Crossover Wavelengths of Natural Metamers
Graham D. Finlayson
Peter M. Morovic
School of Information Systems
University of East Anglia
Norwich NR4 7TJ, United Kingdom
Abstract
For a xed illuminant and observer there is a whole set of reectances resulting in identical response, these reectances are called metamers. It can be shown analytically that all reectances in each such
set must intersect at least three times.
There has been a large amount of literature arguing about the
properties of these sets, in particular about the position and number
of nodes of intersections. The results in literature vary in particular as
a consequence of dierent methods used for generating metamers.
Using a new method based on statistical information from measured
sets, metamers are generated and their nodes of intersection studied.
The results presented here conrm the result of there being three major
wavelengths of intersection. These are around 450nm, 540nm and
610nm.
1
Introduction
Two physical stimuli that induce the same visual response in an observer
are called metamers. Metamerism is obviously extremely important e.g. it
allows one to reproduce a wide range of colours on a monitor using just 3
primaries even though the spectral power distribution of the primary mixtures is usually far from the natural colour signal that is being matched.
That a monitor suÆces to reproduce a wide range of colours actually draws
attention to an important fact: the spectral shape of metamers can be fairly
arbitrary.
However, in nature one nds that metamers are not arbitrary but rather
are very ordered. In particular, many authors have noted that almost all
metamers tend to have three \prime" crossover points located at around
450nm, 540nm and 610nm. In an attempt to understand this circumstance
various statistical studies have been undertaken.
Thornton [1], generated metamer pairs using a basis (linear combinations) of relatively broad gaussian reectance functions. The result is smooth
Appeared in Proceedings of Colour 2000 Conference, Derby, UK
2
reectances which qualitatively look like real reectances. For each metamer
pair he noted the location of crossover points. He observed cross-over points
at 450nm, 540nm and 610nm.
In other studies (starting with Ohta and Wyszecki [2]) numerical methods are used to create metamers that do not have these prime crossover
points. Unfortunately, qualitatively these spectra tend to look unnatural
(they are generally not smooth). Thus, the cross-over debate is split into two
camps. One which observes that natural metamers (or smooth metamers)
have prime crossovers and the other which contends that prime crossovers
are not in fact necessary. In this paper we present a statistical argument
which shows that natural metamers generally have prime crossovers and in
this sense prime crossovers are necessary.
Our result is based on a new mathematical technique for metamer generation; one which generates only those metamers which are statistically
plausible (and physically realisable)[3]. In our method characteristic vector
analysis (CVA) is used to generate a set of basis functions; linear combinations of which suÆce to model a set of real reectances. It follows that
reectances can be represented by their coeÆcients with respect to the CVA
basis. The set of coeÆcients is necessarily convex [4] and the convex hull of
the coeÆcient set bounds, in some sense, on the shape of all natural reectances. In particular, we say that any spectra in the convex set can plausibly
appear in nature. Each point inside the convex hull is a convex sum of a nite set of reectances. We can synthesise any metamer by colouring surfaces
from within this reectance set such that the area they occupy is proportionate to the coeÆcients in the convex sum. Viewed from far enough, this
collage has the desired spectral reectance. In contrast, coeÆcients which
lie outside the plausible set cannot be generated in this way.
Now, given any XYZ (under any illuminant) we can solve for the corresponding set of all metamers. We simply nd the (necessarily) convex subset
of all reectances that, under the same illuminant, which projects onto the
same XYZ. It is then a simple matter to calculate the cross over points for
all metamer pairs belonging to this set. Like Thornton we found these to
be 450nm, 540nm and 610nm.
Section 2 describes the formation of colour and the phenomenon of metamerism, section 3 explores the properties of metameric black reectances.
Section 4 presents the new method for generating natural metamers and
section 5 presents the crossover statistics from experiments conducted. Finally section 6 presents a discussion of research in the area of crossover
wavelengths and concludes this study.
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3
Principles of Colour Formation
The phenomenon of colour is a function of illumination, object reectance
and the spectral sensitivity of our visual system (or of a color camera). The
illuminant is described by its spectral power distribution (spd) e() and the
object by its spectral reectance r() { both being continuous non-negative
functions of wavelength. The spd of the illuminant tells us how much energy
there is present at each individual wavelength and the spectral reectance
describes what percentage of the illuminant is reected at each wavelength
(therefore it is constrained to lie in the interval (0%; 100%)). Light reected
from an object form the colour signal, entering the visual system (the eye
of a human or the lens of a camera) and is integrated with three types of
spectral sensors result in the colour response. It is this colour response that
drives the whole of colour perception.
