We develop a theoretical model of image brightness variation of line-scan imaging systems using an ac-operated light source for object illumination. First, we derive an equation that relates the lamp operating frequency F, the object transport speed v, the image resolution R-o, and the number n(c) of lamp operating cycles within the time interval of one scan line. It is shown that n(c) = F/vR(o) = Ft(exp), where t(exp)=1/vR(o) is the exposure time of an image sensor element and is the time interval within one scan line. Following this, we define a normalized modulation function M(n(c)) that relates to the variation Delta E of the light energy E = integral l(t) dt received by a sensor element over t(exp), where l(t) is the intensity of object illumination. Three types of periodic functions for l(t) are considered far which closed-form solutions for M(n(c)) can be obtained: square-wave, sinusoidal, and triangular-wave. For the square-wave function we show that M(n(c)) is a periodic function of n(c), with zeros occurring at integral values of n(c) and peaks occurring at odd multiples of 1/2, with the heights of these peaks decreasing as 1/2n(c) with increasing n(c). For the sinusoidal and triangular-wave functions, M(n(c)) has similar periodic behavior except that the heights of the peaks decrease as 1/pi n(c) and 1/4n(c) respectively. These results indicate that the square-wave function can be used to represent the worst case for assessing the performance of an imaging system under different conditions. Accordingly, we show numerically that lamps operating at a low frequency F=50 Hz cannot be used for abject illumination because the variation in image brightness is too large, and lamps operating at F=20 kHz are generally applicable except at high object transport speed, typically at 100 in./s. Finally, we point out that this theoretical model may also be applied to area-scan imaging systems, such as for video cameras. For these imaging systems, the normalized modulation function can be used to assess the variation of image brightness between one whole image and another, rather than between one scan line and another, (C) 1998 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(98)03305-4].
