$(15$ points$)$ A rectangular part with a hole in the middle is placed under a digital camera. The

figure below shows the projection of its optical image on the photo sensitive array.

Assume that the camera uses a $3-$bit frame grabber to store the digital image in which the pixel's

gray level value is proportional to the light intensity the photo cell detects at that pixel location.

$($That is$,$ a higher value represents a brighter gray level$).$ Also assume that the back lighting

technique is used and the pixels that make up the inner circle and the left and right edges of the

object are half$-$covered.

$($a$)$ Show the digitized image array. Use the round up rule for assigning gray levels to pixels $($e$.$g$.$

$4\mathrm{.}1=5).$

$($b$)$ Assume that the calibration result shows that $1$ pixel $=1$ inch square $($i$.$e$.,1$ in $x$ in$).$

Estimate the size of this part based on the digitized image array obtained in $($a$).$

$($c$)$ Calculate the true size of the object $($i$.$e$.,$ the size of rectangle $-$ the size of circle$)$ and

compare it with the result obtained in $($b$).$ What is the error in estimating the true size of this

object? $($hint: the error is the difference between the real size of the object and the estimated

size.$)$