a$)$ An image patch, I, generated by a camera, containing pixel values between $0$ through $7,$ is given as:

$I=([1,3,7,3,1,5,5,4],[0,1,0,0,0,0,0,0],[0,0,0,0,5,0,0,7],[0,7,5,1,3,2,4,4],[0,2,3,0,0,1,1,1],[0,5,0,0,0,1,1,2],[0,0,0,6,0,0,1,0],[1,5,3,3,5,4,0,0])$

i$)$ Calculate the probability of pixel values in I.

$5$

ii$)$ Design a Huffman code to compress this image.

$10$

iii$)$ What is the average number of bits needed to encode each pixel using the

$4$

Huffman code you designed in $($ii$)$ above?