In the previous topic we talked about
describing a light beam consisting of several (possibly many) wavelengths in
terms of only three parameters: its hue, saturation and intensity. It is also
possible to describe the color of the beam by specifying the relative
intensities of three colors which can be combined to produce a color that looks
the same (that is, has the same hue, saturation and intensity) as the color of
the original beam. This business can be complicated in practice, since the
perceived color of a light beam depends to some extent both on its intensity
and on the color of the surrounding objects. The fact that only three colors
can be used to match the appearance of almost any other color is not an
accident – it is related to the way the eye perceives color, as we will see
later.
There are a number of different
choices for the colors that can be used to match any other color. It turns out
that no triplet of colors can be used to match every possible color. A good
choice, which can match most colors, is the triplet 650 nm red, 530 nm green
and 460 nm blue. (A somewhat “bluer” blue at 425 nm is sometimes used as an
alternative to the 460 nm blue.) Using these three colors, it is possible to match
many (but not all) other colored beams.
If we combine these three primary
colors in different combinations, we find that:
Red + Green = Yellow
Red + Blue = Purple (called magenta)
Green + Blue= Cyan (a dark blue)
Red + Green + Blue= White
Substituting
each of the first three relationships into the last one, it is also true that
Yellow + Blue= White
Red + Cyan= White
Green + Purple= White
so
that that these pairs of colors can be thought of as “complementary” – each one
of the colors on the left side of these relationships can also be gotten by removing
the other color from a white beam.
Since any light beam of any color can
be described by giving the intensity of each of its constituent monochromatic
frequencies, in principle we could reconstruct this arbitrary light beam by
generating each of these frequencies with the proper intensity and combining
them. Therefore, the process of reconstructing any arbitrary light beam can be
completed successfully if we can use our three primary colors to match any
monochromatic frequency. The job of matching the given light beam would then
consist of adding the appropriate amount of each primary color to match each of
its constituent wavelengths taken one at a time. (Note that while this
complicated process is sufficient, it may not be necessary – there are simpler
ways of matching many colors as we will see below.)
The following figure (taken from the
textbook on page 244) shows the relative percentages of the three primary
colors needed to match any monochromatic color in the visible spectrum.
Note
that there are some portions of the spectrum where a negative percentage of one
primary is required, with another primary requiring a percentage of more than
100% -- obviously both of these are impossible. For example, there is no way of
matching a monochromatic blue beam that is bluer than the color of our blue
primary, which is 460 nm. We just don’t have enough “blue-ness” to do the job,
and the percentages are strange in that region as a result. The best that we can do is to add some green
to the blue so as to decrease its saturation and then to match the combination.
That is the significance of the fact that the percentage of green is shown as
negative for these colors – that means that the green must be added to the
color we are trying to match.
The
same sort of problem happens at about 500 nm, which we would call a
yellow-green. We can’t match a monochromatic color in this region without
adding some red to it. This red reduces the saturation of the initial color,
and we can then match it using a combination of blue and green as shown.
The
following figure (also taken from the textbook on page 245) shows the same
information in a different format.
The
x and y axes show the amount of 650 nm read and 530 nm green that must be added
to match any color. Since the percentages must always add up to 100%, any
remainder between the sum of these two percentages and 100% is made up by our
blue primary at 460 nm.
The
horse-shoe shaped curve shows the visible wavelengths we are trying to match.
The primaries that we have chosen are at the corners of the horse-shoe, which
explains why they were chosen.
The
triangle formed by the lines joining the primary colors represent the boundary
range of colors that can be matched using these 3 primaries: colors inside of
the triangular area can be matched exactly, while those outside of the triangle
can be matched only by adding a “negative” contribution from one of the
primaries – that is, by adding one of the primaries to the color we are trying
to match and then matching this combination using the other two primaries. As
you can see from the figure, the largest region of difficulty comes in the
blue-green portion of the spectrum, where the horseshoe curve is significantly
to the left of the y axis, signifying that significant amounts of red must be
added to any color in this region of the spectrum in order for us to match it
using a combination of green and blue.
In
addition to specifying the combinations of the 3 primaries required to match
any color, this chart also shows what will happen when we combine any
two other colors, even if they are not primaries. Thus the line joining the red
and green primaries gives the colors that will result from mixing these two
colors in various ratios. Likewise, the line joining 570 nm yellow (which is
not one of our primaries) and 460 nm blue shows the various colors that will
result from combining these two wavelengths in any combination. Since Yellow
and Blue are complementary colors (that is, Yellow + Blue = White), the white
point must lie somewhere on this line. The exact position of this line will
depend on what we think of as our standard for yellow.
