Difference between revisions of "ECE-QE CS5-2013" - Rhea

Latest revision as of 17:06, 12 November 2014

Question

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Problem 1. (50 pts)

Consider the 2D discrete space signal x(m,n) with the DSFT of X(e^jμ,e^jν) 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(e^jω) in terms of X(e^jμ,w^jν).

b) Derive an expression P₀(e^jω) in terms of X(e^jμ,e^jν).

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

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

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