d$)$ Let, $F^{\text{'}}=\left[\begin{array}{ccc}-1& 0& 1\\ -1& 0& 1\\ -1& 0& 1\end{array}\right].$ What is the difference in functionality of filter $F^{\text{'}}$ and $F$ of part $($c$)?$

e$)F=\frac{1}{16}\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right]$

Briefly describe what the filter in part $($e$)$ is doing to the image.

f$)F=\frac{1}{9}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

What is the difference in functionality of the filters in part $($e$)$ and $($f$)?$

Problem $2$

a$)$ Show that we can write correlation as a vector dot product,

$G(i,j)=f^T{t}_{ij}$

Where $f$ is the vector representation of the filter and is the vector representation of the

neighborhood patch of the image.

b$)$ Implement the correlation operation in Matlab or Python which takes a filter $F$ and a grayscale image I

and outputs $G,$ where $G=$FoxI. Specifically, implement it using the dot product result of part $($a$),$

otherwise it will be too slow. Remember to implement zero$-$padding to maintain correct sizes.

Test your code using the provided image "parrot.png$"$ and the filters in problem $1.$Manually calculate the output image $Gin{R}^{33}$ as a result of $G=$FoxI when the filter $F$ is: d$)$ Let, $F^{\text{'}}=\left[\begin{array}{ccc}-1& 0& 1\\ -1& 0& 1\\ -1& 0& 1\end{array}\right].$ What is the difference in functionality of filter $F^{\text{'}}$ and $F$ of part $($c$)?$

e$)F=\frac{1}{16}\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right]$

Briefly describe what the filter in part $($e$)$ is doing to the image.

f$)F=\frac{1}{9}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$

What is the difference in functionality of the filters in part $($e$)$ and $($f$)?$

a$)F=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

b$)F=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

c$)F=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ -1& -1& -1\end{array}\right]$

Briefly describe what the filter in part $($c$)$ is doing to the image.