The following figure show a $4-$bit image I of size $8-$by$-8$ in the square, with $x$ and $y$ coordinates

specified:

$0$ How the derivatives are obtained in edge detection during formulation?

$(2)$ Apply the $33$ Sobel edge detector in both horizontal and vertical direction to the following selected

pixels $(2,2),(3,3),(3,5),(6,1),(6,6),$ then calculate their magnitudes.

$3$ Apply the Laplacian kernel to the following selected pixels $(2,2),(3,3),(3,5),(6,1),(6,6).$

Laplacian kernel: $\left[\begin{array}{ccc}0& 1& 0\\ 1& -4& 1\\ 0& 1& 0\end{array}\right]$

$4$ Compare the results obtained by the two operators.

$5$ What are the two image processing processes performed by the LoG operator?

$(6)$ Explain the two following operations:

$gra{d}^2(G**I)$

$(gra{d}^{2}G)**I$

Will the results be the same? justify your answer.

$(2$By analyzing the image I and it's histogram, binarize correctly the image in order to separate the

rectangular shape from the background. Give the value of the chosen threshold and represent the binary

image by $$ and $1.$