NMI TR 8
Uncertainties in Colour Measurements
James L. Gardner
First edition — August 2005
Bradfield Road, Lindfield, NSW 2070
PO Box 264, Lindfield, NSW 2070
Telephone: (61 2) 8467 3600
Facsimile: (61 2) 8467 3610
Web page: http://www.measurement.gov.au
© Commonwealth of Australia 2005
This document was originally prepared for the Korean Institute of
Standards and Science in 2003 while James Gardner was there as a
guest researcher. It had minor revisions up to November 2003.
This document is that latter revision (circulated as Revision 2.7) with
text added to the section dealing with distribution temperature.
CONTENTS
1
Propagation of Uncertainty ....................................................................................................... 1
2
Tristimulus Values .................................................................................................................... 2
3
Sources versus Surfaces ............................................................................................................ 3
4
Transfer Measurement and Component Uncertainties .............................................................. 3
5
Uncertainties (Variances) and Covariances of the Tristimulus Values ..................................... 5
6
Covariance of a Colour Triplet.................................................................................................. 7
7
Methods of Calculation for Colour Triplets .............................................................................. 7
7.1 (x,y,Y) Colour Coordinates ............................................................................................... 7
7.2 (u,v,Y) Colour Coordinates ............................................................................................... 8
7.3 (u′,v′,Y) Colour Coordinates ............................................................................................. 8
7.4 (L*,a*,b*) Colour Coordinates......................................................................................... 9
7.5 (L*,u*,v*) Colour Coordinates ......................................................................................... 9
7.6 (L*h*c*) Colour Coordinates ......................................................................................... 10
8
Uncertainty Components in Colour Measurement .................................................................. 10
8.1 Base Uncertainty of the Reference Standard .................................................................. 11
8.2 Random Measurement Noise in the Transfer ................................................................. 11
8.3 Random Wavelength Accuracy in the Transfer ............................................................. 11
8.4 Wavelength Offset Error in the Transfer........................................................................ 12
8.5 Constant Relative Error in Absolute Value of the Reference or the Transfer Ratio ...... 12
8.6 Constant Offset in the Reference or Sample Channels .................................................. 12
8.7 Scaling of the Reference Value or Transfer Ratio Linear with Wavelength ................. 13
8.8 Source Noise .................................................................................................................. 13
8.9 Bandwidth Effects .......................................................................................................... 14
8.10 Sample Uniformity ......................................................................................................... 14
8.11 Multiple Transfers from the Reference .......................................................................... 14
9
Representative Examples for Surface Colour Uncertainties ................................................... 15
9.1 Typical Uncertainty Components................................................................................... 15
9.2 (x,y,Y) by Component ..................................................................................................... 16
9.3 (L*a*b*) by Component ................................................................................................ 17
9.4 Combined Uncertainty for the Various Surface Colour Quantities ............................... 19
10 Methods for Source Parameters and Uncertainties ................................................................. 20
10.1 Dominant Wavelength.................................................................................................... 20
10.2 Source Correlated Colour Temperature ......................................................................... 21
10.3 Source Distribution Temperature ................................................................................... 22
11 Representative Examples for Source Colour Uncertainties .................................................... 25
11.1 Typical Uncertainty Components................................................................................... 25
11.2 Broadband Sources ......................................................................................................... 26
11.3 Narrow-band Sources ..................................................................................................... 27
12 Conclusion ............................................................................................................................... 29
13 References ............................................................................................................................... 29
iii
SUMMARY
Techniques are presented to estimate uncertainties in the various colour coordinates that may
be reported from measurements of surface reflectance or source spectral power distributions.
The measurements are treated as a transfer from a reference reflector or source. Uncertainties
are then calculated for effects arising in either the base reference or in the transfer
measurement. Uncertainties of the various colour quantities are propagated through
uncertainties in the tristimulus values, which are in turn propagated from uncertainties in the
spectral distribution of the reflectance or source power. Effects contributing to the uncertainty
of a given colour value are treated separately and then combined into the final uncertainty.
The form of the tristimulus integrals leads to simple expressions to calculate their
uncertainties. Many systematic effects can be described by a single parameter. Then for fullycorrelated effects, the uncertainty of each tristimulus value is a linear sum of the signed
spectral uncertainty, weighted by the appropriate colour-matching function. Further, the
tristimulus values themselves are fully correlated and the covariances of the tristimulus values
are products of the uncertainty sums. For effects random between wavelengths the
covariances are the product of the same linear sums and the variances are linear sums of the
spectral variances, again weighted by the appropriate colour-matching functions. For surface
colour uncertainties, the colour-matching functions are replaced by their convolution with the
illuminant spectral power distribution.
Sensitivity coefficients for the dependence of colour quantities on the tristimulus values are
then used to propagate uncertainty from the tristimulus values to those quantities, considered
as colour triplets. Some colour quantities are combinations of simpler quantities; for these,
expressions are given for the propagation of the covariance from the simpler values to the
more complex ones.
Representative calculations are made for both sources and surfaces. Uncertainties vary with
position in colour space and should be estimated for each spectral measurement.
In addition to the various colour triplets, methods for estimating uncertainties in correlated
colour temperature and dominant wavelength of sources are given, as is that for source
distribution temperature (which is not strictly a colour parameter, as it does not depend on the
tristimulus values).
iv
1
PROPAGATION OF UNCERTAINTY
Uncertainty propagation is described in detail in the ISO Guide to the Expression of
Uncertainty in Measurement [1]. The uncertainty in a quantity X formed by
combining measured quantities xi through the relationship X  f ( x1 , x2 ,..xN ) is given
by
2
N 1 N
 f  2
f f
u (X )  
u
(
x
)

2
u ( xi , x j )
(1)



i
i 1  xi 
i 1 j i 1 xi x j
where u ( xi ) is the uncertainty in xi and u ( xi , x j ) is the covariance of xi and x j . For
uncorrelated input quantities, the covariance of pairs of variables is zero and Eq. (1)
reduces to
N
2
2
 f  2
u (X )  
(2)
 u ( xi )
i 1  xi 
the ‘sum of squares’ commonly applied. The derivatives f xi are sensitivity
coefficients for the dependence of X on the various measured quantities. Given that
u 2 ( xi )  u ( xi , xi ) , Eq. (1) can be expressed as
N
2
u 2 ( X )  f x U xf x T
(3)
where
 f
f
f 
fx  
..

xn 
 x1 x2
is a column vector of sensitivity coefficients ( T indicates the transpose) and
U x   u ( xi , x j ) 
(4)
(5)
is the N  N variance–covariance matrix of squares-of-uncertainty (variance) in
diagonal elements, covariance values elsewhere.
If we form another quantity Y by combining the measured quantities xi through the
relationship Y  g ( x1 , x2 ,..xN ) , the uncertainty in Y is given by an expression similar
to that of Eq. (1), but now the quantities X and Y are correlated through dependence
on the common set xi. The covariance between X and Y is given by
N N
f g
u ( X , Y )  
u ( xi , x j )
(6)
i 1 j 1 xi x j
In matrix form, this is simply expressed as
u( X , Y )  f x U xg xT
(7)
where
 g g
g 
(8)
gx  
..

xn 
 x1 x2
Correlation coefficients are normalised covariance values, defined as
u( X , Y )
(9)
r( X ,Y ) 
u ( X )u (Y )
We note that if r ( xi , x j )  1 for all pairs, Eq. (1) reduces to
N
u( X )  
i 1
NMI TR 8
f
u( xi )
xi
(10)
1
In section 5 we see that this linear sum can be used to calculate the uncertainty of
many systematic effects even where the correlation coefficient changes sign through
the spectrum.
Colour quantities can all be expressed in terms of the tristimulus values. Uncertainties
in the derived quantities are then found by propagating uncertainties through
sensitivity coefficients and the variance–covariance matrix for the tristimulus values.
2
TRISTIMULUS VALUES
Tristimulus values are integrals representing convolution of the CIE colour-matching
functions [2], shown in Figure 1, and the spectral power distribution of the light
reaching the detector (the eye). The set of colour-matching functions chosen (2 or
10) depends on the application.
2.0
x
y
z
Response
1.5
1.0
0.5
0.0
400
500
600
700
Wavelength /nm
Figure 1. CIE colour-matching functions (the solid lines with symbols are the
1931 values for a 2º field of view; the dashed lines are for the 1964 10º observer)
The X,Y, Z tristimulus values are carried in the vector
N
N
 N

