Three terms have often been used to describe colour.
The experiment consisted of a two display booth. In the one booth a
monochromatic colour was projected onto the screen. In the other there
was a tri-colour projector that the subject could control. The intensity
of each of the three colours was controled individually. The subject then
had to match the colour that was presented on the monochromatic screen
by adding different amounts of red, green, and blue colours. The initial
results showed that for the most part this could be done. In the following
figure we see the amounts of red, green, and blue colours that are necessary
to match different monochromatic colours.
This diagram is used in various ways. Given two colours A, and
B their CIE chromaticity coefficients can be found and the positions of
the colours plotted on the chromaticity diagram. Once this is done we can
deduce what colours can be achieved by combining the two colours. The colours
that can be achieved by combinations of A and B are precisely the colours
that lie on the line that lies between A and B on the chromaticity diagram.
If we are given three different colours then we can determine
the colours that can be generated with these primaries by plotting their
positions on the chromaticity diagram. The colours that can be generated
by combinations of these three colours lie within the triangle that is
generated by the positions of the three colours.
Each device has its own gamut. The differences between these gamuts
are important if accurate colour reproduction is desired. There may well
be colours that are achievable on one device but not on the other. In such
situations the designer or programmer must ensure that colours are chosen
that can be reproduced on all the relevant devices. There is a good example
of this problem presented in plate II.2 in the book.
Notice that there is an anomality in the diagram. I have highlighted
a region, this indicates a set of monochromatic colours that could not
be matched by simply adding different amounts of red, green, and blue.
The researchers found, however, that the colour could be matched if some
red light was added to the sample. This is represented in the figure by
a negative lobe on the red curve.
Note that these curves are not the spectral distribution of the
X,Y,and
Z colours, rather they are the amounts of X,Y, and Z
colours that are needed to match a particular monochromatic colour.
Full spectrum Light
In general these experiments with monochromatic light are useful for understanding
how our eyes work. However, in the natural world the light that reaches
our eyes is full spectrum light. That is, a colour is made up of multiple
wave lengths. For example a red apple might have the following colour distribution.
Our task now is to determine how these colours can be represented using
the tristimulus theory. Well, it turns out that given a particular colour
spectra we can approximate the perceived colour using the following scheme.
Given a spectral distribution P() we determine three
coefficients X,Y,Z as follows
Where x() x(
) d
) y(
) d
) z(
) d
), y(
), z(
)
are the matching curves presented above. And k is a constant that depends
on the display device. For monitors this is 680 lumens/watt, for reflecting
objects k is chosen so that a bright white has a Y value of 100.
CIE Cromaticity diagram.
Consider a colour C represented in the following manner.
C=XX + YY +ZZ
We can define new terms
Notice that x+y+z = 1. Because of this we can represent the colours
using two of the three coefficients. If x and y are given then we can always
recover z = 1-x-y. The visible colours can then be plotted on the xy plane.
The outline of this plot is presented here. This plot is known as the CI
chromaticity diagram. The wavelengths and positions of the main colours
are outlined.
A colour version of this can be seen in the course textbook on plate
II.1 The monochromatic lights appear on the curved boundary of the diagram.
Gamut
Given a display such as a cathode ray tube we can plot the chromaticity
values of the three phosphors used and thus identify the achievable colours
of the monitor. This set of colours is called the device gamut.
Just noticeable differences
If we were to choose two colours in the CIE chromaticity diagram and measure
their difference numericaly we might get some value deltaV. It
is possible to find other colours that are a similar distance apart but
perceptualy are not the same distance from each other. This problem is
caused by our non-linear perception of colour. In certain situations the
accurate reproduction of colours and of the distance between colours is
important. In these situations CIE LUV space is useful. A non linear device
dependent transformation is used to produce a colour space in which the
distances between colours roughly correspond to the perceptual distances
experienced by a wide set of observers. For more details on this space
and the transformation from the CIE space to the CIE LUV space please see
section 13.2 in the book.
