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 f_k(λ) has negative values, it will result in negative values in F_k. 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] $

Let $ I(\lambda) $ be the light reflected from a surface.

a) Calculate $ (r_e, g_e, b_e) $ the tristimulus values for the spectral distribution $ I(\lambda) $ using primaries $ R, G, B $ and an equal energy white point.

b) Calculate $ (r_c, g_c, b_c) $ the tristimulus values for the spectral distribution $ I(\lambda) $ using primaries $ R, G, B $ and white point $ (r_w, g_w, b_w) $.