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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 | ||
| − | <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>, |
| − | <math>F_2 = \int_{-\infty}^{\infty}f_2(\lambda)I(\lambda)d\lambda</math>, | + | <math>F_2 = \int_{-\infty}^{\infty}f_2(\lambda)I(\lambda)d\lambda</math>, |
| − | <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> |
| − | where < | + | 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, < | + | Furthermore, assume there exists a matrix, <span class="texhtml">''M''</span>, so that |
| + | <br> | ||
| + | a) Why is it necessary that <math>f_k(\lambda) \geq 0</math> for <span class="texhtml">''k'' = 0,1,2</span>?<span style="line-height: 1.5em;" /> | ||
| − | + | 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? | |
| − | + | c) Derive an formula for the tristimulus vector <math>[r, g, b]^t <math> in terms of the tristimulus vector <math> F=[F_1, F_2, F_3]^t <math>. | |
| + | d) Do functions <math> f_k(\lambda) <math> exist, which meet these requirements? If so, give a specific example of such functions. | ||
[[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 19:28, 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 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
a) Why is it necessary that $ f_k(\lambda) \geq 0 $ for k = 0,1,2?<span style="line-height: 1.5em;" />
b) Are the functions, $ 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? c) Derive an formula for the tristimulus vector <math>[r, g, b]^t <math> in terms of the tristimulus vector <math> F=[F_1, F_2, F_3]^t <math>. d) Do functions <math> f_k(\lambda) <math> exist, which meet these requirements? If so, give a specific example of such functions. [[Category:ECE]] [[Category:QE]] [[Category:CNSIP]] [[Category:Problem_solving]] [[Category:Image_processing]] $
