The detection and discrimination of known signals in Gaussian noise is a well-understood problem. Many authors have pointed out that this problem leads to the formation of a test statistic that is a linear function of the data [1,2]. Of more clinical relevance is the more difficult problem of detecting a random signal in a noisy background. Barrett, Myers, and Wagner  and Wagner and Barrett  have shown that the detection of signals with random parameters in general leads to a test statistic that is a nonlinear function of the data. In this paper we will further explore the nonlinear detection strategy of an ideal observer for signals with random parameters. We will show that this strategy is intimately related to the operations the ideal observer would perform when estimating the underlying object in a noisy scene. We will review experimental evidence that shows human observer performance is inferior to the performance of the ideal observer for certain nonlinear tasks, suggesting that the human is unable to utilize nonlinear features in the detection of random objects in these cases.
K. J. Myers,
R. F. Wagner,
"Detection And Estimation: Human Vs. Ideal Performance As A Function Of Information", Proc. SPIE 0914, Medical Imaging II, (27 June 1988); doi: 10.1117/12.968643; https://doi.org/10.1117/12.968643