Background subtraction is a commonly used process in surveillance systems. One difficult problem when using the process is maintaining a correct background image against changing illumination conditions. Most methods for maintaining the background image are based on intuitive definitions about the illumination change and are implemented as somewhat ad hoc algorithms. In contrast, we first define mathematical models representing the relation between the illumination intensity, a reflection index of objects and a pixel value. We also mathematically define an assumption about illumination, which requires that the distribution of the illumination intensity in a small region does not change. Then we formalize the background subtraction problem as a statistical test (\chi² test) based on the models and assumption. The experiments show that our models appropriately express the imaging process of a camera and our method 1 provides stable detection performance for foreground objects.