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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 | ||
| − | + | <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>? | ||
Revision as of 19:33, 1 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_{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.
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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