Image filtering

$($a$)$ Filter image $J$ with the gaussian filter G using zero padding.

$G=\frac{1}{16}\{\begin{array}{ccc}1,& 2,& 1\\ 2,& 4,& 2\\ 1,& 2,& 1\end{array}$

$($b$)$ Is it more efficient to filter an image with two $1D$ filters as opposed to a $2D$ filter? Why? How does the computational complexity relate to the size of the filter kernel $($with $KK$ pixels$)$ in both cases?

$($c$)$ Is the following convolution kernel separable? If so$,$ separate it$.$

For the image I below apply the following filters to the pixel at the center $($marked with a box$).$ Round the results to the nearest integer value.

$($d$)33$ box filter $($i$.$e$.$ averaging in a $33$ neighborhood$).$

$($e$)33$ median filter.

$($f$)$ Why is the Gaussian filter a better smoothing filter than a box filter? How can it be implemented fast?

$($g$)$ Compute the edge direction and magnitude $($that is$,$ the direction and magnitude of image gradient$)$ at the center pixel using the masks of the Sobel edge detector $(S_1$ and $S_2$ below$).$

$S_1=\frac{1}{8}\{\begin{array}{ccc}-1,& 0,& 1\\ -2,& 0,& 2\\ -1,& 0,& 1\end{array},S_2=\frac{1}{8},\{\begin{array}{ccc}-1,& -2,& -1\\ 0,& 0,& 0\\ 1,& 2,& 1\end{array}$

$($h$)$ The binary pixel array on the left below was convolved with an unknown kernel $[?]$to produce the result on the right. The output is limited to the same size as input and zero padding was used at the boundaries. Specify the kernel as an array. What task does it accomplish in computer vision.