Exercise $5.$

In this problem we study a single$-$photon image sensor$.$ First, recall that photons arrive according to a

Poisson distribution, i$.$e$.,$ the probability of observing $k$ photons is

$P[Y=k]=\frac{{}^k{e}^{-}}{k!},$

where $$ is the $($unknown$)$ underlying photon arrival rate. When photons arrive at the single$-$photon detector,

the detector generates a binary response $"1"$ when one or more photons are detected, and $"0"$ when no photon

is detected.

$($a$)$ Let $B$ be the random variable denoting the response of the single$-$photon detector. That is$,$

$B=\{\begin{array}{cc}1,& Y1,\\ 0,& Y=0.\end{array}$

Find the PMF of $B.$

$($b$)$ Suppose we have obtained $T$ independent measurements with realizations $B_1=b_1,B_2=b_2,$dots,

$B_T=b_T.$ Show that the underlying photon arrival rate $$ can be estimated by

$=-log(1-\frac{{\displaystyle \underset{t-1}{\overset{T}{}}}{b}_t}{T}).$

$($c$)$ Get a random image from the internet and turn it into a grayscale array with values between $0$ and $1.$

Write a Python program to synthetically generate a sequence of $T=1000$ binary images. Then use

the previous result to reconstruct the grayscale image.