This step completes the segmentation removing static objects from the grayscale images $T_i$ using Eigen-Background decomposition. First, each image $T_i \in \mathbb{R}^{n_x\times n_y} $ is reshaped into a column vector $I_i \in \mathbb{R}^{n_p\times 1}$. These $n_t$ vectors become the $i-th$ column of a video matrix $A(n_p\, , \, n_t)$, after having removed the average image:
\begin{equation}
A_{(n_s \times n_t)}=\begin{pmatrix} \hat{I}_1 & \hat{I}_2 & \cdots & \hat{I}_{n_t}
\end{pmatrix}
\quad with \quad \hat{I}_i=I_i-\frac 1 {n_t} \sum_{i=1}^{i=n_t} I_i
\end{equation}
The columns of this matrix are usually independent, hence this matrix has a full rank $r=n_t$. However, because static objects appear in all the frames in the same position, their contribution to the information contained in the matrix has a much lower rank representation: a $n_t$ set of images containing nothing but static objects, for example, can be represented by only one picture, i.e. one column of this matrix, i.e. a rank $r=1$ representation. …show more content…
Therefore, its eigenvalue decomposition is $D=\Phi \Lambda \Phi^T$, with $\Lambda$ the diagonal matrix containing the eigenvalues $\lambda_i$ and $\Phi$ the matrix with the eigenvectors. Using the $SVD$ in $D$ reads: $D=A^T A=(U\Sigma V^T)^T(U\Sigma V^T)=(V \Sigma U^T)(U \Sigma V^T)=V\Sigma^2V^T$, from which $\lambda_i=\sigma_i^2$. The technique was in fact named Eigen-backgound because the PCA of the video matrix $A$ was first presented in terms of $\lambda_i$s. } matrix
$D=A^{T} \cdot A$, either by Singular Value Decomposition (SVD) of the matrix $A$. With the second approach, the video can be decomposed in its components $E_k$ as
\begin{equation}
A_{(n_s \times n_t)}=\begin{pmatrix} \hat{I}_1 & \hat{I}_2 & \cdots & \hat{I}_{n_t}
\end{pmatrix}
\quad with \quad \hat{I}_i=I_i-\frac 1 {n_t} \sum_{i=1}^{i=n_t} I_i
\end{equation}
The columns of this matrix are usually independent, hence this matrix has a full rank $r=n_t$. However, because static objects appear in all the frames in the same position, their contribution to the information contained in the matrix has a much lower rank representation: a $n_t$ set of images containing nothing but static objects, for example, can be represented by only one picture, i.e. one column of this matrix, i.e. a rank $r=1$ representation. …show more content…
Therefore, its eigenvalue decomposition is $D=\Phi \Lambda \Phi^T$, with $\Lambda$ the diagonal matrix containing the eigenvalues $\lambda_i$ and $\Phi$ the matrix with the eigenvectors. Using the $SVD$ in $D$ reads: $D=A^T A=(U\Sigma V^T)^T(U\Sigma V^T)=(V \Sigma U^T)(U \Sigma V^T)=V\Sigma^2V^T$, from which $\lambda_i=\sigma_i^2$. The technique was in fact named Eigen-backgound because the PCA of the video matrix $A$ was first presented in terms of $\lambda_i$s. } matrix
$D=A^{T} \cdot A$, either by Singular Value Decomposition (SVD) of the matrix $A$. With the second approach, the video can be decomposed in its components $E_k$ as