contrast. smooth curves were drawn through these, and values tabulated. the resulting data have been used extensively in areas such as lighting engineering and road safety. a separate study by knoll et al investigated thresholds for point sources by requiring subjects to vary the brightness of the source to find the level at which it was just visible. a mathematical formula for the resulting threshold curve was proposed by hecht, with separate branches for scotopic and photopic vision. hecht's formula was used by weaver to model the naked-eye visibility of stars. the same formula was used later by schaefer to model stellar visibility through a telescope. crumey showed that hecht's formula fitted the data very poorly at low light levels, so was not really suitable for modelling stellar visibility. crumey instead constructed a more accurate and general model applicable to both the blackwell and knoll et al data. crumey's model covers all light levels, from zero background luminance to daylight levels, and instead of parameter-tuning is based on an underlying linearity related to ricco's law. crumey used it to model astronomical visibility for targets of arbitrary size, and to study the effects of light pollution.