Difference between revisions of "ECE-QE CS5-2013" - Rhea

Revision as of 18:31, 10 November 2014

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] = \left[ {\begin{array}{*{20}{c}} M \end{array}} \right]\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.