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| − | August 2013(Published on May 2017) | + | August 2013 (Published on May 2017) |
</center> | </center> | ||
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Consider the 2D discrete space signal <span class="texhtml">''x''(''m'',''n'') with the DSFT of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''e''<sup>''j''ν</sup>) given by </span></span> | Consider the 2D discrete space signal <span class="texhtml">''x''(''m'',''n'') with the DSFT of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''e''<sup>''j''ν</sup>) given by </span></span> | ||
| + | <center> | ||
<math>X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} | <math>X(e^{j\mu},e^{j\nu}) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} | ||
x(m,n)e^{-j(m\mu+n\nu)}</math> | x(m,n)e^{-j(m\mu+n\nu)}</math> | ||
| + | </center> | ||
Then define | Then define | ||
| − | + | <center> | |
<math>p_{0}(n) = \sum_{m=-\infty}^{\infty}x(m,n)</math><br> | <math>p_{0}(n) = \sum_{m=-\infty}^{\infty}x(m,n)</math><br> | ||
<math>p_{1}(m) = \sum_{n=-\infty}^{\infty}x(m,n)</math> | <math>p_{1}(m) = \sum_{n=-\infty}^{\infty}x(m,n)</math> | ||
| + | </center> | ||
with corresponding DTFT given by | with corresponding DTFT given by | ||
| − | + | <center> | |
<math>P_{0}(e^{j\omega}) = \sum_{n=-\infty}^{\infty} p_{0}(n)e^{-jn\omega}</math><br> | <math>P_{0}(e^{j\omega}) = \sum_{n=-\infty}^{\infty} p_{0}(n)e^{-jn\omega}</math><br> | ||
| − | <math>P_{1}(e^{j\omega}) = \sum_{m=-\infty}^{\infty} p_{0}(m)e^{-jm\omega}</math><br> a) Derive an expression for <span class="texhtml">''P''<sub>0</sub>(''e''<sup>''j''ω</sup>)</span> in terms of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''w''<sup>''j''ν</sup>)</span>. | + | <math>P_{1}(e^{j\omega}) = \sum_{m=-\infty}^{\infty} p_{0}(m)e^{-jm\omega}</math><br>\\ |
| + | </center> | ||
| + | |||
| + | a) Derive an expression for <span class="texhtml">''P''<sub>0</sub>(''e''<sup>''j''ω</sup>)</span> in terms of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''w''<sup>''j''ν</sup>)</span>. | ||
b) Derive an expression <span class="texhtml">''P''<sub>0</sub>(''e''<sup>''j''ω</sup>)</span> in terms of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''e''<sup>''j''ν</sup>)</span>. | b) Derive an expression <span class="texhtml">''P''<sub>0</sub>(''e''<sup>''j''ω</sup>)</span> in terms of <span class="texhtml">''X''(''e''<sup>''j''μ</sup>,''e''<sup>''j''ν</sup>)</span>. | ||
Revision as of 19:23, 1 May 2017
Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2013 (Published on May 2017)
Question is posted from this link.
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_{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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