OBJECTIVE ASSESSMENT OF IMAGE QUALITY - III - ROC METRICS, IDEAL OBSERVERS, AND LIKELIHOOD-GENERATING FUNCTIONS

We continue the theme of previous papers [J. Opt. Sec. Am. A 7, 1266 ( 1990); 12, 834 (1995)] on objective (task-based) assessment of image q uality. We concentrate on signal-detection tasks and figures of merit related to the ROC (receiver operating characteristic) curve. Many dif ferent expressions for the area under an ROC curve (AUC) are derived f or an arbitrary discriminant function, with different assumptions on w hat information about the discriminant function is available. In parti cular, it is shown that AUC can be expressed by a principal-value inte gral that involves the characteristic functions of the discriminant. T hen the discussion is specialized to the ideal observer, defined as on e who uses the likelihood ratio (or some monotonic transformation of i t, such as its logarithm) as the discriminant function. The properties of the ideal observer are examined from first principles. Several str ong constraints on the moments of the likelihood ratio or the log like lihood are derived, and it is shown that the probability density funct ions for these test statistics are intimately related. In particular, some surprising results are presented for the case in which the log li kelihood is normally distributed under one hypothesis. To unify these considerations, a new quantity called the likelihood-generating functi on is defined. It is shown that all moments of both the likelihood and the log likelihood under both hypotheses can be derived from this one function. Moreover, the AUC can be expressed, to an excellent approxi mation, in terms of the likelihood-generating function evaluated at th e origin. This expression is the leading term in an asymptotic expansi on of the AUG; it is exact whenever the likelihood-generating function behaves linearly near the origin. It is also shown that the likelihoo d-generating function at the origin sets a lower bound on the AUC in a ll cases. (C) 1998 Optical Society of America.