$($a$)$ Describe an algorithm $($other than thresholding$)$ which will convert a grayscale image $(8$ bits per pixel$)$ to a bi$-$level black and white image $(1$ bit per pixel$),$ with the same number of pixels, while retaining as much detail as possible.

$($b$)$ Compare and contrast half$-$toning and error diffusion. Include in your answer an explanation of the situations in which each is superior to the other.