(11)
T    X Y Z      xi Si  yi Si  zi Si 
i 1
i 1
 i 1

where Si is the spectral power distribution reaching the eye (at the ith of N
wavelengths) and xi , yi , zi are the CIE colour-matching functions defined over the
wavelength range from 360 nm to 830 nm.
Chromaticity values involve ratios of tristimulus values and luminance or lightness is
usually expressed as a ratio to that of a reference illuminant; hence only relative
spectral power distributions are important and we can delete  from Eq. (11). (But
note that when we calculate component uncertainties for the tristimulus values using
different wavelength steps we must include a scaling factor for when adding
components.)
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3
SOURCES VERSUS SURFACES
In reflectance colorimetry we measure reflectance under a specified geometry as a
function of wavelength in the visible range. The spectral power distribution reaching
the eye is then a convolution of the reflectance spectrum and a source power
distribution, usually taken as CIE Illuminant A or CIE Illuminant D65 [2], depending
on the application. Illuminant A represents typical incandescent lighting and
Illuminant D65 represents a phase of daylight. These agreed source distributions are
shown in Figure 2 – their values carry no uncertainty.
Spectral Power Distribution
120
100
80
60
40
A
D65
20
0
400
500
600
700
Wavelength /nm
Figure 2. Spectral power distributions for CIE Illuminants A and D65
All the colour quantities for surfaces can be derived using the same expressions as
those for sources provided the colour-matching functions are replaced by their
convolution with the reference illuminant spectrum
xi '  SiIll xi , yi '  SiIll yi , zi '  SiIll zi
(12)
where SiIll is the value of the illuminant spectral power distribution at the ith
wavelength. Values of the spectral power distribution, S i , in Eq. (11) are then
replaced by Ri , the value of the reflectance at the ith wavelength. In the sections
below we take S i to refer to the measured spectrum (source or reflectance) and the
colour-matching functions to be amended as in Eqs (12) if we are referring to a
surface measurement (with the prime notation dropped).
4
TRANSFER MEASUREMENT AND COMPONENT UNCERTAINTIES
Spectral reflectance may be required for various geometric conditions and is
measured against a reference standard. Similarly, measurements of spectral power
distribution are traced back to a reference standard. Hence the spectral value Si can be
written as a transfer from that of a reference standard S iRef
Si  ti SiRef
(13)
Uncertainties in the spectral value Si arise both from those of the reference value and
those introduced by the spectral transfer. Colour values are formed by combining the
spectral measurements at different wavelengths. Systematic effects in a spectral
measurement have a cooperative relationship between wavelengths, that is, the values
NMI TR 8
3
at different wavelengths are correlated. Correlations also exist between the reference
values at different wavelengths, arising from systematic effects in the methods used
to derive them.
Table 1. Uncertainties ui , j in spectral values Si due to different effects j, combined to
estimate the uncertainty u c in the value of a colour quantity; the total uncertainty of
Si at wavelength i is uic and the uncertainty in the colour value due to effect j is u c | j
Wavelength
Effect 1
Effect 2
Effect m
Combined
1
u1,1
u1,2
u1,m
u1c
2
u2,1
u2,2
u2,m
u 2c
n
un ,1
un ,2
un , m
u nc
Combined
u c |1
u c |2
u c |m
uc
Individual sources of uncertainty can be considered as independent effects. Table 1
shows the two ways that uncertainties in spectral values due to these individual
effects can be combined when the spectral values themselves are combined. Because
the individual effect are not correlated, we can form the uncertainties of the combined
effects at a given wavelength by sum-of-squares, Eq. (2). However, these spectral
values are now most often partially correlated, and we must use the full expressions
of Eqs (1) or (7) to calculate the final combined uncertainty, u c .
The final combined uncertainty u c is also found by combining the u c | j values by
sum-of-squares. The advantage of this form of combination is that many of the
systematic effects that can contribute to uncertainty across wavelengths are either
uncorrelated (random) – which means that the uncertainty for the combination across
wavelengths is found by sum-of-squares – or fully correlated, for which the simple
linear sums of Eq. (10) can be used.
For uncertainty components of the reference spectrum, from Eq. (13) we have
u ( Si )  ti u ( SiRef )
(14)
whereas for components in the transfer we have
u ( Si )  SiRef u (ti )
(15)
In both cases we calculate the transfer value as the ratio ti  Si SiRef . The reference
spectrum and the measured spectrum generally are defined at different wavelengths.
To avoid introducing correlations through interpolation procedures [3], we use
different procedures for reference and transfer uncertainty components. When dealing
with reference uncertainty components, we interpolate the measured spectrum (which
has no uncertainty if we are only considering a single component of the reference
spectrum) to the wavelengths of the reference spectrum. When dealing with transfer
uncertainty components, we interpolate the reference spectrum (which has no
uncertainty if we are only considering a single component of the transfer) to the
wavelengths of the measured spectrum.
NMI TR 8
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5
UNCERTAINTIES (VARIANCES) AND COVARIANCES OF THE
TRISTIMULUS VALUES
The reference spectrum values will in general be partially correlated. The reference
contribution to the variance–covariance of the tristimulus values is then calculated
directly using Eq. (7); this is discussed in detail in section 8.1Ошибка! Источник
ссылки не найден..
Many effects systematic across wavelength can each be described by a single
parameter p as
(16)
Si  Si ( p)
Noting that uncertainties are always positive, the uncertainty of Si due to this effect
alone is
S
(17)
u ( Si )  i u ( p )
p
For an effect that is random from wavelength to wavelength, the uncertainty in the X
tristimulus value is propagated through Eq. (2), with the variance given by
N
u 2 ( X ) | p   xi 2u 2 ( Si )
(18)
i 1
with similar expressions for the variance of the Y, Z tristimulus values. Even though
the spectral values are uncorrelated, the common parameter correlates the tristimulus
values. We have u ( Si , S j )  0 for i  j , u ( Si , Si )  u 2 ( Si ) and the covariance
between X and Y, for example, is
N
u ( X , Y ) | p   xi yi u 2 ( Si )
(19)
i 1
For non-random effects, the covariance of spectral values at different wavelengths is
given by
S S j 2
(20)
u (Si , S j )  i
u ( p)
p p
Hence u ( Si , S j )  u ( Si )u ( S j ) and the spectral values are fully correlated for the
effect we are considering. The value of the correlation coefficient r ( Si , S j ) can be
either +1 or –1 at different parts of the spectrum (for example, an offset in
wavelength may cause such an effect) and so we cannot necessarily apply Eq. (10).
However we can rewrite Eq. (20) using the sign function sgn( ) as
 S  S S j 2
 S 
(21)
u (Si , S j )  sgn  i  sgn  j  i
u ( p)
 p 
 p  p p
The covariance of the tristimulus values, due to our single-parameter effect, is given
by expressions of the form
X Y 2
u ( X , Y )| p 
u ( p)
p p
N N
S S j 2
  xi y j i
u ( p)
p p
i 1 j 1
N
N
 S  S
 S  S
 xi sgn  i  i u ( p) y j sgn  j  j u ( p)
i 1
j 1
 p  p
 p  p
NMI TR 8
(22)
5
These are the diagonal terms required for the tristimulus variance–covariance matrix.
If we treat the uncertainty as carrying the sign of the sensitivity component
 S  S
S
(23)
us ( Si )  sgn  i  i u ( p)  i u ( p)
p
 p  p
we have
N
N
i 1
j 1
u( X , Y )| p  xi u s (Si ) y j u s (S j )
(24)
where the s subscript denotes a signed uncertainty.
Using Eqs (1), (11), (17) and (21), the variance of the X tristimulus value is given by
2
N
N 1 N
 S j  Si S j 2
 Si  2
 Si 
2
u ( X )| p    xi
u ( p)

