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 $.

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### 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) $

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