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# 3.7 calculation of tristimulus values from measured reflectance values

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• 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 willbe completely independent of the light shone on the sample.(Many instruments give readings in terms of the percentage oflight reflected, i.e. Rλ ´ 100, but fractions are easier to usein the present discussion.). 2
• 3. Measured Reflectance RλFor example, a white paint will reflectabout 90% of the incident light(i.e. Rλ= 0.9) at, say, 500 nmWhether illuminated with strong daylightor with weak tungsten light.Thus the Rλvalues are independent of the actual light source used in the spectrophotometer.. 3
• 4. Measured Reflectance RλThe actual amount of light reflected will bedifferent for different light sources, however. Suppose that the sample is now viewed under a light sourcefor which the light emitted at each wavelength is Eλ.Then the amount reflected at each wavelength will beEλ x Rλ.Now if we consider only light of wavelength l, one unit ofenergy of l can be matched by an additive mixture of x–l units of [X] together with y–l units of [Y] and z–lunits of [Z] (Eqn 3.8):. 4
• 5. Measured Reflectance RλThen the amount reflected at each wavelength will beEλ x Rλ.Now if we consider only light of wavelength λ,one unit of energy of λ can be matched byan 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 RλThen the amount reflected at each wavelength will beEλ x Rλ.It also follows from the properties of additive mixtures of lights thatthe light reflected at two wavelengths λ1 and λ2,Eλ1 Rλ1 [λ1] + Eλ2 Rλ2 [λ2] can be matched by 6
• 7. Measured Reflectance RλThen the amount reflected at each wavelength will beEλ x Rλ.The total amount of energy reflected over the visiblespectrum isthe 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 wavelengththrough the visible region should be added together,and the limits of λ = 380 and 760 nm are theboundaries 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 intoInfinitesimally small wavelength intervals (dλ) and the total amount of lightgivenbut 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 RλSince the light reflected from our paint sample can also be matched by X[X] + Y[Y] + Z[Z],it follows (Eqn 3.13): 9
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• 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 indicatethe fraction of light reflected by the sample at each wavelength.At the wavelengths around 500 nm the sample is reflectinga 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 nmand 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 CALCULATIONThere 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 seewhy the standard observer is soimportant, Provided that we know the energy distribution of the light sourceand how the tristimulus values can be under which the specification of our paint sample isobtained without actually required,producing a visual match for our colour. 15