Difference between revisions of "QEsLittleHelper" - Rhea

@@ Line 1: / Line 1: @@
-[[Category:ECE]]
-[[Category:QE]]
-[[Category:CNSIP]]
-[[Category:problem solving]]
-[[Category:image processing]]
-<center>
-<font size= 4>
-[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
-</font size>
-<font size= 4>
-Communication, Networking, Signal and Image Processing (CS)
-Question 5: Image Processing
-</font size>
-August 2014
-</center>
-----
-----
-=== Problem 1.(50pt) ===
-Consider an X-ray imaging system shown in the figure below
-[[File:CS-5.jpg|thumbnail]]
-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 <math>T</math> with density <math>u(x)</math> where <math> x </math> is the depth into the object. The number of photons in the beam at depth <math>x</math> is denoted by the random variable <math>Y_x</math> with Poisson density given by
-<math>
-P\left\{Y_x=k\right\} = \frac{e^{-\lambda_x}\lambda_x^k}{k!}</math>
-where <math>x</math> is measured in units of <math>cm</math> and <math> \mu(x)</math> is measured in units of <math> cm^{-1}</math>.
-a) Calculate the <math> E[Y_x]</math>
-b) Write a differential equation which describes the behavior of <math> \lambda_x</math> as a function of <math> x </math>.
-c) Calculate an expression for <math> /lambda_x in terms of u(x) and \lambda_0 by solving the differential equation</math>.
-d) Calculate an expression for the integral of the density, <math> \int_0^T u(x)dx</math>, in terms of the measured values of <math> Y_0</math> and <math> Y_T</math>.
-:'''Click [[CS5_2015_Aug_prob1|here]] to view student [[CS5_2015_Aug_prob1|answers and discussions]]'''
-----
-===Problem 2.(50pt)===
-Consider the 2D difference equation
-<math>
-y(m,n) = bx(m,n) + ay(m-1,n) + ay(m,n-1) - a^2y(m-1,n-1)
-</math>
-where <math> b \in \mathbb{R} </math> and <math> a \in (-1,1)</math> are two constants, and <math> Y(z_1, z_2)</math> and <math> X(z_1,z_2)</math> are the 2D Z-transforms of <math> y(m,n)</math> and <math> x(m,n)</math> respectively.
-a) Calculate <math> H(z_1,z_2) = \frac{Y(z_1,z_2)}{X(z_1,z_2)}</math>, the 2D transfer function of the casual system. Make sure to express your result in factored form.
-b) Calculate, <math>h(m,n)</math>, the impulse response of the system with transfer function <math>H(z_1,z_2)</math>
-c) In an application, <math> x(m,n) </math> is an input image, and <math> y(m,n) </math> is an output filtered image. Specify a relationship between <math> a</math> and <math> b</math> so that the average values of the input and output images remain the same.
-d) For parts d) and e), assume the input, <math> x(m,n)</math>, are i.i.d. Gaussian random variables with mean zero and variance one. Calculate the auto covariance given by
-<math>
-R_x(k,l) = E[x(m,n)x(m+k,n+l)]
-</math>
-and its associated power spectral density <math>S_x(e^{j\mu}, e^{j\nu})</math>.
-e) Calculate <math>S_y(e^{j\mu},e^{j\nu})</math>, the power spectral density of <math>y(m,n)</math>
-:'''Click [[CS5_2015_Aug_prob2|here]] to view student [[CS5_2015_Aug_prob2|answers and discussions]]'''
-----
-[[ECE_PhD_Qualifying_Exams|Back to ECE QE page]]:

Difference between revisions of "QEsLittleHelper" - Rhea

Latest revision as of 14:03, 18 May 2017

Alumni Liaison