While the above interaction of illuminant spd, object reectance and
observer spectral sensitivity can be fairly complex, for a simplied world
of matte lambertian surfaces illuminated by a diuse light it can be well
modeled by.
Z
Z
Z
e
b
e
b
e
b
r()e()x()d
=X
r()e()y ()d
=Y
r()e()z ()d
=Z
(1)
where x(),y() and z () are the CIE 1931 Standard Colorimetric Observer
spectral sensitivities, X , Y and Z is the response of the observer (the tristimulus values ), i.e. the colour of the object whose spectral reectance is
r() as viewed under the illuminant whose spd is e() by an observer whose
spectral sensitivities are x(), y() and z(). The integral in eq. (1) is taken
from b to e which delimit the visible range, while this range varies from
study to study, usually b = 400nm and e = 700nm.
For simplicity all above continuous functions of wavelength are in practice represented by discrete vectors sampling the range b ! e in eqi-distant
steps instead. The dimension of these vectors again varies somewhat, though
a 10nm sampling of the range 400nm ! 700nm resulting in 31 samples is
thought suÆcient and is also the most widely used as well. Let e denote the
31 1 vector representing the illuminant spd, r a 31 1 vector representing
spectral reectance and x, y and z the 31 1 vectors recording the three
XYZ spectral sensitivities. These spectral sensitivities are grouped into the
31 3 matrix X. The continuous colour-equation eq. (1) can now be rewritten in terms of vectors where continuous integration is now replaced by
Appeared in Proceedings of Colour 2000 Conference, Derby, UK
discrete summation.
X r(i)e(i)x(i)
X r(i)e(i)y(i)
X r(i)e(i)z(i)
31
i=1
31
i=1
31
i=1
4
=X
=Y
=Z
(2)
Equation (2) can be further simplied to the following form:
rT D (e)X = xT
(3)
where x is a 3 1 vector of responses (x = [X; Y; Z ]) and the operator
D() maps elements of a 31 1 vector to the diagonal elements in a 31 31
diagonal matrix.
In this paper we wish to understand how sensor response depends on
light surface and sensor. In a trivial sense we have already arrived at this
understanding as eq. (3) informs us of the trichromatic response of a particular light incident on a particular surface. However, our interest is in
the reverse direction. Given knowledge of the tristimulus, what can we say
about the reectance.
Let us suppose that the right hand side (the tristimuli in x) and the
illuminant spectral power distribution is known, then eq. (3) represents an
under{determined set of equations (3 equations of 31 unknowns). Equation
(3) can in fact be solved but it cannot be solved uniquely. Rather, the 3
equations pin down only 3 of the 31 degrees of freedom in reectance (we
have 31 sample points). The other 28 degrees of freedom are unconstrained.
The result is an innite set of reectances all of which induce the same
tristimuli. That many reectances may induce the same tristimuli is well
known; the phenomenon is called metamerism. Two reectances inducing
the same tristimuli are called metamers. The whole gamut of all possible
metamers is called a metamer set [3].
Formally, two spectrally dierent reectances r and r are metamers
for illuminant e if:
rT D (e)X = rT D (e)X
(4)
Metamerism can be viewed as a plus and a minus. The plus is that
the same color response can be induced by spectrally dierent stimuli, the
spectral distributions produced by the monitor are far from those that occur
in nature. The minus is that sometimes we would actually like to know
the reectance e.g. so we could predict its appearance under a change of
illumination. Recent work has shown that even incomplete knowledge about
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1
0.9
0.8
reflectance (%)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
400
450
500
550
wavelength (nm)
600
650
700
Figure 1:
An example of three metameric reectances resulting in the same response [X; Y ; Z ] = [20; 20; 50] for CIE illuminant D65 and for CIE 1931 Standard
Colorimetric Observer.
metamerism is useful. In particular if we wish to map camera RGBs to XYZs
then solving for the metamer set (all possible metamers) for a particular
RGB gives us useful information for mapping that RGB to a corresponding
XYZ [3].