Colour spaces for Computer Graphics
The theory that we have discussed thus far has concentrated on the human
visual system. We now turn our attention to the issue of representing and
manipulating colour on computer graphics systems. There are a wide variety
of colour spaces that have been developed for computer graphics. In these
notes I overview three of these RGB, CMY, and HSV.
RGB
Colour Cathode Ray tubes generate colours by stimulating three different
phosphors on the screen. The resulting intensity of each colour can thus
be independently controlled. The RGB colour space is designed to
enumerate the colours that can be generated on a colour Cathode ray tube.
The following diagram illustrates the RGB colour space.
In the diagram the values of R,G, and B range between 0 and 1. In practice
the values are integers that correspond to discrete values. Most CRTs that
are being used have 256 intensity levels for each gun.
This colour space describes an additive colour space, that is, the
colours are obtained by adding different amounts of the primary colours.
This model works well for CRTs.
CMY
If we consider the manner in which colours are reproduced on paper we must
use a different model. If a white piece of paper is viewed under white
light then we are seeing all wavelengths of the light reflected. However
if we now colour the paper with red ink we observe only the red portion
of the light spectrum. This is because the red ink absorbs all but the
red portion of the light. If we now add green ink to the paper we have
the red ink absorbing all but the red portion of the light and the green
ink absorbing all but the green portion of the light. Clearly the use of
the additive colour model does not work here. What we must use then is
the subtractive model. If we consider red,green, and blue to be the primary
colours then we must use a set of dyes that absorb only one of the primaries.
These dyes are precisely Cyan (absorbs red), Magenta (absorbs green), and
Yellow (absorbs blue). We can now obtain any of the original primary colours
by the combination of the subtractive primaries. For example red is obtained
by
Magenta(absorbs green) + Yellow(absorbs blue).
Given a colour definition in RGB (r,g,b) the transformation to CMY
space is straight forward.
c = 1 - r
m = 1 - g
y = 1 - b
In theory this subtractive colour space is sufficient for the representation
of the same set of colours as the RGB space. However, in practice it is
impossible to generate pure pigments. This means that achieving black (cmy
= 1 1 1) is difficult. Because of this a black dye is also used in
printing. This is known as the CMYK colour space. The transformation from
CMY to CMYK is also straight forward.
First we account for the amount of black present in the colour
k = min(c,m,y)
Then we subtract this from each of the colours.
c = c - k
m = m - k
y = y - k
Both of these colour spaces are useful representing the colours that can
be generated on a particular device. But how would you
HSV
The lack of intuitive feel of either of the above colour spaces led to
the development of a number of more intuitive colour spaces. I will
outline one of these here, namely the HSV space. If one is to look
at the RBG cube such that the line of sight is parallel to the main line
that goes from Black (0,0,0) to white (1,1,1), then one sees a hexagon
as follows:
We can think of the colours on the perimeter of this hexagon as being
fully saturated. We can also represent the hue by an angle of rotation
starting with red = 0. So if we represent the colours in this manner we
have a two-dimensional representation for colours where we represent the
hue by the rotation and the saturation as the distance from the center
of the hexagon.
The claim is that this saturation value is also intuitive since white
is at the centre, thus colours with less saturation will be closer to the
centre. The third parameter of this colour space can be seen if we flatten
the cube so that the three faces of the cube that make up this hexagon
now lie in a shared plane. This is the main idea behind the HSV space.
Now that the colours on the surface of half of the cube have been represented
we must represent in a consistent manner the remaining colours. The following
figure shows how this is achieved in the HSV space. The black colour has
been pulled down and the resulting colour space is an inverted hexagonal
based cone. Black is at the apex of the cone, the gray colours are along
the main axis of the cone.
The book contains procedures for converting from RGB to HSV and from
HSV to RGB.
This representation more closely resembles our intuitive notion of
colour. The textbook contains code for making the transformation from RGB
to HSV and vice-versa. There are other colour spaces but these are mostly
variations on the device dependent colour spaces, or on the LUV space,
or on the intuitive spaces.