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| − | August 2013(Published on May 2017) | + | August 2013 (Published on May 2017) |
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== Problem 1 == | == Problem 1 == | ||
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_{ | + | <math>P_{1}(e^{j\omega}) = \sum_{m=-\infty}^{\infty} p_{1}(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>. | ||
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d) Do the function <span class="texhtml">''p''<sub>0</sub>(''n'')</span> and <span class="texhtml">''p''<sub>1</sub>(''m'')</span> together contains sufficient information to reconstruction the function <span class="texhtml">''x''(''m'',''n'')</span>? If so, provide a reconstruction algorithm; if not, provide a counter example. | d) Do the function <span class="texhtml">''p''<sub>0</sub>(''n'')</span> and <span class="texhtml">''p''<sub>1</sub>(''m'')</span> together contains sufficient information to reconstruction the function <span class="texhtml">''x''(''m'',''n'')</span>? If so, provide a reconstruction algorithm; if not, provide a counter example. | ||
| − | Click [[ | + | Click [[QE637_sol2013_Q1 |here]] to view student [[QE637_sol2013_Q1 |answers and discussions]] <br> |
---- | ---- | ||
| − | <br> '''Problem 2. ''' | + | <br> '''Problem 2. ''' |
| − | Let <span class="texhtml">''r''<sub>0</sub>(λ)</span>, <span class="texhtml">''g''<sub>0</sub>(λ)</span>, and <span class="texhtml">''b''<sub>0</sub>(λ)</span> 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 <span class="texhtml">[''r'',''g'',''b'']</span> be the corresponding CIE tristimulus values. | + | Let <span class="texhtml">''r''<sub>0</sub>(λ)</span>, <span class="texhtml">''g''<sub>0</sub>(λ)</span>, and <span class="texhtml">''b''<sub>0</sub>(λ)</span> 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 <span class="texhtml">[''r'',''g'',''b'']</span> be the corresponding CIE tristimulus values. |
Furthermore, let <span class="texhtml">''f''<sub>1</sub>(λ)</span>, <span class="texhtml">''f''<sub>2</sub>(λ)</span>, and <span class="texhtml">''f''<sub>3</sub>(λ)</span> 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 <span class="texhtml">''F'' = [''F''<sub>1</sub>,''F''<sub>2</sub>,''F''<sub>3</sub>]<sup>''t''</sup></span>, where | Furthermore, let <span class="texhtml">''f''<sub>1</sub>(λ)</span>, <span class="texhtml">''f''<sub>2</sub>(λ)</span>, and <span class="texhtml">''f''<sub>3</sub>(λ)</span> 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 <span class="texhtml">''F'' = [''F''<sub>1</sub>,''F''<sub>2</sub>,''F''<sub>3</sub>]<sup>''t''</sup></span>, where | ||
| − | + | <center> | |
<math>F_1 = \int_{-\infty}^{\infty}f_1(\lambda)I(\lambda)d\lambda</math>, | <math>F_1 = \int_{-\infty}^{\infty}f_1(\lambda)I(\lambda)d\lambda</math>, | ||
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<math>F_3 = \int_{-\infty}^{\infty}f_3(\lambda)I(\lambda)d\lambda</math> | <math>F_3 = \int_{-\infty}^{\infty}f_3(\lambda)I(\lambda)d\lambda</math> | ||
| + | </center> | ||
| − | where <span class="texhtml">''I''(λ)</span> is the energy spectrum of the incoming light and <math>f_k(\lambda)\geq 0</math> for <span class="texhtml">''k'' = 0,1,2.</span> | + | where <span class="texhtml">''I''(λ)</span> is the energy spectrum of the incoming light and <math>f_k(\lambda)\geq 0</math> for <span class="texhtml">''k'' = 0,1,2.</span> |
Furthermore, assume there exists a matrix, <span class="texhtml">''M''</span>, so that | Furthermore, assume there exists a matrix, <span class="texhtml">''M''</span>, so that | ||
| − | + | <center> | |
<math> | <math> | ||
\left[ {\begin{array}{*{20}{c}} | \left[ {\begin{array}{*{20}{c}} | ||
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\end{array}} \right] | \end{array}} \right] | ||
</math> | </math> | ||
| + | </center> | ||
<br> a) Why is it necessary that <math>f_k(\lambda) \geq 0</math> for <span class="texhtml">''k'' = 0,1,2</span>? | <br> a) Why is it necessary that <math>f_k(\lambda) \geq 0</math> for <span class="texhtml">''k'' = 0,1,2</span>? | ||
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d) Do functions <span class="texhtml">''f''<sub>''k''</sub>(λ)</span> exist, which meet these requirements? If so, give a specific example of such functions. | d) Do functions <span class="texhtml">''f''<sub>''k''</sub>(λ)</span> exist, which meet these requirements? If so, give a specific example of such functions. | ||
| − | Click [[ | + | Click [[QE637_sol2013_Q2|here]] to view student [[QE637_sol2013_Q2|answers and discussions]] |
<br> | <br> | ||
[[Category:ECE]] [[Category:QE]] [[Category:CNSIP]] [[Category:Problem_solving]] [[Category:Image_processing]] | [[Category:ECE]] [[Category:QE]] [[Category:CNSIP]] [[Category:Problem_solving]] [[Category:Image_processing]] | ||
Latest revision as of 19:27, 2 May 2017
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.
Click here to view student answers and discussions
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.
Click here to view student answers and discussions