 u ( p)  2  xi x j sgn 
 sgn 
p 
i 1 
i 1 j i 1
 p 
 p  p p
or
2
 N
 S  S 
(25)
u ( X )| p    xi sgn  i  i  u 2 ( p)

p

p
i

1




with similar expressions for Y and Z. In terms of the signed uncertainties we have
2
u ( X )| p 
N
 x u (S )
i 1
i s
i
(26)
Comparing Eqs (24) and (26) shows
u ( X , Y )  u ( X )u (Y )
(27)
that is, the tristimulus values are fully correlated. The sum terms in Eq. (24) are in
fact the signed uncertainties of the X and Y tristimulus values; the covariance values
are a product of these signed uncertainties.
The complete variance–covariance matrix for the tristimulus values, for our effect
being considered, is
 u 2 ( X ) u( X , Y ) u( X , Z )


U XYZ | p   u ( X , Y ) u 2 (Y ) u (Y , Z ) 
(28)
2
u ( X , Z ) u (Y , Z ) u ( Z ) 


and the uncertainty in any colour quantity is found by propagating these values
through Eq. (7) and the appropriate sensitivity coefficients.
An important consequence of systematic effects is that all colour quantities calculated
from fully-correlated spectral values are also fully correlated; the correlation is
positive for systematic effects that are positively correlated, but may vary for
systematic effects whose correlation can be positive or negative.
The variances and covariances of the tristimulus values for all effects are found as
sums of those of the independent individual effects. Here we must take into account
the wavelength step used in calculating the tristimulus values. In calculating the base
reference spectrum uncertainties, the wavelength step may be different from that of
the measured spectrum used to calculate the systematic effects. If so, the tristimulus
variances and covariances from the base reference spectrum must be scaled by
 
Ref
 spectrum  .
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6
Metameric colours are those for which their spectral distribution is different but their
tristimulus values and hence appearance are identical. The uncertainties of the
tristimulus values depend on the spectrum itself, and so uncertainties will be different
for the parameters of metameric colours.
6
COVARIANCE OF A COLOUR TRIPLET
Three variables are required to describe colour. In the case of the simple chromaticity
values (x,y) or (u,v), the third variable is the Y tristimulus value itself (luminance),
often quoted as a ratio to that of an illuminant. These values may be combined to
form other colour quantities, and we then need to know not only their uncertainty (or
variance), but also the relation between them (covariance), which may be quoted as a
matrix of correlation coefficients. Instead of calculating individual variance or
covariance values using Eq. (7), we form a 3x3 matrix of sensitivity coefficients for
each desired quantity in terms of the X,Y and Z tristimulus values. If S is such a
matrix with colour quantities ordered by columns and tristimulus values by rows, the
variance–covariance matrix for the colour quantities is given by
T
U  S
U XYZ S
(29)
Using x,y,Y as an example of a colour-triplet, the matrix S xyY is given by
 x
 X

 x
 Y

 x
 Z
7
y
X
y
Y
y
Z
Y 
X 

Y 
Y 

Y 
Z 
(30)
METHODS OF CALCULATION FOR COLOUR TRIPLETS
With the exception of source temperature parameters and dominant wavelength, the
quantities of interest are colour-triplets, for which uncertainty is propagated from the
variance–covariance matrix of the tristimulus values, the appropriate sensitivity
matrices and Eq. (29). In the following sections we derive the sensitivity matrices for
the various colour triplets. Some of these require a second stage of propagation,
because they in turn depend on other colour quantities.
7.1
(x,y,Y) Colour Coordinates
The (x,y) chromaticity values are given simply as
Y
X
, y
, with Txy  X  Y  Z
(31)
x
TXY
Txy
The reported value of Y is usually Y/YN for a source, where YN is the Y tristimulus
value of the reference illuminant, or 100 Y/YN for reflectance. In the source case, the
sensitivity matrix is
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7
Y  Z
 2
 Txy
 X
S xyY   2
 Txy

 X
 Txy2
with the obvious modification for reflectance.
Y
Txy2
X Z
Txy2
Y
Txy2

0

1

YN 

0

(32)
The linear dependence of the numerator for the x,y sensitivity coefficients on the
tristimulus values has two important consequences. Any spectral uncertainty
component which is a constant multiplied by the spectral value leads to zero
uncertainty in the x and y chromaticity values (and also u, v, u and v). This is true for
an uncertainty in the relative value of the reference or in the transfer, as expected
since the chromaticity values are ratios of the tristimulus values to their sum. In the
case of reflectance colorimetry, often the reference spectrum has a value that is
approximately constant; in such cases, we have from Eq. (13)
Si
u  Si   Ref
u  SiRef   Si u  SiRef 
(33)
Si
and uncertainty in the offset in the reference spectrum also leads to zero uncertainty
in the (x,y) chromaticity values.
A further general conclusion can be drawn for red LED x,y chromaticites. Here the Z
tristimulus value is effectively zero and the chromaticity is close to the
monochromatic boundary. Hence x  y  1 and any sensitivity coefficients for x are
the negative of those for y and any uncertainties for x and y will be equal.
7.2
(u,v,Y) Colour Coordinates
In a manner similar to the treatment for (x,y,Y) above
4X
6Y
u
, v
with Tuv  X  15Y  3Z
Tuv
Tuv
In the source case reporting Y/YN, the sensitivity matrix is
 60Y  12 Z

6Y
0

2
2
Tuv
Tuv


 60 X
6 X  18Z 1 
SuvY  

Tuv2
Tuv2
YN 

 12 X

18Y
0

2
2
Tuv
Tuv


7.3
(34)
(35)
(u′,v′,Y) Colour Coordinates
These are given as a simple scaling of (u,v), u '  u, v '  3v / 2 . This scaling is applied
to the middle column of Eq. (35).
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7.4
(L*,a*,b*) Colour Coordinates
L*,a* and b* are calculated as
1
Y 3
L*  116    16
 YN 
 X  1 3  Y  1 3 
a*  500 
   
 X N 
 YN  

 Y  13  Z  13 
b*  200    
 
ZN  
 YN 



(36)
where Xn, Yn and Zn are the tristimulus values for a perfect reflector (i.e. the
illuminant distribution alone – these carry no uncertainty).
The sensitivity matrix is
S L*a*b*
7.5

0


116 13  2 3

Y Y
 3 N


0

500 13  2 3
XN X
3
500 13  2 3

YN Y
3
0



200 13  2 3 
YN Y

3

200 13  2 3 

ZN Z

3
0
(37)
(L*,u*,v*) Colour Coordinates
These are defined as
1
Y 3
L*  116    16 , u*  13L *(u ' u ' N ) , v*  13L *(v ' vN ')
 YN 
(38)
where u′, v′ are CIE 1976 chromaticity coordinates and u′N, v′N are similar quantities
for the illuminant alone.
First we calculate the covariance matrix U Lu ' v ' for the quantities L*, u′ and v′, for
which the sensitivity matrix in terms of the tristimulus values is

60Y  12Z
9Y 
0


Tuv2
Tuv2


116 13  2 3
60 X
9 X  27 Z 
(39)
S L*u ' v '  
YN Y

2
2
3
T
T
uv
uv



12 X
27Y 
0


Tuv2
Tuv2


where Tuv  X  15Y  3Z .
The covariance matrix U L*u ' v ' for the quantities L*, u′ and v′ is then
U L*u ' v '  S TL*u ' v ' U XYZ S L*u ' v '
NMI TR 8
(40)
9
The sensitivity matrix for the quantities L*, u* and v* in terms of L*, u′ and v′ is
1 13(u ' uN ') 13(v' vN ') 

