Please code this in Julia. $3.($Compressing a Grayscale Image$)$ Download the file 'lion.jpg$\text{'},$ and import it into Julia using 'img$=$load$("$lion$.$jpg$")\text{'}.$ Convert the image into a grayscale image with 'img$2=$Gray.$($img$)\text{'}.$ Then convert the image into a matrix A$.$

$($a$)$ After level shifting the matrix, and applying the DCT to each $8$ x $8$ block, apply a low pass filter by letting $\backslash ($ y$\_\{$i j$\}=0\backslash )$ whenever $\backslash ($ i $\backslash$pm j $\backslash$geq $4\backslash ).$ Use the Inverse DCT and then undo the level shifting to construct an approximation matrix $\backslash (\backslash$bar$\{$A$\}\backslash ).$ Compare the compressed image with the original image.

$($b$)$ Construct another approximation matrix $\backslash (\backslash$overline$\{\backslash$bar$\{$A$\}\}\backslash )$ by using the singular value decomposition to compute a rank $30$ approximation of $\backslash ($ A $\backslash ).$ Repeat with a rank $20$ approximation. Compare both compressed images with the original image.