Communication, Networking, Signal and Image Processing (CS)
Question 5: Image Processing
August 2015
Question
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Problem 1. (50 pts)
Consider the emissive display device which is accurately modeled by the equation
$ \left[ \begin{array}{c} X \\ Y\\Z \end{array} \right] = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \times \left[ \begin{array}{c} R^{\alpha} \\ G^{\alpha} \\ B^{\alpha} \end{array} \right] $
Where R, G, and B are red, green, and blue inputs in the range 0 to 255 that are used to modulate physically realizable color primaries.
a) What is the gamma of the device?
b) What are the chromaticity components (x"r", y"r").
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. (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. </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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