Karhunen$-$Loeve Transform $($KLT$)$ is used in an image compression scheme. A $6\backslash$times $6$ image is

given as follows:

$B=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 1& 1& 1& 2\\ 1& 1& 2& 2& 1& 2\\ 1& 1& 2& 4& 4& 4\\ 1& 2& 2& 3& 4& 4\\ 1& 2& 3& 3& 4& 4\end{array}\right]$

$($a$)$ Partition the image into $922$ blocks $($subimages$),$ and order them lexicographically to

form $941$ column vector $($x$).$ Calculate the mean vector, and show that its covariance

matrix is given by:

$C=\left[\begin{array}{cccc}1\mathrm{.}6944& 1\mathrm{.}7361& 1\mathrm{.}4028& 1\mathrm{.}4167\\ 1\mathrm{.}7361& 2\mathrm{.}0278& 1\mathrm{.}6944& 1\mathrm{.}6667\\ 1\mathrm{.}4028& 1\mathrm{.}6944& 2\mathrm{.}1111& 1\mathrm{.}9167\\ 1\mathrm{.}4167& 1\mathrm{.}6667& 1\mathrm{.}9167& 2\mathrm{.}2500\end{array}\right]$

$($b$)$ Determine the corresponding eigenvectors and eigenvalues for the covariance matrix $C$

using MATLAB $/$ other programming platforms.