DigitalVideoAndHDTVAlgorithmsAndInterfaces.pdf

To understand the mathematical
details of color transforms,
described in this section, you should
be familiar with linear (matrix)
algebra. If you are unfamiliar with
linear algebra, see Strang, Gilbert,
Introduction to Linear Algebra,
Second Edition (Boston: Wellesley-Cambridge,
1998).
Table 22.6 Example primaries
are used to explain the necessity
of signal processing in
accurate color reproduction.
Eq 22.1 This matrix is based upon
R, G, and B components with
unusual spectral distributions. For
typical R, G, and B, see Eq 22.8.
Eq 22.2
because light power cannot go negative. So we cannot
build a real display that responds directly to XYZ. But as
you will see, the concept of negative SPDs – and
nonphysical SPDs or nonrealizable primaries – is very
useful in theory and in practice.
There are many ways to choose nonphysical primary
SPDs that correspond to the X(λ), Y(λ), and Z(λ) colormatching
functions. One way is to arbitrarily choose
three display primaries whose power is concentrated at
three discrete wavelengths. Consider three display
SPDs, each of which has some amount of power at
600 nm, 550 nm, and 470 nm. Sample the X(λ), Y(λ),
and Z(λ) functions of the matrix given earlier in Calculation
of tristimulus values by matrix multiplication, on
page 218, at those three wavelengths. This yields the
tristimulus values shown in Table 22.6:
Red, 600 nm Green, 550 nm Blue, 470 nm
X 1.0622 0.4334 0.1954
Y 0.6310 0.9950 0.0910
Z 0.0008 0.0087 1.2876
These coefficients can be expressed as a matrix, where
the column vectors give the XYZ tristimulus values
corresponding to pure red, green, and blue at the
display, that is, [1, 0, 0], [0, 1, 0], and [0, 0, 1]. It is
conventional to apply a scale factor in such a matrix to
cause the middle row to sum to unity, since we wish to
achieve only relative matches, not absolute:
⎡X
⎤ ⎡0.
618637 0. 252417 0. 113803⎤
⎡R
⎤
⎢ ⎥ ⎢
⎥ ⎢ 600nm⎥
⎢Y
⎥ = ⎢0.
367501 0. 579499 0. 052999⎥
• ⎢G550nm⎥
⎢
⎣
Z ⎥ ⎢
⎥ ⎢ ⎥
⎦ ⎣
0. 000466 0. 005067 0. 749913
⎦ B
⎣⎢
470nm⎦⎥
That matrix gives the transformation from RGB to XYZ.
We are interested in the inverse transform, from XYZ to
RGB, so invert the matrix:
⎡R
⎤
600nm ⎡ 2. 179151 −0. 946884 −0.
263777⎤
⎡X
⎤
⎢ ⎥ ⎢
⎥ ⎢ ⎥
⎢G550nm⎥
= ⎢−1.
382685 2. 327499 0. 045336⎥
• ⎢Y
⎥
⎢ ⎥
B ⎢
Z
⎣⎢
470nm⎦⎥
⎣
0. 007989 −0.
015138 1. 333346⎥
⎢ ⎥
⎦ ⎣ ⎦
CHAPTER 22 COLOR SCIENCE FOR VIDEO 241