Communication, Networking, Signal and Image Processing (CS)

Question 5: Image Processing

August 2014

### Problem 1.(50pt)

Consider an X-ray imaging system shown in the figure below

Photons are emitted from an X-ray source and columnated by a pin hole in a lead shield. The collimated X-rays then pass in a straight line through an object of length $T$ with density $u(x)$ where $x$ is the depth into the object. The number of photons in the beam at depth $x$ is denoted by the random variable $Y_x$ with Poisson density given by

$P\left\{Y_x=k\right\} = \frac{e^{-\lambda_x}\lambda_x^k}{k!}$

where $x$ is measured in units of $cm$ and $\mu(x)$ is measured in units of $cm^{-1}$.

a) Calculate the $E[Y_x]$

b) Write a differential equation which describes the behavior of $\lambda_x$ as a function of $x$.

c) Calculate an expression for $\lambda_x$in terms of $u(x)$ and $\lambda_0$ by solving the differential equation.

d) Calculate an expression for the integral of the density, $\int_0^T u(x)dx$, in terms of the measured values of $Y_0$ and $Y_T$.

### Problem 2.(50pt)

Consider the 2D difference equation $y(m,n) = bx(m,n) + ay(m-1,n) + ay(m,n-1) - a^2y(m-1,n-1)$

where $b \in \mathbb{R}$ and $a \in (-1,1)$ are two constants, and $Y(z_1, z_2)$ and $X(z_1,z_2)$ are the 2D Z-transforms of $y(m,n)$ and $x(m,n)$ respectively.

a) Calculate $H(z_1,z_2) = \frac{Y(z_1,z_2)}{X(z_1,z_2)}$, the 2D transfer function of the casual system. Make sure to express your result in factored form.

b) Calculate, $h(m,n)$, the impulse response of the system with transfer function $H(z_1,z_2)$

c) In an application, $x(m,n)$ is an input image, and $y(m,n)$ is an output filtered image. Specify a relationship between $a$ and $b$ so that the average values of the input and output images remain the same.

d) For parts d) and e), assume the input, $x(m,n)$, are i.i.d. Gaussian random variables with mean zero and variance one. Calculate the auto covariance given by

$R_x(k,l) = E[x(m,n)x(m+k,n+l)]$

and its associated power spectral density $S_x(e^{j\mu}, e^{j\nu})$.

e) Calculate $S_y(e^{j\mu},e^{j\nu})$, the power spectral density of $y(m,n)$