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ECE Ph.D. Qualifying Exam in Communication Networks Signal and Image processing (CS)

Question 5, August 2013, Problem 2

Problem 1 ,Problem 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.

Solution 2:

a) f₁(λ), f₂(λ) and f₃(λ) are the spectral functions for the three color outputs of color camera. It must be positive because we cannot produce negative spectrum. 

b) No. r_o(λ),g_o(λ)a'n'd'b_o(λ) are CIE color matching. It takes negative value in order to substract some color to be saturated. 

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_2(\lambda)\\ f_3(\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] $

d)

Yes. They exist. If there is a matrix M that exist to satisfy this equation  $ \left[ {\begin{array}{*{20}{c}} f_1(\lambda)\\ f_2(\lambda)\\ f_3(\lambda) \end{array}} \right] = M \left[ {\begin{array}{*{20}{c}} r_0(\lambda)\\ g_0(\lambda)\\ b_0(\lambda) \end{array}} \right] $. 

Related Problem

1. 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₍λ), g₍λ), and b₍λ) 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(λ) be the light reflected from a surface.

a) Calculate (r_e,g_e,b_e) the tristimulus values for the spectral distribution I(λ) 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(λ) using primaries R,G,B and white point (r_w,g_w,b_w).

(Refer to ECE637 2004 Final Problem 4.)

2. Consider the two channel sensors with response function Q_S(λ) and Q_L(λ). Suppose that we have two primaries P_L(λ) = σ(λ − 0.6) and P_S(λ) = σ(λ − 0.5).

Find the color matching function $ \bar{l}(\lambda) $ and $ \bar{s}(\lambda) $ for these two primaries.

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