(41)
S L*u*v*'  0
13L *
0

0
0
13L * 
and the final uncertainties and correlations are carried in the variance–covariance
matrix
U L*u*v*  S TL*u*v*U Lu ' v 'S L*u*v*
(42)
7.6
(L*h*c*) Colour Coordinates
The quantities hue and chroma are calculated from either (a*,b*) or (u*,v*)
chromaticity pairs. Taking the (a*,b*) example
 b*
(43)
h*  tan 1   , c*  a *2 b *2
 a*
We first calculate the variance–covariance matrix U L*a*b* for L*a*b* from that of the
tristimulus values using the sensitivity matrix in section 7.4.
U L*a*b*  S TL*a*b*U XYZ S L*a*b*
(44)
The sensitivity matrix for the quantities L*, h* and c* in terms of L*, a* and b* is


1
0
0 


b * a *

(45)
S L*h*c*'  0  2

c* c*

a*
b*
0

c *2
c*

and the final uncertainties and correlations are carried in the variance–covariance
matrix
U L*h*c*  S TL*h*c*U La*b*S L*h*c*
(46)
Hue and chroma are also calculated from (u*,v*) chromaticities; uncertainties in these
are found by substituting (u*,v*) for (a*,b*) in the above equations.
Saturation s* may be required in place of chroma; s*=c*/L* and it is a simple matter
to modify the sensitivity matrix S L*h*v* to accommodate this change.
8
UNCERTAINTY COMPONENTS IN COLOUR MEASUREMENT
For each independent component we calculate the transfer ratio at each wavelength
and hence the spectral uncertainties using Eq. (14) or (15). We then propagate those
uncertainties to the tristimulus values, and from those to the desired colour quantities
as a full variance–covariance matrix. The component matrices are then summed to
obtain the combined uncertainties and correlations. Such correlations are required for
estimating uncertainties of combinations of the colour quantities, such as in
determining colour-differences, correlated colour temperature and dominant
wavelength.
The tristimulus values, and hence their uncertainties, vary strongly throughout colour
space; it is not possible to provide accurate colour uncertainties as a single value
applicable over the whole of colour space.
NMI TR 8
10
8.1
Base Uncertainty of the Reference Standard
The spectral reference standard is likely to be measured at a limited number of
wavelengths, with some systematic errors in the process. The reference values then
will be at least partially correlated. In this case the full expressions of Eq. (1) or its
matrix form Eq. (7) must be used to calculate the tristimulus uncertainties and
correlations. If the correlation coefficient for the reference values is constant then this
calculation can be split into fully-correlated and uncorrelated parts and the simpler
expressions of Eqs (2) and (10) used to calculate the variance. At the highest levels of
accuracy, reference spectra uncertainties are dominated by systematic effects and they
are highly correlated.
8.2
Random Measurement Noise in the Transfer
Assuming that the monochromator efficiency and the source spectral power
distribution are significant at all wavelengths through the visible range, the
uncertainty in the transfer can generally be defined in terms of two components.
The first is a fixed fraction p of the transfer value, which can be related to source
noise. In the formalism of section 5Ошибка! Источник ссылки не найден., we
have at any wavelength, ti  pti' with p=1 (if it is not, a correction should be applied),
where ti' is the measured transfer ratio. Hence Si  pti ' SiRef  pSi ' , where S i' is the
measured spectral value, and u (ti )  ti u ( p) . From Eq. (13) we then have
(47)
us (Si )  Si u( p)
The second is fixed offsets in the sample and reference signal channels, which can be
related to electronic noise in the measurement – offsets due to scattered light are
S p
treated in section 8.6. Here ti  Refi
with p, pR  0 but u ( p), u ( pR )  0 . The
Si  pR
offsets being treated here are considered uncorrelated, specified as fractions of the
respective maximum sample or reference signal value. Hence
2
2
 1 
 S 
u (ti )   Ref  u 2 ( p)   Refi 2  u 2 ( pR )
 Si 
 ( Si ) 
2
(48)
and
2
 Si  2
(49)
us ( Si )  u ( p)   Ref
 u ( pR )
 Si 
As both these effects are uncorrelated between wavelengths we use Eq. (18) to
calculate the variance of the tristimulus values and Eq. (19) to calculate the
covariances.
2
8.3
Random Wavelength Accuracy in the Transfer
We assume that the same wavelength setting is used to measure the sample and
reference signals. If we have incorrectly set the wavelength by an amount p, the
transfer function becomes
NMI TR 8
11
ti 
Si  p
Srefi  p

1 Si 
Si  1 
p
Si  

1 Si
1 Siref 



t
'
1

p

p
i 
ref
ref
S


S





S
1
i
i


i
Siref 1  ref
p
 Si
 

(50)
to first order. We now have p  0 , u ( p )  0 and
 1 Si
1 S ref 
(51)
us ( Si ) 
 ref i  Siref u ( p)
 
 Si  Si
The derivatives are calculated numerically. As these values are uncorrelated between
wavelengths we use Eq. (18) to calculate the uncertainties and Eq. (19) to calculate
the covariance of the tristimulus values.
8.4
Wavelength Offset Error in the Transfer
This can arise if the spectral lamp(s) used for calibrating the wavelength scale have
different alignment to the broad-spectrum lamp used for measurement. The
uncertainty treatment for each spectral value is identical to that of the previous
section but now the uncertainties of different wavelengths are fully correlated. Hence
we use Eq. (26) to calculate the uncertainties and Eq. (27) to calculate the covariances
of the tristimulus values.
8.5
Constant Relative Error in Absolute Value of the Reference or the Transfer Ratio
These two effects are equivalent. They may rise from a number of instrumental
effects. In a double-beam spectrometer the beam alignment on the detector may be
different in the two internal paths, for example. Variation in the distance setting of the
reference and sample sources may also produce such an effect. The magnitude of this
effect can be estimated by reversing the reference and sample positions. (A better
estimate of the relevant spectral value is then made by taking the geometric mean of
the two sets of readings.) Whether the effect is in the reference or measured spectrum,
we have
Si  pSi'
(52)
where Si' is the uncorrected value. We have p  1 (if it is not a correction must be
applied) and we estimate its uncertainty u(p). This effect is fully correlated between
wavelengths and we use Eq. (26) to calculate the uncertainties and Eq. (27) to
calculate the covariance of the tristimulus values. In section 7.1 we note the
consequence of such an uncertainty on (x,y) chromaticity values.
8.6
Constant Offset in the Reference or Sample Channels
These two effects are not equivalent. Scattered light in either reference or sample
beam can cause such an offset, as can electronic offset in an amplifier. For an offset
S p
in the sample channel, ti  i Ref and
Si
u ( Si )  SiRef u (ti )  u ( p)
S
For an offset in the reference beam we have ti  Ref i
, hence
Si  p
NMI TR 8
(53)
12
u (ti )  
Si
S 
Ref
i
2
u ( p ) and
Si
u ( p)
(54)
SiRef
Note that if the reference spectral value is constant, a common occurrence in
reflectance spectroscopy, and the source intensity is approximately constant, an
uncertainty of the offset in the reference spectrum is equivalent to an uncertainty in
the relative value of the transfer or reference value, discussed in the last section. The
conclusions of section 7.1 then also apply.
u ( Si )  ti u ( SiRef )  
The effects are fully correlated between wavelengths and we use Eq. (26) to calculate
the uncertainties and Eq. (27) to calculate the covariance of the tristimulus values.
An offset that is common to both channels is correlated. An example is incorrect
background subtraction of room light reaching a detector from outside the signal
paths. In such a case the two effects should be treated together, with a correlation
coefficient of –1.
8.7
Scaling of the Reference Value or Transfer Ratio Linear with Wavelength
Such an effect may be due to an alignment that shifts with wavelength, or a reference
standard which ages at different rates at different wavelengths. If we assume that
known effects have been corrected and that the uncertainty of the correction is linear
with wavelength, we have Si  Si 1  p(i  1 )  , where p  0 is the fractional
differential error, and
(55)
us (Si )  (i  1 ) Siu( p)
Again we use Eqs (26) and (27) to calculate the tristimulus variance–covariance
matrix. The resultant uncertainties (in all except (x,y) or (u,v) chromaticities) depend
on the wavelength chosen for the reference. This should be where the true values are
known for the reference or the transfer ratio.
8.8
Source Noise
Source noise is treated through random and correlated effects in the transfer. The
treatment depends on the time scale of the fluctuations, and is generally different for
source and reflectance colorimetry.
Reflectance colorimetry usually is carried out in a double-beam instrument, with
relatively rapid switching between sample and reference paths. In the visible range the
source is usually a thermal one, with relatively slow fluctuations; hence source
fluctuations are likely to correlate the transfer between sample and reference
measurements, but not the transfer ratio between wavelengths. Short-term source noise is
likely to be important if a discharge lamp (e.g. a deuterium source) is used, leading to
increased random fluctuations in the transfer.
Long-term drift may be important for source colorimetry, because the reference and
sample beams are usually measured sequentially. Long-term drift or fluctuation leads
to correlations between spectral values at different wavelengths, if spectra are
recorded on a time scale short compared with the rate of the fluctuation. These
correlations can be measured experimentally by recording a number of repeat spectra
and calculating the correlation between values.
NMI TR 8
13
Correlation between wavelengths may also occur if the gain of a detector changes
randomly but slowly; such effects have been seen in a scanning system where the
detector was a photomultiplier driven from a DC–DC converter and a power supply
of only moderate stability.
8.9
Bandwidth Effects
Unresolved or poorly resolved spectral shapes can affect the calculated values of
colour coordinates – this is important in measuring LED colour parameters, for
example. Such effects are systematic, but not covered here. We take the reference
value as the one applicable for the resolution being used in our measurement. In most
cases the spectral distribution of the reference is relatively smooth and bandwidth
effects are small. For reflectance colorimetry we also take the source spectral power
distribution to be constant over the bandwidth and consider any fluctuation effects to
have been treated in the transfer uncertainties. The measurement equation is
 R S d  Ref

