Spontaneous pupillary behaviour in darkness provides information about a subject's level of vigilance. To establish infrared video pupillography (IVP) as a reliable and objective test in the detection and quantification of daytime sleepiness, the definition of numerical parameters is an important precondition characterising spontaneous pupil behaviour adequately for further statistical procedures. The correct measurement of the pupil size, even if the lid or eyelashes are occluding the pupil, is of particular concern when testing vigilance. In this case many edge points of the pupil are detected and a fitting procedure is described that fits these edge points to a circle and excludes outliers. The first step of data preparation consists of a mathematical artefact management consisting of blink detection and elimination, followed by interpolation. Second, a fast Fourier transformation is carried out for frequencies from 0.0 to 0.8 Hz for each time segment of 82 s. Results are given in absolute and relative power of each frequency band per time segment and mean values over the entire record of 11 min. Third, the changes of the mean pupillary diameter per data window against time are shown graphically. An additional parameter referring to the pupil's tendency to instability, the pupillary unrest index (PUI), is defined by cumulative changes in pupil size based on mean values of consecutive data sequences. These mathematical procedures provide a high level of quality in both data collection and evaluation of IVP as an objective test of vigilance. In a pilot study, the pupillary behaviour of two groups were measured. One group rated themselves as alert (ten men), the other group as sleepy (12 men). The power and PUI were compared using the Mann-Whitney U-test. Both parameters show significant differences between the two groups.