Spectral interpolation errors
N Sándor, T Ondro and J Schanda
University of Veszprém, Colour and Multimedia Laboratory
Abstract
Several CIE Technical Committees discussed during the past years the advantages and
disadvantages of different spectral interpolation methods. Linear, Lagrange, cubic spline and Sprague
methods were under discussion, without experimentally investigating the amount of the difference if
one method or the other was used. We have investigated the colorimetric differences caused by using
the different interpolation methods, and show in the present paper that Lagrange, cubic spline and
Sprague methods provide very similar results for most well behaving spectra. As a further result we
found that the 5 nm sampling and using the 5 nm tables of the CIE Technical Report “Colorimetry” is
almost as good as an interpolation to 1 nm sampling and colorimetric calculation using the 1 nm tables
of the CIE colorimetric standards.
1. Introduction
A CIE Draft Technical Report 1 groups spectral functions used in colorimetric calculations into the
categories of “CIE (standard and recommended) data” if they are standardized or recommended in a
CIE standard or technical report and “measured data” if they are obtained by measuring reflecting or
transmitting specimens or radiant power. CIE data are either defined by theoretical functions (e.g. CIE
standard illuminant A) or are empirical data defined by their tabulated values. Values of theoretical
functions can be calculated for any wavelength required. CIE empirical data are usually tabulated at
1 nm intervals in CIE Standards 2,3 and to obtain values at intermediate wavelengths linear
interpolation provides high enough accuracy and is the recommended method. Exceptions are only
the spectral power distribution functions of CIE Test Illuminants (fluorescent and high-pressure lamp
spectra) 4, where the data are provided only at 5 nm intervals and the interpolation of these values is
not recommended.
Measured data can be subdivided into two categories: spectra of reflecting or transmitting
materials and radiant power spectra. If the radiant power spectra are derived from sources that contain
both continuous and line power, special care has to be taken to avoid sampling errors 4,51. In the
present paper we would like to concentrate on the question of interpolating measured data of
reflecting materials (some transmitting materials contain narrow band spectra and their proper
colorimetric measurement resembles that of mixed line plus continuum emission spectra), as the
recommendations for these have been changed since the detailed report by Billmeyer and Fairman,
who summarized the recommendations of CIE 15.2 6.
The definition of the tristimulus values of reflecting or transmitting samples (secondary light
sources) can be written as follows:

Ti  k  S   ( )t i ( )d
(1)
0
where i = 1 stands for x, i = 2 stands for y, i = 3 stands for z, and
1
The CIE draft Technical Report states: For colour stimulus functions of primary light sources that
vary rapidly with wavelength, the spectral bandwidth at half power of the measuring instrument used to
determine the colour stimulus function should be equal to, or an integer multiple of, the measuring
interval. For smoothly varying functions this restriction is less important. For the highest accuracy, a
bandwidth of 1 nm may be used for the measuring instrument, but for most practical purposes a
bandwidth and measurement interval of 5 nm may be used. The use of bandwidths of 10 nm, or 20 nm
is not recommended; it can lead to considerable loss of accuracy, and if applied, should be checked
on typical spectra. For practical measurements a trade-off between bandwidth and signal-to-noise
ratio has to be found, as the later will influence photometric accuracy. A signal-to-noise ratio of 1:10-4
when measuring an ideal white sample is adequate for most applications.
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k
100

 S t
2
(2)
( ) d 
0
and
S is the spectral power distribution of the irradiating illuminant,
ρ() is the spectral reflectance factor (or spectral radiance factor or spectral reflectance) of the
sample (similar equations can be written for transmitting samples with  (), the spectral
transmittance of the object colour instead of ρ()).
In practice the integral is substituted by summation:

Ti  k  S   ( )t i ( )
(3)
0
and similarly for k. The CIE draft Technical Report 4 states that for the purpose of calculating
tristimulus values, the quantities involved for secondary light sources shall be defined as points of a
continuous distribution, each point representing an infinitely narrow range of wavelengths. But
equation 3 obviously raises the question what the proper value of Δ ., i.e the sampling interval is. CIE
recommendations 4 require for highest accuracy a summation between 380 nm and 780 nm (outside
this range the t i values are practically equal to zero) with Δ = 1 nm, but the measured ρ() values
very often are not available at every nanometer. Many instruments measure reflectance spectra at
every 2 nm, or 5 nm, or 10 nm, or even only at larger wavelength increments. Long debates were
conducted in CIE technical committees how to interpolate reflectance spectra measured at wavelength
intervals larger then 1 nm to the required 1 nm interval data. Linear, Lagrange, cubic spline and
Sprague interpolation methods were under discussion, but to our best knowledge only theoretical
discussions were conducted, without checking how large the difference is if the four methods are
compared. In the present paper we would like to show on some practical examples the colorimetric
differences encountered by using one method or the other.
2. Interpolation methods tested
In early days of CIE colorimetry linear interpolation was the general practice, thus e.g. in calculating
the daylight illuminant spectra it is still the official method to interpolate the standardized values from
every round 10 nm value (from 300 nm up to 830 nm). Even the 1 nm standard values of CIE standard
illuminant D65 have been tabulated by this method 4.