Si 
Si
(56)
Ref
 R S d 

where R is the spectral response function of the instrument and the integration is
over the bandwidth. If we also consider the reported spectral value for the sample to
be one applicable for the resolution being used, that is, an average over the bandwidth
of the measurement, this becomes
Si  R d 
Si  Ref
SiRef  ti SiRef
(57)
Si  R d 

as in Eq. (13). Bandwidth effects are then treated as correction factors and not
uncertainty components. Ignoring the bandwidth effects can lead to correction factors
that are larger than the propagated uncertainties – it is important to state the
measurement conditions when reporting results.
8.10 Sample Uniformity
This component is estimated from repeat measurements on a given sample.
8.11 Multiple Transfers from the Reference
The spectral result of one measurement may be the reference for a subsequent one, as
we progress from base standards through various levels of working standards. Here
the base reference spectrum may be used to calculate uncertainties provided the
transfer uncertainty components are combined for all the transfers. The usual rules for
such combination apply; if the uncertainties for a given component are correlated, the
combined transfer uncertainty is found by sum-of-squares, Eq. (2). If the effects are
correlated, they are added linearly, Eq. (10). Care must be taken here. For example, if
the wavelength scale was recalibrated between transfers, and the lamp repositioned,
the offset applicable for the second transfer is not correlated to that of the first,
whereas it is if the system is undisturbed between transfers.
NMI TR 8
14
9
REPRESENTATIVE EXAMPLES FOR SURFACE COLOUR UNCERTAINTIES
9.1
Typical Uncertainty Components
Uncertainty components for a double-beam reflectance measurement are shown in
Table 2. These are chosen to be large enough to show relativity between the
components, yet be representative of a routine measurement. The reference
reflectance value is taken to be unity and strongly correlated between wavelengths.
Scattered light can produce reasonable offsets in either the reference or sample
beams, and these components have been set to 0.1% of the maximum signals, taken to
be independent effects. An uncertainty of 1% in the transfer absolute value can be
caused by misalignment of the sample and reference beams, or a non-uniform
detector response if there is a lateral or angular shift of the beams at the detector. The
random wavelength setting of 0.03 nm can be achieved by careful calibration using a
number of spectral lines. The wavelength offset of 0.1 nm represents 1/50 of the
typical 5 nm bandwidth used for spectral measurements.
Table 2. Uncertainty components used for representative calculations for
surfaces (standard uncertainties); the component number
identifies the uncertainty source for the subsequent plots
Component
Identification
Uncertainty
1
Base standard
0.5% of the value
Correlation coefficient 0.8
20 points used
2
Reference slope uncertainty
1% over 400 nm
3
Reference offset uncertainty
0.1% of the maximum
4
Transfer random uncertainty (relative)
1%
5
Transfer random uncertainty (signal offsets) 0.1% of the signal maxima
6
Transfer scaling factor uncertainty
1.0%
7
Spectrum signal offset uncertainty
0.1% of the maximum
8
Wavelength random uncertainty (nm)
0.03 nm
9
Wavelength offset uncertainty (nm)
0.1 nm
10
Sum of all components
All sample spectra were calculated over a 5 nm grid from 360 nm to 830 nm, using
the 10 colour-matching functions and Illuminant D65. A grey sample surface was
simulated as one with a uniform reflectance of 50%. A green surface with broad
spectral distribution was simulated by the V( distribution scaled by 0.9 on a 10%
grey background; red and blue surfaces were simulated by shifting the V(
distribution by 75 nm and –150 nm, respectively, both scaled by 0.9 and on a 10%
grey background. Examples of surface colour uncertainty estimates are also given in
[4].
NMI TR 8
15
9.2
(x,y,Y) by Component
Grey surface: x = 0.3138, y = 0.3310, Y = 50.00
Grey
0.0005
0.6
u(x)
u(y)
0.5
Uncertainty
Uncertainty
0.0004
Grey
0.0003
0.0002
0.0001
u(Y)
0.4
0.3
0.2
0.1
0.0
0.0000
1
2
3
4
5
6
7
8
9
1
10
2
3
4
5
6
7
8
9
10
Component
Component
Red surface: x = 0.5169, y = 0.3689, Y = 34.91
0.0008
Red
Red
0.4
u(Y)
Uncertainty
Uncertainty
0.0006
u(x)
u(y)
0.0004
0.0002
0.3
0.2
0.1
0.0000
0.0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
Component
5
6
7
8
9
10
Component
Green surface: x = 0.3880, y = 0.4924, Y = 71.72
0.0004
Uncertainty
Green
Green
0.8
u(Y)
u(x)
u(y)
Uncertainty
0.0005
0.0003
0.0002
0.6
0.4
0.2
0.0001
0.0
0.0000
1
2
3
4
5
6
Component
NMI TR 8
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Component
16
Blue surface: x = 0.1963, y = 0.1430, Y = 14.83
0.0008
0.20
Blue
u(x)
u(y)
0.15
Uncertainty
Uncertainty
0.0006
Blue
0.0004
u(Y)
0.10
0.05
0.0002
0.00
0.0000
1
2
3
4
5
6
7
8
9
1
10
2
3
4
5
6
7
8
9
10
Component
Component
For the routine uncertainty components chosen, the uncertainty of the reference scale
is relatively minor in all cases. In part this is due to the high level of correlation
between reference values. If the correlation coefficient is reduced from 0.8 to 0.3, the
reference uncertainty component approximately doubles in value. The effect of
wavelength errors varies significantly with the surface colour. For the grey surface,
wavelength errors are zero, as expected. The effect of random wavelength error is
negligible, but wavelength offset is highly significant for the green surface, less so for
red and blue, and the effect is quite different for the x and y chromaticities – the effect
is strongest where the overlap between the spectrum and colour-matching function is
strongest, i.e. where the colour-coordinate itself is strongest.
Offsets in the signals have a significant effect on chromaticity. Here the reference
spectrum is constant and we have considered the source strength to be constant, and
so it is the offset in the sample signal that is important. The effect of the offset
uncertainty is more pronounced as the 10% general background set for the
calculations above is reduced, particularly for the red and blue surfaces; the effect of
the offset (scaled from the maximum signal value) is more significant if the
reflectance is small for much of the visible range.
Scaling of the transfer value doesn’t affect the chromaticity, but is highly significant
for the luminance.
9.3
(L*a*b*) by Component
Grey surface: L* = 76.07, a* = 0, b* = 0
Grey
0.35
a*
b*
0.15
Grey
0.30
Uncertainty
Uncertainty
0.20
0.10
0.05
0.25
L*
0.20
0.15
0.10
0.05
0.00
0.00
1
2
3
4
5
6
7
Component
NMI TR 8
8
9
10
1
2
3
4
5
6
7
8
9
10
Component
17
Red surface: L* = 65.68, a* = 48.97, b* = 47.75
0.4
Red
Red
0.3
a*
b*
Uncertainty
Uncertainty
0.3
0.2
L*
0.2
0.1
0.1
0.0
0.0
1
2
3
4
5
6
7
8
9
1
10
2
3
4
5
6
7
8
9
10
Component
Component
Green surface: L* = 87.83, a* = –26.77, b* = 69.92
0.4
Green
0.4
Uncertainty
Uncertainty
L*
a*
b*
0.3
Green
0.2
0.1
0.3
0.2
0.1
0.0
0.0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
8
9
10
Component
Component
Blue surface: L* = 45.41, a* = 34.73, b* = –66.36
0.3
Blue
a*
b*
Uncertainty
Uncertainty
0.3
Blue
0.2
0.1
0.0
1
2
3
4
5
6
7
Component
8
9
10
0.2
L*
0.1
0.0
1
2
3
4
5
6
7
Component
In contrast to the (x,y) chromaticities, the uncertainty components for (a*,b*) are
more even in their effect – no component particularly dominates. Wavelength effects
are generally small, except for the green surface, where the signal is concentrated
more towards the central part of the visible range and the overlap with the colourmatching functions is larger. Scaling uncertainties affect both a* and b* through the
non-linear dependence on the Y tristimulus value.
The uncertainty in L* is important in determining the accuracy of colour differencing.
NMI TR 8
18
9.4
Combined Uncertainty for the Various Surface Colour Quantities
Blue
Blue
y
x
Green
Red
Red
Grey
Grey
0.0000
v'
u'
Green
0.0005
0.0010
0.0000
0.0005
Uncertainty
Blue
Blue
Green
Green
Red
Red
b*
a*
Grey
0.0
0.1
0.2
0.3
0.4
v*
u*
Grey
0.5
0.0
0.2
Uncertainty
Blue
Green
Green
h*(u*v*)
h*(a*b*)
0.005
0.8
c*(u*v*)
c*(a*b*)
Grey
Uncertainty
0.6
Red
Grey
0.000
0.4
Uncertainty
Blue
Red
0.0010
Uncertainty
0.010
0.0
0.2
0.4
0.6
0.8
Uncertainty
The uncertainty in chromaticities for all components combined is most evenly
distributed over the different colours for (a*,b*), although even here the difference
between uncertainty in a* and that in b* can be significant. Note that for a grey
surface, hue is infinite and its uncertainty has no meaning.
The main conclusions to be drawn for surface colour uncertainties are:

