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== Question == | == Question == | ||
| − | '''Problem 1. ''' (50 pts) | + | '''Problem 1. ''' (50 pts) <br> |
| + | |||
| + | 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> | ||
| + | |||
| + | <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> | ||
| + | |||
| + | Then define | ||
| + | |||
| + | <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> | ||
| + | |||
| + | with corresponding DTFT given by | ||
| + | |||
| + | <span class="texhtml">''P''<sub>0</sub>(''e''<sup>''j''ω</sup>)</span> | ||
<br> | <br> | ||
| + | |||
| + | <br> <br> | ||
'''Problem 2. ''' (50 pts) | '''Problem 2. ''' (50 pts) | ||
| − | Let < | + | 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. </span> |
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 | ||
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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 [[QE637 2014 Pro2|here]] to view student [[QE637 2014 Pro2|answers and discussions]] | + | Click [[QE637 2014 Pro2|here]] to view student [[QE637 2014 Pro2|answers and discussions]] |
[[Category:ECE]] [[Category:QE]] [[Category:CNSIP]] [[Category:Problem_solving]] [[Category:Image_processing]] | [[Category:ECE]] [[Category:QE]] [[Category:CNSIP]] [[Category:Problem_solving]] [[Category:Image_processing]] | ||
Revision as of 19:44, 10 November 2014
Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2013
Question
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
P0(ejω)
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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