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\title{QR-DECOMPOSITION-BASED BACKGROUND SUBTRACTION}

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\author{\underline{M. Amintoosi}$^{1,2}$, F. Farbiz$^{1}$, M. Fathy$^{1}$,
M.Analoui$^{1}$, N.Mozayani$^{1}$\\
        (1)  Computer Engineering Department,\\
Iran University of Science and Technology, IRAN\\
        (2)  Mathematics Department,\\
Sabzevar Teacher Training University, IRAN, 397 \\(email:
mAminToosi@iust.ac.ir)
       }

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\begin{abstract}
\noindent In this paper a new background subtraction algorithm
based on QR-Decomposition has been proposed. Gaussian Mixture
Model\footnote{ GMM (Gaussian Mixture Model)} has been used for
modeling the background in many literatures. In the GMM method for
background subtraction, background segmentation involves a binary
classification problem based on P(B$\vert $x), where x is the
pixel value at time t, and B represents the background class. The
posterior probability can be expressed in terms of the mixture
components P(G$_{k})$ and P(x$\vert $G$_{k})$ and a density
estimate P(B$\vert $G$_{k})$. The estimation of P(B$\vert
$G$_{k})$, (or in other words distinguishing which kernel of GMM
belongs to the background and which is that of the foreground) is
one of the most concentrating aspects of background subtraction in
the literatures. This problem is eliminated in the proposed
approach with QR-Decomposition, a known method in Linear Algebra.
In a sequence of frames in which the scene is not empty at any
moment, each small block of background can be seen in some frames.
The key idea of the proposed method lies in the identification of
the background based on these blocks with QR-Decomposition method.
R-values produced with QR-Decomposition can be applied to
decompose a given system and indicate the degree of the
significance of the decomposed parts. Selection of background
blocks is conceptually obtained by choosing those parts which have
weak contribution, according to the assigned R-values. The
background model is constructed with these selected blocks.
Simulation results showed that the proposed method produced better
estimation of background with respect to some others.
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\medskip\noindent{Key Words:Background Subtraction, QR Decomposition, Gaussain Mixture Model,
Singular Value Decomposition}

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