Question

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’s electron gun.

Short answer

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

Write the matrix using the MATLAB command

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

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

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

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