In this paper we wish to extend this previous work by looking deeper
at the metamer sets. Is there any structure that is common to all metamer
sets and if such a structure exists does this shed useful light on our understanding of colour response and does it help to solve practical problems. A
priori we are minded to believe that signicant structure exists. Many authors have established that typical metamers tend to crossover in particular
regions of the visible spectrum (around 450nm, 540nm and 610nm). Although it is easy to criticise these studies. They are based on small numbers
of reectances (often synthetically generated). In this paper we re-address
this \structure" problem. We make two technical contributions. First we
show how reectances can be interpolated to produce metamers. Basically,
any convex sum of real reectances can be used to synthesise a new natural
metamer. A convex sum can be viewed as a collage of reectances with
the area covered by each reectance being in proportion to the weight of
the convex sum. Viewed from far enough away this collage will generate
precisely the interpolated metamer. Thus, we are examining what happens
in terms of metamerism as an observer moves about a world where this
type of averaging is commonplace. Our second contribution is to calculate a
very large number of metamers for many tristimuli. The result is metamer
sets with many members; suÆcient members to make statistically meaningful comments about crossover wavelengths. Our ndings are that natural
metamers statistically cross-over around 450nm, 540nm and 610m.
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0.04
0.03
0.02
0.01
0
−0.01
−0.02
400
450
500
550
600
650
700
Figure 2:
An example of a metameric black for CIE illuminant D65 and for the
CIE 1931 Standard Colorimetric Observer.
3
Metameric Blacks
The question arises as to what it is that allows for metamerism and the key
to its answer is in the \invisible". Metameric blacks are reectances that
satisfy:
rTB D (e)X = 0T
(5)
where 0 is a 3 1 column vector of zeros. Note also that a pre-multiplication
of eq. (5) by a real constant would aect the values of the metameric black,
but it would not aect the response.
Studying metameric blacks it becomes clear that these are not \ordinary"
reectances in the sense that they necessarily have negative values (except
for the trivial metameric black which is a zero vector). In fact it is possible to
prove that under certain conditions, which apply for most cases in practice,
metameric blacks need change sign at least three times [5, 6].
One way of generating metameric blacks is to calculate the dierence
between the reectance spectra of two metamers r and r :
rT D (e)X rT D (e)X = 0T
(r r )T D (e)X = 0T
(6)
Given that a metameric black rB changes sign at least three times and
that it is the dierence between the two metamers r and r , it follows that
at those wavelengths , and (or indices in the vectors representing
them) where rB is zero, r and r are equal { they cross.
Many authors have set out to study the inner structure of metamer
sets, and in particular the wavelengths at which metamers tend to cross
(or wavelengths at which metameric blacks change sign) as well as their
number and frequency. These studies vary in particular in the method used
to generate metamers. We join the argument about crossover wavelengths
by presenting a new way to generate sets of natural metamers.
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2
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1
1
2
1
1
2
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2
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7
Natural Metamers
Given a tristimulus vector how can we generate the corresponding metamer
set? The answer to this question cannot be simply mathematical (though
mathematically the problem is straightforward to solve). Rather, we must
take into account the statistics of naturally occurring spectra (a monochromatic reectance exists in mathematics but not in nature). So, to start we,
in line with many previous studies [7, 8], aim to represent reectances using
a low dimensional space. Suppose we describe a reectance r as a weighted
sum of m basis functions bi : r = b + b + : : : + mbm. Given that the
basis functions are constant, reectances can be represented uniquely by the
weights.
Let us group the m basis functions bi into a 31 m matrix B. Reectances are dened relative to the basis: B = r where is an m 1 weight
vector of -s. Clearly for the best possible representation the aim is to minimise the error occurring when reectances are substituted by weights. Let
W be a m k matrix of weights corresponding to the 31 k matrix U of
representative reectances, then minimising the expression in eq. (7) yields
an optimal set of basis functions.
kU BWk2
(7)
For many de{saturated colours reectances are well enough described by
three basis functions. This is understandable as these colours have generally
smooth reectances. Let the three basis functions be grouped into the 31 3
matrix B . The relationship between tristimulus values and weights becomes
linear, and solving the colour equations under these conditions consists of
solving three linear equations of three unknowns:
(8)
xT = T BT D (e)X
The linearity of this relationship can be further expressed by a 3 3
matrix known as the lighting matrix = BT D (e)X [9, 10].
xT = T (9)
The variation present in a larger set of reectances, containing not only
de{saturated smooth samples but also reectances with high frequency components representing saturated colours, is not well enough modeled by three
basis functions. Often these colours when represented in the three dimensional basis set turn out to have negative values or values above one
{ they reect or absorb more light than was incident. Statistical studies
[11, 12, 13, 14] have shown that more basis functions are needed in order to
facilitate a good representation of more general reectance sets.