Later the Lagrange and cubic spline interpolation 7 methods became favoured, although
already Judd used more extensive methods in determining interpolated values of the spectral
luminous efficiency function 8. This method used by Judd is practically equivalent with the so-called
Sprague interpolation method 9, recommended by CIE TC 1-38 1.
One of the authors (T Ondro) developed a computer program that enables the interpolation of
measured data sets using the linear, Lagrange, cubic spline and Sprague interpolation formulae 10.
While the Lagrange and cubic spline formulae are standard textbook methods, for the use of the
Sprague method see 1,11.
3. Test samples, illuminants and calculations
For our tests we used the 18 chromatic samples of the Macbeth Color Checker Chart 12, because
the colour patches of this chart represent a reasonable selection of colour samples. Table 1 shows the
determined Y, x and y values for D65 illuminant and the CIE 2° standard Observer. Actual
measurements were done using a PhotoResearch 705 spectroradiometer, using 0°/45°measuring
geometry and a white PTF reflectance standard as white reference. The instrument delivers the
spectrum with 2 nm increments, these were interpolated by the Sprague method to get the reflectance
spectra of the samples (tables of these are available at 13). These tables were regarded as the input
dataset of our investigation. Figure 1 shows the chromaticity of the 18 samples in the CIE u’, v’
diagram for CIE standard illuminant D65 and CIE 2°colorimetric observer. As can be seen the samples
cover a reasonable part of the diagram that can be produced as surface colours.
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We selected two illuminants for our investigation: CIE standard illuminant A, a continuous
spectrum, without any discontinuities in its first and second derivative spectra, and a spectrum that
shows several smaller maxima and minima, and where the spectrum is constructed from linear
segments 10 nm long. CIE standard illuminant D65 has such a spectrum. Thus we used in our test
these two illuminants.
As the next step the 1 nm tables were sampled at every 5 nm and 10 nm. Table 2 shows e.g.
the Y, x, y values for the sampled spectra for D65 illumination and 2° standard observer. Comparing
these data with the data in Table 1, small but for the 10 nm sampling non-negligible differences can be
seen.
The 18 test sample spectra sampled at every 5 nm and 10 nm were interpolated using four
interpolation methods: The Sprague method as recommended by CIE TC 1-38 1, a cubic spline, a
Lagrange and a linear interpolation. The tristimulus values and the CIELAB co-ordinates were
calculated for the original (1 nm sampled), the 5 nm and 10 nm sampled spectra, as well as the backinterpolated spectra to the 1 nm step size, for both CIE standard illuminant D65 and CIE standard
illuminant A and CIE 2° standard Observer. From these colour differences were calculated between
the original and the sampled and back-interpolated spectra.
4. Results
Tables 3 and 4 show the results for the 5 nm calculation, i.e. CIELAB colour differences between the
original 1 nm sampled spectra and the 5 nm sampled and then back-interpolated to 1 nm increments
spectra, with the four interpolation methods for the two selected illuminants.
As can be seen, the Sprague and the cubic spline methods provide practically identical
results, interesting enough the Lagrange method (column 4) works for the maximum occurring error
even slightly better then the other two non-linear methods. The simple 5 nm sampling is better then a
linear interpolation, and for D65 illumination on the average better then the non-linear interpolation
methods, the maximum error by using 5 nm sampling is only 10 % larger then the maximum error if
the 5 nm data were back-interpolated. The advantage of 5 nm sampling and calculation is even more
striking in case CIE standard illuminant A: On the average its results come nearest to the real 1 nm
sampling results, and even for maximum error only the Lagrange method delivers better results.
Our results for the sampling down to 10 nm and interpolating back to the 1 nm sampling
increments can be seen in Table 5 and 6. Also in this case, as can be seen, the Sprague and the
cubic spline methods provide practically the same accuracy. The Lagrange method is for some
samples slightly better, for others slightly worse then the above two ones. Compared with linear
interpolation the non-linear ones are considerably better, both for the average and maximum errors.
Interesting enough the simple 10 nm sampling is only slightly worse then the sophisticated non-linear
interpolations and for CIE standard Illuminant A only the Lagrange method gives better average
results.
As a further analysis step it seemed to be interesting to see where and why some interpolation
methods fail, especially as for the two illuminants the largest differences occurred at different samples
(see values in bold in tables 3 to 6).
Our first example is the MCC No. 18 test sample. With this sample for illuminant D 65 and
10 nm sampled spectra, the non-sampled, the spline and Sprague interpolation formulae gave
approximately the same size of error from the 1 nm sampled spectrum, and the linear interpolation
gave smaller deviation then the Lagrange interpolation, which gave the largest deviation (see bold
figures in Table 5). Figure 2 shows a critical part of the reflection spectrum. One can see that the
original spectrum shows small irregularities, and these are smoothed out by the non-linear
interpolation formulae. (It is an open question whether the original 1 nm data – if regarded as a true
measured spectrum – correctly reflects the spectral reflection of the material, or it is distorted by some
noise; but this is outside of the current investigation.) Just by chance the original spectrum shows a
small ditch in the wavelength range of maximum reflection, and this gets to advantage for the linear
interpolation. This is the reason that for this sample the linear interpolation does not give much worse
results then the other interpolation methods.