Using
the chart above, we can see that combinations of red and blue would lie along
the x-axis of the plot above. These combinations make various shades of what we
call the color “purple,” but there is no wavelengths that correspond to these
colors. Thus the various shades of purple cannot be produced using any single
monochromatic wavelength. The color purple therefore has a special, unique
place in the spectrum – it can only be produced by adding red and blue in some proportion and cannot be made
as a purely saturated color. At least in principle, any other color around the
horseshoe could be produced in a monochromatic, 100% saturated form.
Since
there is no unique wavelength associated with any purple color, the best that
we can do is to specify its wavelength in terms of its complement, which is
some shade of green, which we can get by drawing a line from the purple through
the white point to the other side of the horseshoe).
The
“white point” – the point on the diagram corresponding to a completely
unsaturated white is near (but not exactly at) the point at which the fractions
of all of the contributing primaries would be the same (that is, 1/3, 1/3,
1/3). Part of this difference comes from the definitions of optical intensity,
which include the variation of the response of the eye to monochromatic beams
with the same physical intensity and part comes from exactly what we mean by a
completely “white” color.
From
the definition above for the complement of a color (that is, the color that
combines with this color to form white), the complement of any color is found
from this diagram by drawing a line from the color through the white point to
the opposite side of the horseshoe.
Since
the sides of the horseshoe above are curved, some colors cannot be matched by
any three primaries, since a triangle joining any three primaries must lie
inside of some part of the horseshoe. A triangle that was completely outside of
the horseshoe could match any color, but such a triangle would require using primaries
that do not actually exist as real colors. There is a standard set of such
imaginary primary colors which lie outside of the entire horseshoe curve and
can therefore match any color. These are called the tristimulus colors,
and any color can be matched using only positive contributions from these three
imaginary primaries. The resulting fractions are called the chromaticity of a
color.
The principle of forming colors by
adding these 3 primaries is used in color television, in computer monitors and
in similar applications. All of these systems share the principle deficiency of
this method, namely that it is impossible to match all colors, especially those
that are almost 100% saturated and therefore lie outside of the triangular
lines joining the three primaries in the figure above. In addition to this
fundamental limitation, real color displays are limited by the fact that the
materials that are used to display the various colors do not produce exactly
the wavelengths specified above for the primary colors; the colors are also not
completely saturated. These limitations are not noticeable for standard
television displays, because the colors that are transmitted are not saturated
anyway. Some of these limitations may be addressed in the newer transmission
formats for “high-definition” television (HDTV), which will be available in a
few years.
When color television was first
introduced, the format used for transmitting the color information was chosen
so that the signals could also be received and processed by black and white
receivers. The method that was chosen to provide this compatibility was a
variation of the hue, saturation, intensity format described above. In TV
parlance, the black-and-white portion of the signal is called luminance
and the color portion is called chrominance. The luminance is roughly
the same parameter as what we have called intensity, and the chrominance is a
clever encoding of the relative amplitudes of the three primary colors that
describe the color at each point on the screen. The luminance signal is
identical to the older black and white transmission format, so that a black and
white receiver can process the luminance signal and recover a complete black
and white version of the picture. The chrominance signal uses a transmission
method that is not processed by the black and white system, and black and white
televisions simply ignore this portion of the transmission.
Computer monitors generally employ
much simpler encoding systems, since there is no need to transmit the information
over a wireless channel or to remain compatible with previous standards. Many
systems simply transmit the amplitudes of the 3 primary colors for each dot on
the screen – in the older VGA standards each dot could have only 3 intensities:
off, on/low, and on/high. This format gives 16 distinct colors for each dot (8
on/off combinations of the 3 colors and 2 overall intensities).
Finally, note that this description of
color does not tell the whole story, because the curves above specify the
fraction of each of the 3 primary colors that matches some given color.
However, the color that we perceive from a mixture of the primaries depends on
the overall intensity as well as on the percentages of each component. For
example, as we decrease the intensity of a light beam (keeping the fractional
contributions the same), the color “white” darkens and becomes various shades
of gray. Likewise, orange darkens and becomes brown. This change in perceived
hue does not affect the monochromatic colors – they look the same whether the
beam is bright or dim. However, almost all of the unsaturated colors change hue
as the intensity changes. To provide a complete description, the curves shown
above would have to be repeated for every possible intensity.