Let Bm be a set of m basis functions (3 < m 31) satisfying the
condition in eq. (7), derived using characteristic vector analysis from a
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1
2
2
3
3
3
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representative sample of measured reectances and let Xe = BTm D (e)X be
the m 3 lighting matrix for this m dimensional basis. As m > 3 the system
of linear equations we are faced with is no longer solvable uniquely { it is an
under-determined system of linear equations to which a whole set of possible
solutions correspond. For a given tristimulus value we end up with a set of
reectances, all being solutions to the same colour equations { forming the
metamer set we want.
Following principles of linear algebra [15], the solution () to such an
under-determined system of linear equations can be split up into two parts:
a particular solution x and an arbitrarily scalable corresponding black solution .
(10)
xT = xT Xe
0T = T eX
(11)
T
T
e
(12)
x = (x + ) X
One way to solve for the particular solution x is to solve for it in the
least squares sense:
xT [[Xe ]T eX ] eX [Xe ]T = xT
0
0
0
1
+
x [[ ] ] eX =
xT
x [ ] =
xT
T
e T e
X
X
1
+
(13)
where [[Xe ]T eX ] eX is the pseudo-inverse denoted by [eX ] [16]. This
means that we choose x to be a linear combination of the 3 columns of Xe .
For a given tristimulus vector x, x is constant for the whole metamer set.
The lighting matrix Xe , being an m3 matrix, spans a three dimensional
subspace of m space only { leaving m 3 degrees of freedom. These form
the basis to the black solution. Let be an m (m 3) matrix satisfying:
[ ]T Xe = 0 m m
(14)
&
T
det ([ ] [ ]) 6= 0
(15)
where 0m m is a m (m 3) zero matrix and det() is the determinant function. The condition in eq. (14) tells us that the space spanned
by columns of is orthogonal to the space spanned by columns of Xe ,
therefore making the columns of the -s in eq. (11), and the condition
in eq. (15) says that the columns of are linearly independent, therefore
forming a basis for the complementary space to Xe .
T
e
X
+
1
+
0
0
0
(
(
3)
)
0
3)
0
0
0
0
1
1
the basis of an m 3 dimensional subspace of m
independent m dimensional vectors
space is a set of m
3 linearly
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In order to nd we need to dene the projector of a n k (n k)
matrix A:
P (A) = A[AT A] AT
(16)
From the denition of the projector it follows that:
P (eX )Xe = eX
(17)
e
P (X ) = 0m m
(18)
(19)
(I P (eX )) = e
P ( ) = I P (X )
(20)
where I is a m m identity matrix.
Equation (20) is particularly important for us, as it shows that we can
obtain the projector of from the projector of eX . The columns of P ( )
span the same space as , but since P ( ) is m m and it spans a m 3
dimensional sub-space of m space, it necessarily contains linearly dependent vectors. It is enough now to select m 3 linearly independent vectors
from P ( ) in order to arrive at . These can be found, for example, by
choosing the rst m 3 principal components of P ( ). Now we can dene
the black solutions as:
= (21)
where is an arbitrary n 3 vector (containing the scalars i ).
Surface reectances must be non-negative (no less than no light is reected by a surface) and less than or equal to one (no more than all light
is reected). To guarantee this property we need to constrain the -s we
recovered earlier so as to result in reectances which satisfy this feasibility
constraint.
0 B 1
(22)
Another constraint we wish to impose on surface reectances is a constraint on the weights themselves. We can recover the weights for the set we
used to derive the basis for, and explore the contribution of each individual
basis function to the whole composite. We nd that i , the weight of the
i-th basis function, lies in an interval (imin ; imax ), therefore we make the
fourth basis equally important in the reectances we recover in our metamer
set. So we further constrain :
min max
(23)
where min and max are m dimensional column vectors of minimal and
maximal weights respectively.
We have described analytically the particular solution as well as the basis
to the black solution. As this set is unbounded (due to the scaling vector in eq. (21)) and our aim is to recover reectances close to \real", measured
0
1
0
(
0
3)
0
0
0
0
0
0
0
0
0
0
0
0
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reectances, we derived a set of constraints imposing properties found in
these natural sets. The fact that these constraints are formulated as linear
inequalities means that the set they describe is convex.
