We measured human observers' detectability of aperiodic signals in noise with two components (white and low-pass Gaussian). The white-noise component ensured that the signal detection task was always noise limited rather than contrast limited (i.e., image noise was always much larger than observer internal noise). The low-pass component can be considered to be a statistically defined background. Contrast threshold elevation was not linearly related to the rms background contrast. Our results gave power-law exponents near 0.6, similar to that found for deterministic masking. The Fisher-Hotelling linear discriminant model assessed by Rolland and Barrett [J. Opt. Soc. Am. A 9, 649 (1992)] and the modified nonprewhitening matched filter model suggested by Burgess [J. Opt. Soc. Am. A 11, 1237 (1994)] for describing signal detection in statistically defined backgrounds did not fit our more precise data. We show that it is not possible to find any nonprewhitening model that can fit our data. We investigated modified Fisher-Hotelling models by using spatial-frequency channels, as suggested by Myers and Barrett [J. Opt. Soc. Am. A 4, 2447 (1987)]. Two of these models did give good fits to our data, which suggests that we may be able to do partial prewhitening of image noise.