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Chapter 2: Q2.7-21Q (page 93)

Exercises 21 and 22 concern the way in which color is specified for display in computer graphics. A color on a computer screen is encoded by three numbers (R, G, B) that list the amount of energy an electron gun must transmit to red, green, and blue phosphor dots on the computer screen. (A fourth number specifies the luminance or intensity of the color.)

21. The actual color a viewer sees on a screen is influenced by the specific type and amount of phosphors on the screen. So, each computer screen manufacturer must convert between the (R, G, B) data and an international CIE standard for color, which uses three primary colors, called X, Y, and Z. A typical conversion for short-persistence phosphors is

\(\left[ {\begin{array}{*{20}{c}}{.61}&{.29}&{.150}\\{.35}&{.59}&{.063}\\{.04}&{.12}&{.787}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}R\\G\\B\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}X\\Y\\Z\end{array}} \right]\)

A computer program will send a stream of color information to the screen, using standard CIE data (X, Y, Z). Find the equation that converts these data to the (R, G, B) data needed for the screen鈥檚 electron gun.

Short Answer

Expert verified

The equation is \(\left[ {\begin{array}{*{20}{c}}R\\G\\B\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{2.26}&{ - 1.04}&{ - 0.35}\\{ - 1.35}&{2.34}&{0.07}\\{0.09}&{ - 0.30}&{1.28}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}X\\Y\\Z\end{array}} \right]\).

Step by step solution

01

Write the matrix using the MATLAB command

Use the MATLAB command to write the matrix equation\(\left[ {\begin{array}{*{20}{c}}{.61}&{.29}&{.150}\\{.35}&{.59}&{.063}\\{.04}&{.12}&{.787}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}R\\G\\B\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}X\\Y\\Z\end{array}} \right]\).

\( > > {\rm{A}} = \left[ {{\rm{.61 }}{\rm{.29 }}{\rm{.150; }}{\rm{.35 }}{\rm{.59 }}{\rm{.063; }}{\rm{.04 }}{\rm{.12 }}{\rm{.787}}} \right]{\rm{;}}\)

\(A = \left[ {\begin{array}{*{20}{c}}{.61}&{.29}&{.150}\\{.35}&{.59}&{.063}\\{.04}&{.12}&{.787}\end{array}} \right]\)

02

Obtain the inverse of matrix A

Compute theinverse of matrix A by using the MATLAB command shown below:

\(\begin{array}{l} > > {\rm{A}} = \left[ {{\rm{.61 }}{\rm{.29 }}{\rm{.150; }}{\rm{.35 }}{\rm{.59 }}{\rm{.063; }}{\rm{.04 }}{\rm{.12 }}{\rm{.787}}} \right]{\rm{;}}\\ > > B = {\rm{A}}\^ - 1\end{array}\)

\(B = {A^{ - 1}} = \left[ {\begin{array}{*{20}{c}}{2.26}&{ - 1.04}&{ - 0.35}\\{ - 1.35}&{2.34}&{0.07}\\{0.09}&{ - 0.30}&{1.28}\end{array}} \right]\)

Thus, the required equation is \(\left[ {\begin{array}{*{20}{c}}R\\G\\B\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{2.26}&{ - 1.04}&{ - 0.35}\\{ - 1.35}&{2.34}&{0.07}\\{0.09}&{ - 0.30}&{1.28}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}X\\Y\\Z\end{array}} \right]\).

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Most popular questions from this chapter

Suppose A, B, and Care \(n \times n\) matrices with A, X, and \(A - AX\) invertible, and suppose

\({\left( {A - AX} \right)^{ - 1}} = {X^{ - 1}}B\) 鈥�(3)

  1. Explain why B is invertible.
  2. Solve (3) for X. If you need to invert a matrix, explain why that matrix is invertible.

In exercise 11 and 12, mark each statement True or False.Justify each answer.

a. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{\bf{1}}}}&{{A_{\bf{2}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_{\bf{1}}}}&{{B_{\bf{2}}}}\end{array}} \right]\), with \({A_{\bf{1}}}\) and \({A_{\bf{2}}}\) the same sizes as \({B_{\bf{1}}}\) and \({B_{\bf{2}}}\), respectively then \(A + B = \left[ {\begin{array}{*{20}{c}}{{A_1} + {B_1}}&{{A_{\bf{2}}} + {B_{\bf{2}}}}\end{array}} \right]\).

b. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{{\bf{11}}}}}&{{A_{{\bf{12}}}}}\\{{A_{{\bf{21}}}}}&{{A_{{\bf{22}}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_1}}\\{{B_{\bf{2}}}}\end{array}} \right]\), then the partitions of \(A\) and \(B\) are comfortable for block multiplication.

Let Abe an invertible \(n \times n\) matrix, and let B be an \(n \times p\) matrix. Show that the equation \(AX = B\) has a unique solution \({A^{ - 1}}B\).

Let \(X\) be \(m \times n\) data matrix such that \({X^T}X\) is invertible., and let \(M = {I_m} - X{\left( {{X^T}X} \right)^{ - {\bf{1}}}}{X^T}\). Add a column \({x_{\bf{0}}}\) to the data and form

\(W = \left[ {\begin{array}{*{20}{c}}X&{{x_{\bf{0}}}}\end{array}} \right]\)

Compute \({W^T}W\). The \(\left( {{\bf{1}},{\bf{1}}} \right)\) entry is \({X^T}X\). Show that the Schur complement (Exercise 15) of \({X^T}X\) can be written in the form \({\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}\). It can be shown that the quantity \({\left( {{\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}} \right)^{ - {\bf{1}}}}\) is the \(\left( {{\bf{2}},{\bf{2}}} \right)\)-entry in \({\left( {{W^T}W} \right)^{ - {\bf{1}}}}\). This entry has a useful statistical interpretation, under appropriate hypotheses.

In the study of engineering control of physical systems, a standard set of differential equations is transformed by Laplace transforms into the following system of linear equations:

\(\left[ {\begin{array}{*{20}{c}}{A - s{I_n}}&B\\C&{{I_m}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\bf{x}}\\{\bf{u}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{y}}\end{array}} \right]\)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(C\) is \(m \times n\), and \(s\) is a variable. The vector \({\bf{u}}\) in \({\mathbb{R}^m}\) is the 鈥渋nput鈥� to the system, \({\bf{y}}\) in \({\mathbb{R}^m}\) is the 鈥渙utput鈥� and \({\bf{x}}\) in \({\mathbb{R}^n}\) is the 鈥渟tate鈥� vector. (Actually, the vectors \({\bf{x}}\), \({\bf{u}}\) and \({\bf{v}}\) are functions of \(s\), but we suppress this fact because it does not affect the algebraic calculations in Exercises 19 and 20.)

Explain why the columns of an \(n \times n\) matrix Aspan \({\mathbb{R}^{\bf{n}}}\) when

Ais invertible. (Hint:Review Theorem 4 in Section 1.4.)
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