Furthermore, as the constraints are linear we may make use of linear
programming [17], a fast optimisation technique to nd the minima or maxima of a linear objective function subject to linear constraints. It is dened
formally as:
min cT (24)
subject to A b
(25)
where A is a k m matrix of the left side of the inequalities, is a m 1 column vector of the unknown weights dening the reectance, b is a
k 1 column vector of the right sides of the inequalities and c is a m 1
column vector dening the objective function (m is the dimension of surface
reectance).
The constraint in eq. (22) can be expressed as two inequalities for each
of the 31 wavelengths (62 inequalities per sample) and the constraints on the
weights in eq. (23) can be written as two inequalities per dimension of the
basis (2m inequalities per sample). Hence we have k = 62 + 2m inequalities
combined into matrix A. By denition all inequalities are of \less than or
equal to" type, but we need \larger than or equal to" as well. This can be
easily achieved by multiplying the appropriate rows of A and b by 1.
We want to solve for those weights which satisfy all constraints represented in A, that is we seek those weights which result in meaningful
reectances. In order to nd a lower and upper bound on each weight, the
objective function was chosen to minimise and maximise each dimension
of the weights, i.e. rst c has a 1 in the rst dimension and zeros elsewhere, than c has a 1 in the second dimension and zeros elsewhere, etc.
Consequently we perform m maximisations and minimisations resulting in
m intervals. These intervals give us an idea of what part of weight space is
spanned by the metamers { we can construct a m dimensional cube (the two
diagonal extreme vertices of which are the minimal and maximal weights)
in this space and consider it an upper-bound of the metamer set.
Clearly not every point in this cube is feasible. We need to get a better
idea of the metamer set and therefore we perform a quantisation of each
weight dimension { chopping up the cube into smaller cubes. Browsing
systematically through this mesh of weight vectors, we examine whether the
combination results in the same tristimulus we seek metamers for. If this is
the case we found a metamer and store it, if the weight vector results in a
dierent tristimulus then it is out of the boundary of the metamer set and
is of no interest. We arrive at a discrete convex set of weights, all of which
result in a reectance producing identical tristimuli { the metamer set.
11
Appeared in Proceedings of Colour 2000 Conference, Derby, UK
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
400
450
500
550
wavelength (nm)
120
600
650
700
0.25
110
0.2
100
0.15
90
0.1
80
0.05
70
60
400
450
500
550
wavelength (nm)
600
650
700
0
400
450
500
550
wavelength (nm)
600
650
700
Figure 3: CIE XYZ colour matching functions, CIE illuminant D65 and a uorescent illuminant.
5
Experiments and Results
The aim of our experiments was to explore the structure of the metamer sets
from the crossover point of view. We took a number of measured natural
reectances. For a given set we went through each reectance it contained,
produced the corresponding metamer set and found the intersections of the
original reectance with the generated ones.
Experiments were carried out on several sets of measured reectances:
the Macbeth ColorChecker Chart (24 samples), Munsell colour atlas chips
(426 samples)[12], 170 object reectances [14], a collection of 134 saturated
colours and a set of 41 uniformly distributed colours in chromaticity space.
Each set was examined separately for the CIE 1931 Standard Colorimetric
Observer and for two illuminants: CIE illuminant D65 and a uorescent
illuminant { see Fig. (5).
The results are shown in Fig. 7 for CIE illuminant D65 and for the
uorescent illuminant for each set separately and in Fig. 5 cumulative results
for all data sets under CIE illuminant D65 and in Fig. 5 cumulative results
for all data sets under the uorescent illuminant. The crossover wavelengths
are then summarised in Tables 5, 5.
The results clearly show major peaks in the vicinity of Thornton's prime
wavelengths, and on average only these are signicant. The results also show
that more than three crossovers are common.
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4000
3500
number of crossovers
3000
2500
2000
1500
1000
500
0
400
450
500
550
wavelength (nm)
600
650
700
Figure 4: Cumulative histogram of crossover wavelengths (for CIE illuminant D65)
data set (# of samples)
Macbeth (24)
Munsell (426)
Object (170)
Satref (134)
Uniref (41)
cumulative (795)
1
(nm)
2
(nm)
456 546 10
452
546
454
536
454
526
454 536 20
453 1 548 1
3
(nm)
606
604
602
608
606
604
Table 1: Prime crossover wavelengths for each dataset (for CIE illuminant D65)
6000
number of crossovers
5000
4000
3000
2000
1000
0
400
450
500
550
wavelength (nm)
600
650
700
Figure 5: Cumulative histogram of crossover wavelengths (for a uorescent illuminant)
data set (# of samples)
Macbeth (24)
Munsell (426)
Object (170)
Satref (134)
Uniref (41)
cumulative (729)
1
(nm)
2
(nm)
452
548
452
550
438 10 546
452
544 6
450
546
451 1
550
3
(nm)
608
606
606
608
608
606
Table 2: Prime crossover wavelengths for each dataset (for a uorescent illuminant)
13
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Discussion
Many authors have explored the phenomenon of metamer crossovers. Thornton [1] rst tackled this problem, producing a set of smooth metamers using
three to four broad gaussian components. He found that there are three
statistically signicant intervals of crossovers: 448 4nm, 537 3nm and
612 8nm and at that more than three crossovers are common.
