| Line 40: | Line 40: | ||
f_1(\lambda)\\ | f_1(\lambda)\\ | ||
f_1(\lambda) | f_1(\lambda) | ||
| − | \end{array}} \right] = | + | \end{array}} \right] = [ {\begin{array}{*{20}{c}} |
M | M | ||
| − | \end{array}} | + | \end{array}} ]\left[ {\begin{array}{*{20}{c}} |
r_0(\lambda)\\ | r_0(\lambda)\\ | ||
g_0(\lambda)\\ | g_0(\lambda)\\ | ||
| Line 48: | Line 48: | ||
\end{array}} \right] | \end{array}} \right] | ||
</math> | </math> | ||
| + | |||
| + | |||
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;" /> | 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;" /> | ||
Revision as of 18:31, 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
$ \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?<span style="line-height: 1.5em;" />
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=[F_1, F_2, F_3]^t $.
d) Do functions $ f_k(\lambda) $ exist, which meet these requirements? If so, give a specific example of such functions.
