Bt+1 (x, y) = (1− α)Bt (x, y) + αIt (x, y) (1.1) where α is the learning rate which is a constant in [0, 1]. Bt and It are the background and the current image at time t, respectively. The main disadvantage of this scheme is that the value of pixels classified as foreground are used in the computation of the new background and so polluted the background image. To solve this problem, some authors used a selective maintenance scheme that consists of updating the new background image with different learning rate depending on the previous classification of a pixel into foreground or background:

Bt+1 (x, y) = (1− α)Bt (x, y) + αIt (x, y) (1.2) if (x, y) is background

Bt+1 (x, y) = (1− β)Bt (x, y) + βIt (x, y) (1.3) if (x, y) is foreground


Here, the idea is to adapt very quickly a pixel classified as background and very slowly a pixel classified as foreground. For this reason, β << α and usually β = 0. So the Equation (1.3) becomes:

Bt+1 (x, y) = Bt (x, y) (1.4)

But the problem is that erroneous classification may result in a permanent incorrect background model. This problem can be addressed by a fuzzy adaptive scheme which takes into account the uncertainty of the classification. This can be achieved by graduating the update rule using the result of the foreground detection such as in [18].