Uncertainty in the base reference reflectance spectrum is relatively unimportant
for routine measurements. It becomes a significant component for
measurements of the highest accuracy, close to the primary standards.

Uncertainties can vary strongly with colour, and may be significantly different
for the members of a chromaticity pair. They should be estimated for each
measurement.
NMI TR 8
19
10
METHODS FOR SOURCE PARAMETERS AND UNCERTAINTIES
10.1 Dominant Wavelength
Dominant wavelength of a source is given as that of the monochromatic locus on a
1931 (x,y) chromaticity diagram at its intersection by a line passing through the (x,y)
chromaticity of the source and that of the equal-energy source (xE = yE = 0.3333). By
equating gradients of lines of the point on the locus (xm,ym) and that of the source, we
have
ym  y E y  y E

(58)
xm  xE x  xE
and we can find the point on the monochromatic locus by interpolating and searching
on a grid. However, the gradients are discontinuous for x  xE (which occurs
between 550 nm and 555 nm for the monochromatic locus) and it is better to search
for a minimum in the value of  ym  yE  x  xE    y  yE  xm  xE  . This is carried
out on a relatively coarse regular grid, typically 5 nm using the tabulated colourmatching functions x , y , z to calculate (xm,ym) at each grid point. To avoid finding
the complementary wavelength, the search end is limited to 560 nm if xm  xE and is
begun at 545 nm if xm  xE . Once the minimum is found, we fit xm,ym as quadratic
functions, using the minimum point and the point either side. Using xm as an example,
we have
xm  a  bd  cd 2
(59)
where d is the wavelength referenced to that at the grid minimum point and scaled by
the wavelength interval. Hence the values of d for the three points are –1,0 and +1
and it is a simple matter to find the coefficients a, b and c. We repeat this process to
find a similar fit for ym.
We now use Newton–Raphson iteration to refine the grid-minimum dominant
wavelength to a more accurate one. We have
(60)
f   ym  yE  x  xE    y  yE  xm  xE 
dy
dx
df
  x  xE  m   y  y E  m
dd
dd
dd
(61)
with
dxm
 b  2cd
dd
and a similar expression for the ym derivative from its quadratic fit.
Beginning with d0=0, an improved value of d is found as d  d 0 
(62)
f
; repeated
df dd
iteration rapidly finds d to the required accuracy. The dominant wavelength is then
found by adding d  to the wavelength of the grid-search minimum, where  is
the wavelength increment during the grid search.
The sensitivity coefficients required to calculate the uncertainty of the dominant
wavelength are obtained from Eq (62) and its ym analogue at the final value of
dominant wavelength as
d
 dx 
   m 
x
 dd 
NMI TR 8
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(63)
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The uncertainty of the dominant wavelength is given by
2
  
 
  
(64)
u (d )   d  u 2 ( x)   d  u 2 ( y )  2 d d u ( x, y )
x y
 x 
 y 
Note that for near-monochromatic sources in the long-wavelength region where the Z
tristimulus value is zero, x  y  1 and the sensitivity coefficients have the same
magnitude but opposite sign; for fully, positively correlated components the
covariance term in Eq. (64) effectively cancels the first two terms and the uncertainty
in dominant wavelength is small.
2
2
10.2 Source Correlated Colour Temperature
Given (u,v), correlated colour temperature is defined as the blackbody temperature
which minimises the distance d in (u,v) space from the blackbody locus (uT,vT), where
d 2  (uT  u ) 2  (vT  v) 2
(65)
At the minimum,
du
dv
f  (uT  u ) T  (vT  v) T  0
(66)
dT
dT
Although this expression is strictly true only at the minimum, we can use Newton–
Raphson iteration to solve for T. The fitting is much more reliable in (u,v) space
rather than directly in terms of temperature. The function f is then modified to
dv
g  (uT  u )  (vT  v) T  0
(67)
duT
Given an initial estimate (T = 3000 K for practical cases encountered in photometry),
we calculate uT , vT and their derivatives with respect to T. Then an improved value of
T is given as
g
T T 
duT dT
(68)
dg du T
where
d 2 vT  dvT 
dg
 1  (vT  v)


duT
duT 2  duT 
2
(69)
with
dvT dvT duT d 2 vT d 2vT d 2uT

,

duT dT dT
duT 2 dT 2 dT 2
Repeated iteration until the steps are below the required accuracy, yields the
correlated colour temperature. This method is described in [5].
(70)
The uncertainty in CCT is given by
T T
 T  2
 T  2
u 2 (T )  
u (u )u (v)
(71)
 u (u )  
 u (v)  2ruv
u v
 u 
 v 
At the minimum, f = 0 and implicit differentiation [6] yields the required sensitivity
coefficients
T
f f T
f f


