a$)$ JPEG uses the discrete cosine transform $($DCT$)$ for image compression.

$($i$)$ What is the value of F$(0,0)$ if the image f $($i$,$ j$)$ is as below?

$($ii$)$ Which AC coefficient $|$F$($u$,$v$)|$ is the largest for this f$($i$,$ j$)?$ Why? Is

this F$($u$,$ v$)$ positive or negative? Why?

$2020202020202020202020202020202080808080808080808080808080808080$

$140140140140140140140140140140140140140140140140200200200200200200200200200200200200200200200200$

$($b$)$ Show in detail how a three$-$level hierarchical JPEG will encode the image above, assuming that

$($i$)$ The encoder and decoder at all three levels use Lossless JPEG.

$($ii$)$ Reduction simply averages each $2\backslash$times $2$ block into a single pixel value.

$($iii$)$ Expansion duplicates the single pixel value four times.