ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 5: Image Processing

August 2013 (Published on May 2017)

Problem 1

Consider the 2D discrete space signal x(m,n) with the DSFT of X(ejμ,ejν) given by 

$ X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} x(m,n)e^{-j(m\mu+n\nu)} $

Then define

$ p_{0}(n) = \sum_{m=-\infty}^{\infty}x(m,n) $

$ p_{1}(m) = \sum_{n=-\infty}^{\infty}x(m,n) $

with corresponding DTFT given by 

$ P_{0}(e^{j\omega}) = \sum_{n=-\infty}^{\infty} p_{0}(n)e^{-jn\omega} $

$ P_{1}(e^{j\omega}) = \sum_{m=-\infty}^{\infty} p_{1}(m)e^{-jm\omega} $

a) Derive an expression for P0(ejω) in terms of X(ejμ,wjν). 

b) Derive an expression P0(ejω) in terms of X(ejμ,ejν).

c) Derive an expression  for $ \sum_{n = -\infty}^{\infty}p_0(n) $ interms of X(ejμ,ejν).

d) Do the function p0(n) and p1(m) together contains sufficient information to reconstruction the function x(m,n)? If so, provide a reconstruction algorithm; if not, provide a counter example.

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Problem 2.

Let r0(λ), g0(λ), and b0(λ) be the CIE color matching functions for red, green, and blue primaries at 700 nm, 546.1 nm, and 435.8 nm, respectively, and let [r,g,b] be the corresponding CIE tristimulus values.

Furthermore, let f1(λ)f2(λ), and f3(λ) be the spectral response functions for the three color outputs of a color camera. So for each pixel of the camera sensor, there is a 3-dimensional output vector given by F = [F1,F2,F3]t, where

$ F_1 = \int_{-\infty}^{\infty}f_1(\lambda)I(\lambda)d\lambda $,

$ F_2 = \int_{-\infty}^{\infty}f_2(\lambda)I(\lambda)d\lambda $,

$ F_3 = \int_{-\infty}^{\infty}f_3(\lambda)I(\lambda)d\lambda $

where I(λ) is the energy spectrum of the incoming light and $ f_k(\lambda)\geq 0 $ for k = 0,1,2.

Furthermore, assume there exists a matrix, M, so that

$ \left[ {\begin{array}{*{20}{c}} f_1(\lambda)\\ f_1(\lambda)\\ f_1(\lambda) \end{array}} \right] = {\begin{array}{*{20}{c}} M \end{array}} \left[ {\begin{array}{*{20}{c}} r_0(\lambda)\\ g_0(\lambda)\\ b_0(\lambda) \end{array}} \right] $

a) Why is it necessary that $ f_k(\lambda) \geq 0 $ for k = 0,1,2?

b) Are the functions, $ r_0(\lambda) \geq 0 $, $ g_0(\lambda) \geq 0 $, and $ b_0(\lambda) \geq 0 $? If so, why? If not, why not?

c) Derive an formula for the tristimulus vector [r,g,b]t in terms of the tristimulus vector F = [F1,F2,F3]t.

d) Do functions fk(λ) exist, which meet these requirements? If so, give a specific example of such functions.

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Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009