ECE Ph.D. Qualifying Exam in Communication Networks Signal and Image processing (CS)
Question 5, August 2013, Part 2
part1, part 2
Solution 1:
a) If the color matching functions fk(λ) has negative values, it will result in negative values in Fk. In this case, the color can not be reproduced by this device.
b) The CIE color matching functions are not always positive. r0(λ) takes negative values. This is the case because, to match some reference color that is too saturated, colors have to be subtracted from the R,G, and B primaries. This results in negative values in tristimulus values r, g, and b. So the color matching functions at the corresponding wavelength have negative values.
c)
$ \left[ {\begin{array}{*{20}{c}} F_1\\ F_2\\ F_3 \end{array}} \right] = {\begin{array}{*{20}{c}} \int_{-\infty}^{\infty} \end{array}} \left[ {\begin{array}{*{20}{c}} f_1(\lambda)\\ f_1(\lambda)\\ f_1(\lambda) \end{array}} \right] I(\lambda)d\lambda = {\begin{array}{*{20}{c}} \int_{-\infty}^{\infty} \end{array}} M \left[ {\begin{array}{*{20}{c}} r_0(\lambda)\\ g_0(\lambda)\\ b_0(\lambda) \end{array}} \right] I(\lambda)d\lambda = M {\begin{array}{*{20}{c}} \int_{-\infty}^{\infty} \end{array}} \left[ {\begin{array}{*{20}{c}} r_0(\lambda)\\ g_0(\lambda)\\ b_0(\lambda) \end{array}} \right] I(\lambda)d\lambda = M \left[ {\begin{array}{*{20}{c}} r\\ g\\ b \end{array}} \right] $
So that, [r,g,b]t = M − 1[F1,F2,F3].
d) It exists. CIE XYZ is one example. However, XYZ has problems with its primaries, since, the primary colors are imaginary.
Related problem
In a color matching experiment, the three primaries R, G, B are used to match the color of a pure spectral component at wavelength λ. Here the color matching allows for color to be subtracted from the reference color. At each wavelength λ, the matching color is given by
$ \left[ {\begin{array}{*{20}{c}} R, G, B \end{array}} \right] \left[ {\begin{array}{*{20}{c}} r(\lambda)\\ g(\lambda)\\ b(\lambda) \end{array}} \right] $
where $ r_{\lambda} $, $ g_{\lambda} $, and $ b_{\lambda} $ are normalized to 1.
Further define the white point
$ W = \left[ {\begin{array}{*{20}{c}} R, G, B \end{array}} \right] \left[ {\begin{array}{*{20}{c}} r_w\\ g_w\\ b_w \end{array}} \right] $
