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'''Problem 2. ''' (50 pts) | '''Problem 2. ''' (50 pts) | ||
| − | Let | + | Let <math> r_0(\lambda)</math>, <math>g_0(\lambda)</math>, and <math>b_0(\lambda)</math> 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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</math> | </math> | ||
| − | <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>? |
b) Are the functions, <math> r_0(\lambda) \geq 0</math>, <math>g_0(\lambda) \geq 0</math>, and <math>b_0(\lambda) \geq 0</math>? If so, why? If not, why not? | b) Are the functions, <math> r_0(\lambda) \geq 0</math>, <math>g_0(\lambda) \geq 0</math>, and <math>b_0(\lambda) \geq 0</math>? If so, why? If not, why not? | ||
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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. | ||
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| + | Click here to view student 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 19:34, 10 November 2014
Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2013
Question
Problem 1. (50 pts)
Problem 2. (50 pts)
Let $ r_0(\lambda) $, $ g_0(\lambda) $, and $ b_0(\lambda) $ 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.
Click here to view student answers and discussions
