ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 5: Image Processing

August 2013


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Problem 1. (50 pts)

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_{0}(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. (50 pts)

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. </span>

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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