As a second example we have chosen the MCC 13 sample and again D65 illumination,
because here the linear interpolation gave far worse results then any other method. Part of the
reflection spectrum is seen in Figure 3. From this it is obvious that in the from above seen highly
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convex region of the spectrum, where also the colour matching functions have large values, the
deviation of the linearly interpolated curve from the 1 nm sampled curve, causes the relatively large
deviation in the CIELAB co-ordinates, leading to the large colour differences.
Table 6 shows that for test sample MCC 11 CIELAB co-ordinates calculated from the
Lagrange interpolated spectrum approximates the CIELAB co-ordinates of the original spectrum quite
well, the colour difference is much smaller than the colour differences calculated with the spectra using
the spline or Sprague interpolation. ΔE*ab = 0,0098 for Lagrange interpolated but is 0,055 for spline or
Sprague interpolated spectra. To find out the cause of this we first checked the reflection spectra (see
Figure 4), but the curves follow each other so closely that from a direct comparison the differences are
not visible. Figure 5 shows the difference spectra: We subtracted at every nanometer from the 10 nm
sampled and linearly, Lagrange-, spline- and Sprague-interpolated spectrum values the values of the
original spectrum. The large difference between the linearly interpolated and the original spectrum is
obvious. But also the large positive and negative deviations of the Lagrange interpolated curve can be
seen. In the ΔE*ab calculation these positive and negative deviation cancel each other and thus – just
by chance – the Lagrange interpolation method delivers better results then the cubic spline or Sprague
methods, for which the differences from the original are generally much smaller.
5. Conclusions and summary
Our experiments have proven the correctness of the statement formulated in the new CIE technical
report on colorimetry 4 that “for smoothly varying functions … for most practical purposes a
bandwidth and measurement interval of 5 nm may be used. The use of bandwidths of 10 nm, or 20 nm
is not recommended, it can lead to considerable loss of accuracy, and if applied, should be checked
on typical spectra.”
Billmeyer and Fairman 6 compared the Lagrange and cubic spline interpolation formulae and
found no major advantage for either of them. We have now extended the investigation including also
the Sprague formula recommended by CIE TC 1-38 1. This formula might be of advantage in some
special cases, but as seen from the present study, for average smooth reflection spectra it does not
give better results as the generally used cubic spline interpolation formula.
A further conclusion of our investigation is that irrespective of the non-linear interpolation
technique used, the to 1 nm intervals interpolated values do not provide tristimulus values that
approximate those obtained by using the real 1 nm measured values much better, than if one takes
the 5 nm tables, both for the reflection spectrum and the illuminant and colour matching spectra. (This
does not mean that a proper bandwidth correction should not be applied if the bandwidth of the
monochromator is not negligibly small compared to the spectral band to be measured 14,15, just as
the use of weighting tables 16.)
That the 5 nm, and even the 10 nm sampled direct calculation provides reasonably good
tristimulus values might be – partly – a consequence of the fact that the D65 illuminant is itself a 10 nm
to 10 nm piecewise linear construction, thus bends at other then the full 10 nm values do not occur in
the illuminating spectrum and do not modify the stimulus function further. One really finds a slightly
smaller average colour difference ratio between the interpolated and 10 nm sampled colour
differences. The next step in our investigations will be the application of the model calculations in case
of other illuminants, particularly white LED sources.
References
1
CIE TC 1-38, Recommended practice for tabulating spectral data for use in colour computations, Draft. 9.
2004.
2
CIE standard illuminants for colorimetry (1999), joint ISO/CIE standard ISO 10526/CIE
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3
CIE Colorimetric observers (1991), joint ISO/CIE standard ISO/CIE 10527(E):1991, CIE S0021986.
4
CIE TC 1-48, Colorimetry, Draft 18. 2004.
5
The spectroradiometric measurement of light sources, CIE Publ. 63:1984.
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6
Billmeyer FW and Fairman HS (1987), CIE method for calculating tristimulus values, Color Res. & Appl. 12/1
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7
See e.g. in Stoer J, Bulirsch (1993) Introduction to Numerical Analysis, Translated by Bartels R, Gautschi W
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8
Judd DB (1931) Extension of the standard visibility function to intervals of one millimicron by third difference
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9
Sprague TB (1880), Explanation of a new formula for interpolation, J. Inst. Actuaries, 22, 270-285.
10 The interpolation program can be downloaded from the following WEB page: xxxx
11 Seve R, Duval B (2003). Interpolation procedure: Proposals and comments. Proc. CIE 25th Session Vol. 1.
D1-74-77 CIE 152:2003.
12 McCamy CS, Marcus H, Davidson JG (1976). A color-rendition chart. Jr. Photographic Engng. 2/3, 95-99.
13 Detailed data and background material of the present paper are to be found at the following URL: xxx
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15 Venable WH (1989). Accurate tristimulus values from spectral data. Color Res. Appl. 14 260-267.
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