Ohta and Wyszecki [2] tried to prove that there are no prime wavelengths
of crossover, by producing two sets of reectances, each containing a large
number of pairs of metamers generated in two dierent ways. The rst set
was generated using a non-linear optimisation technique modifying a fourth
order polynomial tted to ve random points to result in given tristimulus
values, and the second set was generated by combining three randomly selected dyes from a set of 12, again to result in identical tristimulus values.
In the rst case four and in the second case three dominant wavelength
bands of intersection were found, though an abundance of metamers with
ve intersections was present. They demonstrated in the rst place that the
method used to generate metamers strongly aect the frequency distribution
of the intersections.
Brill [18], in an attempt to support Thornton's conjecture, derived the
most signicant black colour signal based on Judd's [19] daylight spectra. As
the visual system is tri-chromatic, Brill concluded that the metameric black
based on the rst four principal components of the daylight spectra (the
fourth being orthogonal to the colour matching functions) is statistically the
most signicant one. The derived black exhibits zero-crossing wavelengths at
450nm, 540nm and 610nm { similar to Thornton's. Brill therefore concluded
that considering Judd's daylight spectra a good representation of what is
\natural", Thornton's results are valid in this context.
Ohta [20] took the approach of studying three zero-crossing metameric
blacks only. Varying the zero-crossing wavelengths ; and , and the
gamuts for feasible blacks were obtained by maximising a metamerism index
he dened for this purpose (the sum of the absolute value of the metameric
black over the visible range). The maximum value of the index was achieved
at 450nm, 540nm and 610nm. Ohta concluded that Thornton's results are
valid for a high degree of metamerism.
The importance of the degree of metamerism of two colour signals, the
impact of the illuminant for which the metamers are generated as well as
the wavelength range of calculation were examined in a study by Berns
and Kuehni [21]. It was questioned whether there are other inuences on
the crossover wavelengths of metamers apart from the colour matching functions. Two sets of data were used { one being mathematically derived (a
logarithmic absorption model of a single maximum) and one being dyed textile samples, both sets representing a basis for a Kubelka{Munk colourant
formulation model. The degree of metamerism for the generated metamer
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sets was quantied by the Eab colour dierence for CIE illuminant A. The
authors again arrive at crossover wavelengths around 450nm, 540nm and
610nm.
Our results conrm Thornton's original conjecture on the necessity of
three prime wavelengths of metamer crossovers, and the position we arrive
at are similar. The studies which arrived at dierent results, were generating
metamers using numerical procedures and optimisation techniques without
a link to measured sets. One could use these very techniques to produce
metamers which cross almost anywhere. The aim here was to say something
about natural metamers, rather than \what is possible".
The statistical signicance of our study lies in the fact that the metamers
we generated indeed exhibit properties of natural reectances, as well as the
fact that we examined crossovers of large metamer sets, rather than pairs
only.
7
Conclusions
In this paper we presented the reader with the phenomenon of metamerism
{ a mechanism inherent in the colour formation process. We proposed a
method to describe the convex set of natural, smooth metamers. We studied
the inner structure of these sets of reectances from the point of view of the
wavelengths of intersection.
The results presented support the theory of three prime wavelengths
of intersection, though they also shows that more crossovers are often the
case. One might conclude, as has been concluded previously, that indeed
it is the method used to generate the metamers, that aects the crossover
wavelengths the most.
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Appeared in Proceedings of Colour 2000 Conference, Derby, UK
160
350
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Figure 6:
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Crossover wavelength statistics for CIE illuminant D65 (left) and the
uorescent illuminant (right) of data sets (from top to bottom): 24 Macbeth ColorChecker Chart samples, 41 uniform samples, 134 saturated samples, 170 object
reectances, 426 Munsell samples.