Question $4.$ Computing Gradients $(2$ points each $=$ total $8$ points$)$

If I make a mistake and use cross$-$correlation instead of convolution when computing $2$D

gradient vectors of an image using the central finite difference filter $[1/2,0,-1/2],$ the

computed values of my gradient vectors will not necessarily be correct. Answer the following

True$/$False questions. $($Ignore border pixels, only think about pixels in the interior of the

image.$)$ Hint: to answer these questions, think first about the numeric effects of applying this

specific filter differently, then think about what implications the different values have when

interpreted geometrically as the gradient of an intensity surface.

a$)$ The computed gradient vector will point in the opposite direction $($e$.$g$.$ downhill instead

of uphill$).($T or F$)$

b$)$ The computed gradient vector will be perpendicular to the correct gradient. $($T or F$)$

c$)$ The computed gradient will still be $[0,0]$ at local minima $/$ maxima in the image. $($T of F$)$

d$)$ The magnitude $($length$)$ of the computed gradient vector will still be correct. $($T or F$)$