,
(72)
u T v
v T
u
2
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where
2
d 2uT  duT 
d 2vT  dvT 
f
 (ut  u )


(
v

v
)

t



T
dT 2  dT 
dT 2  dT 
2
(73)
and
du
dv
f
f
 T ,
 T
(74)
u
dT
v
dT
The procedure to calculate the derivatives is as follows. Given the colour-matching
functions xi , yi , zi at air wavelengths i , and the relative spectral radiance on the
blackbody locus given by Planck’s law
1
(75)
Pi 
c2
i5 (e n  T  1)
a i
(where na is the refractive index of air), we have chromaticity values
4 xi Pi
6 yi Pi
, vT 
uT 
 ti Pi
 ti Pi
(76)
where ti  xi  15 yi  3zi .
Hence
duT

dT
4 xi
Pi
P
P
P
 uT  ti i
6 yi i  vT  ti i
dv
T
T , T 
T
T
t
P
dT
t
P
i i
i i
(77)
and
 2 Pi
 2 Pi
u
4 xi
 uT  ti
2 T
2
2
d 2uT
T
T
T

dT 2
t
P
i i
t
i
Pi
T
(78)
 2 Pi
 2 Pi
Pi
vT
6 yi
 vT  ti
2
 ti T
d 2 vT
T 2
T 2
T

dT 2
 ti Pi
The derivatives of the Planck function are
Pi
c2 Pi
(79)

c
 2
T
na iT 2 (1  e na iT )
and



 Pi
 2 Pi
c2
c2
1

(80)

 Pi  

c2
c2
2



T

T
T

na iT 2 (1  e na iT ) 
na iT 2 (1  e na iT )  

To be consistent with the CIE Illuminant values, we must use the value of c2 as
1.4388  10–2 m.K and ignore the correction for the refractive index of air.
10.3 Source Distribution Temperature
Distribution temperature is defined as the value of T that minimises the integral
2

S ( ) 
 1  aP(t  , T )  d 
NMI TR 8
(81)
22
where St ( ) is the spectral irradiance, and wavelengths are taken in the range 400 to
750 nm. CIE in its definition give the value of c2 as 1.4388  10–2 m.K and do not
clearly define whether air- or vacuum-wavelengths are used. The current value of c2
is 1.4387752  10–2 m.K, and we can take na = 1.00029 to convert from the airwavelengths at which measurements are made to the vacuum wavelengths needed for
the calculation of photon energies assumed in the derivation of Planck’s law.
However to be consistent with the CIE Illuminant values, we must use the value of c2
as 1.4388  10–2 m.K and ignore the correction for the refractive index of air.
Distribution temperature only has meaning for sources close to the Planckian locus.
Woeger [7] demonstrated a general method for least-squares fitting and uncertainty
estimation, based on a generalised Newton–Raphson technique [8]. Determination of
distribution temperature was used as an example. In that example, he considered a
weighted fit, where the weighting included correlations in the spectral points. The
strict definition of distribution temperature requires an unweighted fit; the method
below follows Woeger, but removes the weighting and includes the constant 1 in the
model definition.
If the spectral data are present at regular intervals, the integral can be written in sum
form over the N spectral points
2
 aS n

  
 1    f n 2
(82)
 Pn (T ) 
where the normalisation constant a has been moved to the numerator for convenience.
Note
If the data are not present at regular intervals, the separation between points
must be included in the summation – distribution temperature is not found by
merely fitting a Planckian form to the spectral energy distribution.
Minimisation with respect to a and T leads to the set of two equations
(83)
F  M y TM  0
where
 f1 f1 
 a T 


 f 2 f 2 
M y   a T 
(84)




 f n f n 


 a T 
is a N  2 matrix of the derivatives of the function with respect to the parameters
being fitted and
 f1 
 
f
M  2
(85)
 
 
 fn 
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is a N  1 matrix containing the model form being fitted. The derivatives are
P
 Sn n
f n Sn f n
T

,
(86)

2
a Pn T
Pn
P
where Pn , n are given above in (75) and (79), respectively.
T
The generalised Newton–Raphson technique is an iterative procedure to solve
Eq. (83) by expanding the equations F in a Taylor series about the initial point,
retaining only first-order derivatives, then setting the value at the new point (a better
approximation to the solution) to zero; that is
(87)
F(y   y)  F(y 0 )  J. y  0
Here J is the Jacobian matrix
 F 
J  i 
(88)
 y j 


where i =1,2 for our two equation rows and j=1,2 for our two parameters. Hence
 
f   f f 
J 
fn n     n n 
(89)

 y j n
yi   n y j yi 

where only first-order terms are retained, or
(90)
J  M y TM y
a
It follows that, given the parameter matrix with some initial estimates, y 0    ,
T 
improved estimates are given by
y  y 0  (M y TM y )1 F(y 0 )
 y0  (M y TM y )1 M y TM
This process is iterated until a required accuracy is reached.
(91)
To calculate the uncertainties in the parameters a and T (input parameters y) from
those in the spectral values Si (input parameters x) we need the sensitivity coefficients
f
f a f i T
a
T

and
. Given that i  i
, the sensitivity coefficients are
S i
S i
Si a Si T Si
carried in the matrix Q where M x  M y Q and
 f1
 S
 1

