Detection performance of the ideal decision function and its McLaurin expansion: signal position unknown

Although optimal decision functions for many simple detection/discrimination tasks can be cast in a form linear in the signal data, more complicated tasks require the addition of higher-order terms. This is typically the case when parameter uncertainty is allowed, in imaging for example, for the detection of a target of known size and shape but unknown position in a noise field. The simple task of detecting signals known exactly except for position, specifically detection of a "boxcar" shaped signal on a uniform data trace, has been studied in order to elucidate the relative importance of the first-, second-, or higher-order terms of the likelihood ratio decision rule. Analytical expressions have been developed to describe signal-to-noise ratios relevant for performance evaluation at low signal contrast levels, and computer simulations have been used to evaluate performance at higher contrast. It was found that for this task the first-order term (which corresponds to measuring the mean value of the data) dominates for low contrast signals but is superseded by higher-order terms (which is jth order correspond to the jth-order correlation of the data match filtered with the jth-order correlation of the signal) as contrast is increased. The quadratic term is found to be inferior to the linear term for small contrast and to the cubic for all values of signal contrast if the background is held constant. When the background level is allowed to vary, the performance of the odd-order terms decreases relative to that of the quadratic (and other even-order ones). Various measures of decision function efficiency are compared, demonstrating the severe limitations of using the simple signal-to-noise ratio (SNR) formalism for processes with non-Gaussian-distributed probability density functions. These results are valuable for guiding approaches to computational observers of signal data by showing the range of validity of suboptimal decision functions that are much easier to compute than the exact likelihood ratio solution.