Question $5(20$ points, $10$ points each part$)$

$($a$)$ Consider the following with top left pixel indexed as $(0,0)$ with increasing indices to the right

and bottom and common property we are interested in as being the value $1$ :

$1,0,0,0$

$0,1,1,0$

$1,a,1,1$

$0,0,0,0$

$1,0,0,0$

What should the value of $a$ be such that pixels $(1,1)$ and pixels $(3,3)$ are $($i$)4-$adjacent, $($ii$)8-$adjacent,

$($iii$)$ m$-$adjacent.

$($b$)$ Consider the following candidate for a distance function between pixels $p=[p_x,p_y]$ and $q=$

$[q_x,q_y]$:

$D(p,q)=[(p_x-q_x-k{)}^s+(p_y-q_y-l{)}^s)$

Is this a valid distance function for general values of $k,l$ and $s.$ If your answer is yes, prove your

answer. If your answer is no$,$ prove your answer and specify the conditions on these three parameters

that would result in a valid distance function.