 0
Mx  


 0


NMI TR 8
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f 2
S 2
0

0 


0 



f n 
S n 
(92)
24
is a diagonal sensitivity matrix with
f n
a
 . Hence Q  My1M x where M y1 is the
S n Pn
non-unique generalised inverse of M y (= M y T  M y M y T  ). The uncertainty matrix
1
for the parameters y is given by
U y  QU x Q T
 M y1M x U x (M y1M x )T
 M y1M x U x M x TM y1T
(93)
 M y1KM y1T
The inverse is given by
U y 1   M y1KM y1T 
1
 M y T K 1M y
(94)
where K is square and so we can calculate its inverse, form the last product and take
its inverse to get the covariance matrix between the fit parameters.
Distribution temperature is calculated using spectral values only in the wavelength
range 400 to 750 nm. For a given set of input spectral values and their correlations,
only the subset in the required spectral range is used for this calculation.
Note Added in Revision 2.8
The variance–covariance matrix for highly correlated data is ill-conditioned to the
point where the inverse matrices in Eq. (94) may not be calculated accurately in a
routine procedure, particularly for large numbers of spectral data points. Also the
inverse  M y M y T 
1
required for the sensitivity coefficients can not be reliably
calculated. However, we can directly calculate sensitivity coefficients for the
dependence of distribution temperature on the spectral value at each wavelength. The
generalised Newton–Raphson method rapidly converges. Using the calculated value
of distribution temperature found for the true spectrum, we increment each spectral
value in turn by 1%, then recalculate the distribution temperature and hence the
sensitivity coefficient as the change in distribution temperature divided by the change
in spectral value. We can then propagate both random and systematic uncertainty
components in the spectral value as quadratic and linear sums, respectively. Note that
the linear sum may be negative because it contains the sign of correlation. For
example, a background level added to blackbody curve reduces the distribution
temperature.
11
REPRESENTATIVE EXAMPLES FOR SOURCE COLOUR UNCERTAINTIES
11.1 Typical Uncertainty Components
The reference for colorimetry of broad-band sources is traced to spectral irradiance or
radiance standards, usually derived by estimating the temperature of a blackbody. For
the calculations here, a blackbody at 2800 K is assumed, with a temperature
uncertainty of 1 K. All values of spectal irradiance are assumed fully correlated. The
remaining components were set as in Table 2, as representing a routine measurement.
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11.2 Broadband Sources
Representative calculations were made for a blackbody at 2856 K (CIE Illuminant A),
one at 5500 K and for the distribution CIE Illuminant D.
11.2.1
(x,y) by Component
0.0010
0.0010
Illuminant D
0.0008
x
y
0.0004
Uncertainty
Uncertainty
x
y
0.0006
0.0005
0.0002
x
y
0.0006
0.0004
0.0002
0.0000
0.0000
1
2
3
4
5
6
7
8
9
0.0000
10
1
2
3
4
Component
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
Component
Component
Again the contribution of the reference is small – systematic effects in the
measurement process dominate. The effect of offsets in the transfer (component 7) is
significant for Illuminant A, because its spectral power is low at the shorter
wavelengths. Similarly, an offset in the 2800 K reference (component 3) is significant
at the higher source temperatures. Random errors are also significant in all cases.
Because the distributions are relatively smooth, wavelength errors have small effect.
Scaling errors (component 6) do not affect (x,y) chromaticity.
11.2.2
(a*,b*) by Component
0.15
5500K blackbody
Illuminant D
a*
b*
Illuminant A
a*
b*
0.2
a*
b*
0.4
0.10
Uncertainty
Uncertainty
0.4
Uncertainty
Uncertainty
0.0008
0.0010
5500K blackbody
Illuminant A
0.05
0.2
0.00
1
0.0
1
2
3
4
5
6
7
Component
8
9
10
2
3
4
5
6
7
Component
8
9
10
0.0
1
2
3
4
5
6
7
8
9
Component
Similar comments to those for uncertainty in (x,y) apply. Differences between the
uncertainties for the chromaticity pair are more marked than for (a*,b*) than for (x,y). An
important point to note is that uncertainties in a* and b* are dependent on the luminance
of the source. The distributions for Illuminants A and D used for these calculations are
those of the CIE tabulations, scaled to have a common luminance value and their L*
values are 100. The scaling of the 5500 K blackbody spectrum was such that its L* value
was 13.9 and the (a*,b*) uncertainties are smaller than for the other sources.
While L*a*b*, and the other colour parameters normalised to the reference
illuminant, provide better agreement with perception than the simple (x,y) or (u,v)
chromaticities, they have little meaning based on the spectrum alone, as the values
(and their uncertainties) depend on the luminance. They are useful quantities for
displays, less so for general sources.
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10
10
11.2.3
Source Temperatures by Component
50
12
Illuminant A
6
4
2
5500K blackbody
DT
CCT
30
20
1
2
3
4
5
6
7
8
9
10
Component
20
10
10
0
0
0
DT
CCT
30
Uncertainty /K
DT
CCT
8
40
Illuminant D
40
Uncertainty /K
Uncertainty /K
10
1
2
3
4
5
6
7
Component
8
9
10
1
2
3
4
5
6
7
8
9
10
Component
As the temperature of the source rises away from that of the reference 2800 K, the
systematic offset in the reference signal dominates the uncertainty. Contribution from
the base reference uncertainty is small. The Planckian distribution is non-linear, and
offsets in the transfer are significant. Distribution temperature is only meaningful for
sources whose spectral distribution is approximately Planckian; this is not true for
CIE Illuminant D. Offset in the transfer is significant in all cases – elimination of
stray light is important for accurate measurements. Systematic scaling factors
(component 6) have no affect, as expected. While the different sensitivity coefficients
(those for correlated colour temperature are weighted towards 555 nm) means that
different components have slightly different effects, the overall uncertainty is similar
for distribution temperature and correlated colour temperature.
11.3 Narrow-band Sources
The most important narrow-band sources are light emitting diodes. Coloured LEDs
have narrow-band emission spectra and their chromaticities are close to the
monochromatic locus. Bandwidth for these sources is typically 35 nm (full width at
half maximum) and a wavelength resolution of order 1 nm is required for the
measurement of the spectrum; the typical 5 nm used for broad-band surfaces or
sources leads to significant errors. Hence calculations were made using the
components listed in Table 2, except that the reference was the same 2800 K
blackbody with 1 K uncertainty used for broadband sources and the wavelength offset
uncertainty was reduced to 0.02 nm. A Gaussian model with a sharpened peak [9] and
35 nm bandwidth was used to calculate LED spectra with peak wavelengths of
470 nm (blue), 524 nm (green), 585 nm (amber) and 605 nm (red).
NMI TR 8
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11.3.1
(x,y) by Component
0.0010
0.0010
LED y
LED x
0.0008
470 nm
524 nm
585 nm
605 nm
0.0006
Uncertainty
Uncertainty
0.0008
0.0004
0.0002
470 nm
524 nm
585 nm
605 nm
0.0006
0.0004
0.0002
0.0000
0.0000
1
2
3
4
5
6
7
8
9
10
1
2
Component
3
4
5
6
7
8
9
10
Component
Here x and y uncertainties are plotted on the same scale in different graphs grouped
by wavelength. For such narrow-band sources, the reference component uncertainty is
negligible. Offsets in the transfer (stray light constant across the spectrum, or
amplifier offset) dominate the uncertainty of both x and y. For the results shown here
the wavelength range was restricted to  2 times the bandwidth about the central
wavelength. If the range was extended to the full visible region, with the same offset
uncertainty scaled to the peak signal levels, the transfer offset uncertainties totally
dominate and the uncertainties for both x and y are of order 0.002, with that in y for
the green led approximately 0.005. Systems for routine measurements are often
programmed to record spectra for the full visible range, then perform a colour
calculation. This is not good LED measurement practice, particularly if signals are
weak (e.g. for fibre-coupled systems viewing a diffusing screen). It is far better
practice to delete from the calculation ranges where the LED signal is known to be
zero.
Wavelength /nm
605
585
LED dominant wavelength
524
470
0.0
0.1
0.2
0.3
0.4
Uncertainty
11.3.2
Dominant Wavelength
Uncertainty in dominant wavelength is calculated from that in (x,y). The increased
value in the green comes not so much from the larger uncertainties in x and y there as
the larger difference between them, because of the effectively complete correlation
between the uncertainties of x and y. The values shown here are increased by a factor
of 6 if measurements are made over the full visible wavelength range with the same
uncertainty components
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12
CONCLUSION
Systematic effects in the measurement system generally dominate the uncertainties of
colour quantities. The uncertainties depend on position in colour space and on the
reference spectrum and its uncertainties as well as the spectral power distribution of
the source. They should be calculated for each measurement of a colour quantity. If
we propagate uncertainties through those of the tristimulus values, the calculation of
the systematic uncertainty components involves single sum expressions and not a full
matrix multiplication with many terms. A simple 3x3 matrix calculation, or series of
such calculations, can then propagate the uncertainties to those of any desired colour
quantities.
Uncertainties of colour quantities for a given effect scale with that of the effect. The
representative examples given here can be used as a guide to uncertainties to be
expected from given experimental conditions, in the different regions of colour space.
13
REFERENCES
[1]
Guide to the Expression of Uncertainty in Measurement (1993) International
Organisation for Standardisation
Colorimetry (1970) Publication CIE 15, International Commission on
Illumination, Vienna
JL Gardner (2003) Uncertainties in Interpolated Spectral Data J. Res. Natl Inst.
Stand. Technol. 108, 69–78
EA Early and ME Nadal (2004) Uncertainty Analysis of Reflectance
Colorimetry, Color Research and Application 29, 205-216
JL Gardner (2000) Correlated Colour Temperature – Uncertainty and
Estimation Metrologia 37, 381–384
J Fontecha, J Campos, A Corrons, A Pons (2002) An Analytical Method for
Estimating the Correlated Colour Temperature Uncertainty Metrologia 39, 531–
536
W Woeger (2001) Uncertainties in Models with more than One Output
Quantity, Proceedings of the CIE Expert Symposium, pp12–17, International
Commission on Illumination, Vienna
WH Press et al. (1988) Numerical Methods in C: The Art of Scientific
Computing, p 381, Cambridge University Press
Y Ohno (personal communication)
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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