$w=\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right],f=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

$($a$)$ Give the $(x,y)$ coordinates of each point on the filter kernels $w$ and $w$ rotated to $180,$ respectively.

$($b$)$ Compute the convolution conv$(w,f)$ using the minimum zero padding needed. Show the details of your computations when the kernel is centered on point $(2,3)(2$nd row, $3$rd col$)$ of $f$; and then show the final full convolution result.

$($c$)$ Repeat $($b$),$ but for correlation corr$(w,f).$

$($d$)$ What may you conclude in terms of the above analysis and computations?