1) The document discusses measured reflectance (Rλ) values, which represent the fraction of light reflected by a sample at each wavelength, and how these values are independent of the light source used to measure them. 2) It explains that to calculate the actual amount of light reflected at each wavelength, the measured Rλ values need to be multiplied by the energy (Eλ) of the light source at that wavelength. 3) The total amount of light reflected across the visible spectrum is calculated by summing the amounts reflected (Eλ x Rλ) at each wavelength between 380-760nm.

1. 1

2. Measured Reflectance Rλ
Suppose that we have a sample, such as a painted surface,
and that we have measured
the fraction of light reflected
at each wavelength, Rλ.
Provided that the sample is not fluorescent, the Rλ values will
be completely independent of the light shone on the sample.
(Many instruments give readings in terms of the percentage of
light reflected, i.e. Rλ ´ 100, but fractions are easier to use
in the present discussion.)
.
2

3. Measured Reflectance Rλ
For example, a white paint will reflect
about 90% of the incident light
(i.e. Rλ= 0.9) at, say, 500 nm
Whether illuminated with strong daylight
or with weak tungsten light.
Thus the Rλvalues are independent of
 the actual light source used in the spectrophotometer.
.
3

4. Measured Reflectance Rλ
The actual amount of light reflected will be
different for different light sources,
however.
Suppose that the sample is now viewed
under a light source
for which the light emitted at each wavelength is Eλ.
Then the amount reflected at each wavelength will be
Eλ x Rλ.
Now if we consider only light of wavelength l, one unit of
energy of l can be matched by an additive mixture of x–
l units of [X] together with y–
l units of [Y] and z–
l
units of [Z] (Eqn 3.8):
.
4

5. Measured Reflectance Rλ
Then the amount reflected at each wavelength will be
Eλ x Rλ.
Now if we consider only light of wavelength λ,
one unit of energy of λ can be matched by
an additive mixture of
xλ units of [X]
together with yλ units of [Y]
and z–λ units of [Z]
(Eqn 3.8):
5

6. Measured Reflectance Rλ
Then the amount reflected at each wavelength will be
Eλ x Rλ.
It also follows from the properties of additive mixtures of lights that
the light reflected at two wavelengths
λ1 and λ2,
Eλ1 Rλ1 [λ1] + Eλ2 Rλ2 [λ2]
can be matched by
6

7. Measured Reflectance Rλ
Then the amount reflected at each wavelength will be
Eλ x Rλ.
The total amount of energy reflected over the visible
spectrum is
the sum of the amounts reflected at each wavelength.
This can be represented quite simply mathematically (Eqn 3.10):
where the sigma sign (Σ) means that the Eλ x Rλ. values for each wavelength
through the visible region should be added together,
and the limits of λ = 380 and 760 nm are the
boundaries of the visible region.
7

8. Measured Reflectance Rλ
The total amount of energy reflected over the visible
spectrum is
the sum of the amounts reflected at each wavelength.
This can be represented quite simply mathematically (Eqn 3.10):
Strictly the spectrum should be divided into
Infinitesimally small wavelength intervals (dλ) and the total amount of light
given
but in practice the summation form is used.
Representing the amounts of [X], [Y] and [Z] in a similar manner,
the light reflected from our paint surface can be matched by Eqn 3.12:
8

9. Measured Reflectance Rλ
Since the light reflected
from our paint sample
can also be matched
by
X[X] + Y[Y] + Z[Z],
it follows (Eqn 3.13):
9

10. 10

11. 11

12. Reflectance CALCULATION
The calculation can be illustrated by reference to
Figure 3.5.
Suppose we have
measured the reflectance curve
of a sample
and obtained the results shown in Figure
3.5(a).
The R values indicate
the fraction of light reflected by the sample at each wavelength.
At the wavelengths around 500 nm the sample is reflecting
a high proportion of the light that is shone on it,
no matter how much or how little light that may be.
Similarly the fraction reflected at 600–700 nm is low,
again irrespective of the amount of light shone on to the surface.
To calculate how much light is actually reflected, we need to know
how much light is shone on the surface.
12

13. Reflectance CALCULATION
Suppose that the surface is illuminated by a source
whose energy distribution is
shown in Figure 3.5(b),
i.e. the source contains
relatively less energy at the short-wavelength
end of the visible region
and relatively much more at the longer wavelengths.
The amount of light reflected by the sample
at each wavelength
will be EλRλ
and this is also plotted against wavelength in Figure 3.5(b).
We can see that while the curve resembles the Rλ curve,
with a maximum at 540 nm
and a minimum at 680 nm,
the balance between the longer and shorter wavelengths is quite different.
The R values are roughly the same at 400 and 620 nm,
While the EλRλ value at 620 nm is almost ten times the corresponding value at 400 nm.
13

14. Reflectance CALCULATION
There are two quite distinct parts
to the curve
with maxima
around 460 and 600 nm,
but the relative sizes of the two
peaks have changed.
Again the curves roughly resemble the
but the relative sizes have changed.
14

15. curves are proportional to
the X, Y and Z tristimulus values
respectively.
It is obvious that Z is considerably
smaller than X or Y.
(In fact the Eλ curve corresponds to
tungsten light
and the approximate
tristimulus values are
X = 38, Y = 45 and Z = 21.)
We can now see
why the standard observer is so
important, Provided that we know
the energy distribution of the light source
and how the tristimulus values can be under which the specification of our paint sample is
obtained without actually required,
producing a visual match for our colour.
15