$[100$ p$]1.$ Perform the subsequent computations: Fig. $1$ illustrates the intensity values of a small image represented as a matrix. $($a$)$ Determine the histogram table for the image coded with $4$ bits. $($b$)$ Determine the cumulative histogram table for the image. To implement histogram equalization on the image depicted in Fig. $1.($c$)$ Find the target cumulative histogram. $($d$)$ Establish the transfer function in the form of a look$-$up table that will transform the image's histogram $($Fig$.1)$ to the desired target histogram for the purpose of histogram equalization. $($e$)$ Obtain the histogramequalized image. $($f$)$ Apply the transfer function depicted in Fig. $2$ to the image illustrated in Fig. $1$ to acquire the changed image.

Fig. $1$