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Colour is a human sensation which has little to do with plain physics
and a lot with biology. Every very small (part of an) object around us
emits electro magnetic energy, which meets the eye. Its image is
projected onto the retina. Because of the rapid elliptical movements
of the eye, the image describes an elliptical trajectory on the
retina. Let be the energy density at wavelength
of the image, so
is the total amount of energy between wave lengths and
. Due to the ellitical movements of the eye, the image
meets different sensors for electromagnetic radiation, that have
different sensitivities for the different wave lengths. For
simplicity's sake we here assume that the signal output of the sensor
is proportional to the input. Let be the energy signal
conversion of sensor at wavelength , so
is the signal of
sensor , if all the separate contributions of the different wave
lengths just add up. The actual signal is approximately
hyperbolically transformed, like
, but this is
at present of no importance.
It seems that most of us have got three different types of
sensors. So, the total signal we receive can be represented as
, which we call sensation.

The lower end of the sensitivity range is around 420nm, where we
have a sensation which we call violet, and an upper limit of 67nm,
where we have a sensation which we call red. The range has no sharp
limits. Every possible sensation can be represented by a point in the
three dimensional sensation space. The sensors have a considerable
overlap in sensitivities. Therefore it is not possible to stimulate
only one sensor, without stimulation the others to some extend. This
means that not all points in the sensation space can be reached in
practice. Suppose that there exists a fixed minimum threshold of the
signal we can detect, the range depends on the energy input , which
we experience as intensity. The relative firing rate of the three
sensors we experience as colour.
Any choice of three different colours (i.e. sensations)

can in principle be used as a basis for the sensation space, as long
the corresponding three vectors are not in the same plane. So all
colours, like white, can be obtained by a linear combination of that
three colours. That is to say, when we admit subtraction. The
sensation
in fact means
when we use the
sensors , , as a basis. It is
equivalent to the sensation
, which means
when we use the three
colours , , as a basis.
Because both expressions are equivalent, we must have that
, or
, where

However, not every choice of the colours as a basis is equally
usefull. In the first place, subtraction is cumbersome from a
physical point of view. Small errors in the signals of the
three sensors are enlarged when the colours are close to each
other.
It is useful to realize that the eye reduces the energy density
function , consisting of an infinite number of "data-
points" to just three values collected in . Therefore many
different energy density functions are mapped into the same sensation
(i.e. colour). It also implies that there are many ways to combine
different colours to produce a particular one. The study of different
types of colour blindness has contributed a lot in our understanding
of colour perception. A particular type is the absence of one sensor,
which means that the perception is in two rather than three
dimensions. Some species of animals have more than three sensors for
electro magnetic radiation. They must be able to distinguish between
different sources, that look the same for us.

Filename: ex018; Date 1991/04/24; Author: Bas Kooijman

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Theoretische Biologie
2002-05-01