic and on face, fingerprint, and iris data.