# What is the reason of using Gaussian model for Background Modeling?

I am a new learner to Background Subtraction (Background Modeling) and I am reading a few previous works on how it is performed. Starting with Wikipedia (by searching "Background Subtraction"), I could understand the first few simple methods, until the part "Running Gaussian average". I simply have no idea why a Gaussian probabilistic density function was proposed to fit on the most recent n frames. Reading from the original paper by Wren et al., the reason for using the Gaussian PDF is "computational convenience", which baffled me since the Gaussian PDF does not look simple to me.

In addition, many later methods were based on the Gaussian Mixture Model by Stauffer and Grimson, which was even more complicated. But there seems no explanation about why a Gaussian PDF was used, and why the use was justified.

Can anyone throw some light to me on these questions? Thanks!

• Welcome to Computer Science Stack Exchange. Please give complete references to the documents you have been using, such as link to the wikipedia page, and full referencce and hoefully link to other papers. – babou Jul 20 '15 at 20:15

The Gaussian function is ubiquitous in statistics and many other fields. One reason is the Central Limit Theorem, which says that the means of independent identically distributed random variables themselves have a normal (Gaussian) distribution. This is why you see the bell curve in nature so often.

The Gaussian PDF may not look simple at a first glance, but it is actually very simple to differentiate and integrate. Remember that exp(x) is its own derivative and its own indefinite integral. It is also one of very few functions whose Fourier Transform can be computed analytically.

Finally, the reason Stauffer and Grimson and others used the Gaussian Mixture Model, is that it is an easy way of approximating a complicated multi-modal distribution.

• Thank you for your answer. May I ask a further but more preliminary question: In the papers mentioned, is the Gaussian PDF trying to "simulate" (or in the formal words of the authors, "model") the frequency of the different intensities of a pixel in the previous frames? – GreenPenguin Jul 21 '15 at 15:31
• That's right. If you assume that the background is static, then a normal distribution is a reasonable model for it. Essentially, you are assuming that a pixel has some "real" intensity of color, and any deviation from it that you see in the video frames is caused by Gaussian noise. However, if you have non-static background (flickering lights, rustling leaves, surface of the water), then the distribution is multi-modal, which can be modeled by a mixture of Gaussians. – Dima Jul 21 '15 at 15:40