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Color processing methods can be divided into methods based on human color vision and spectral based methods. Human vision based methods usually describe color with three parameters which are easy to interpret since they model familiar color perception processes. They share however the limitations of human color vision such as metamerism. Spectral based methods describe colors by their underlying spectra and thus do not involvehuman color perception. They are often used in industrial inspection and remote sensing. Most of the spectral methods employ a low dimensional (three to ten) representation of the spectra obtained from an orthogonal (usually eigenvector) expansion. While the spectral methods have solid theoretical foundation, the results obtained are often difficult to interpret.
Non-euclidean structure of spectral color space
Reiner Lenz a and Peter Meer
a) Dept. Science and Engineering, Campus Norrköping,
Linköping University,SE-601 74 Norrköping, Sweden,reile©
b) ECE Department,Rutgers University,94 Brett Road
Piscataway, NJ 08854-8058, USA,meer©
Color processing methods can be divided into methods based on human color vision and spectral based methods.
Human vision based methods usually describe color with three parameters which are easy to interpret since they model
familiar color perception processes. They share however the limitations of human color vision such as metamerism.
Spectral based methods describe colors by their underlying spectra and thus do not involve human color perception.
They are often used in industrial inspection and remote sensing. Most of the spectral methods employ a low
dimensional (three to ten) representation of the spectra obtained from an orthogonal (usually eigenvector) expansion.
While the spectral methods have solid theoretical foundation, the results obtained are often difficult to interpret.
In this paper we show that for a large family of spectra the space of eigenvector coefficients has a natural cone
structure. Thus we can define a natural, hyperbolic coordinate system whose coordinates are closely related to
intensity, saturation and hue. The relation between the hyperbolic coordinate system and the perceptually uniform
Lab color space is also shown. Defining a Fourier transform in the hyperbolic space can have applications in pattern
recognition problems.
Keywords: Color spectra, eigenvector decomposition, non-euclidean geometry, Mehler-Fok transform
Color processing methods can roughly be divided into two groups: methods based on human color vision and methods
working directly on the color spectra. The human vision 1)ased methods describe color usually by three parameters.
Typical examples are the CIE—systems (such as XYZ-, Luv and Lab) and the more sophisticated color appearance
systems (see' 2)
. These
systems have the advantage that many of their features are easy to understand since they
model familiar color perception processes. In the CIE-Lab system, for example, the L-component describes the
lightness, and the length and the orientation of the vector in the ab-plane describe the saturation and the hue of
the spectrum. The euclidean distance between two Lab-vectors corresponds roughly to the perceptual similarity of
two colors. Since the systems are designed to model the human visual system they also share the limitations of
human color vision. One of the most serious restrictions is the fact that the system consists of three different types
of detectors. All colors are thus described by only three different parameters although the space of real color spectra
has certainly more than three dimensions. The best-known consequence of this limitation is probably metamerism:
two different reflectance spectra produce the same sensor output under one illumination but different sensor outputs
under a different light source. This is not only a problem for systems based on human color vision but also for similar
technical systems such as RGB cameras.
Spectral based color processing methods try to avoid these limitations by working on the color spectra directly.
They require the measurement of color spectra with instruments like color photospectrometers or multi-channel
cameras which allow a better reconstruction of the underlying color spectra than the usual RGB-cameras. Examples
of such systems and some of their applications are described in.3'4 Most of these methods do not use the raw spectral
data but they describe the spectra with parameters in a low-dimensional space. Typically eigenvector expansions are
used and three to ten eigenvector coefficients describe the space of reflectance spectra which are relevant for human
color vision. These methods are effective as long as these coefficients are only used for relatively simple numerical
calculations. Results obtained in this way may however be difficult to interpret (and to process further) since the
raw numbers have no perceptual correlates which are similar to the lightness, hue and saturation interpretations of
color version of the paper is available under: http:
//www. itn. liu.
se/reile/prints/noneucol .ps
of the EUROPTO Conference on Polarization and Color Techniques in Industrial Inspection
Munich, Germany . June 1999 SPIE Vol. 3826. 0277-786X/99/$1 0.00
Downloaded from SPIE Digital Library on 30 Sep 2010 to Terms of Use:
the Lab-values. An illustration of the problems encountered is the following: assume you want to describe the color
distributions of two different images. The eigenvector expansion (based on n eigenvectors) allows the description
of the color distribution of an image as a probability distribution in n-dimensional space. From these distributions
statistics such as the means and the correlation coefficients can be computed but these values are difficult to interpret
since the raw coefficients have no perceptual correlates. For Lab-based descriptions this is no problem since different
mean values for the L-components have the obvious interpretation of being lighter/darker on average. Since the
raw-measurements lack simple interpretations it is also difficult to define which statistics are best suited for a given
In this paper we derive a framework in which most of the these problems can be accessed. We will first show
that the space of eigenvector coefficients has a natural cone structure. This implies that there is a natural coordinate
system in which these coefficients can be described. It is then shown that the first coordinate axis is related to
intensity, the second to saturation and the last, circular coordinate corresponds to hue. The exact relation between
the conical coordinates and the Lab system is more complicated due to the characteristics of human color vision and
will mainly be illustrated with the help of some diagrams. Finally we sketch an application in which we show that
the existence of the natural conical coordinate system in the coefficient space allows the introduction of a hyperbolic
Fourier Transform which should be useful in pattern recognition applications. We restrict us in this paper to the case
where spectra are approximated as linear combinations of three eigenvectors. This restriction allows us to describe
the essential features of the approach while avoiding further technical difficulties. The nature of the data allows
investigations of higher order approximations along the same lines described below.
It is well known that the reflectance spectra measured from the Munsell chips are all linear combinations of a few
basis vectors.1'59 Usually the eigenvectors, computed from the set of measured reflectance spectra, are taken as
these basis vectors but other selections are also possible and have been used earlier.1'7'°
In the following we use A as the variable describing wavelength and s(A) is a vector denoting an arbitrary spectral
distribution. The eigenvectors (shown in Figure 1(a)) are written as bk(A) and for the coefficients in the eigenvector
expansion of the spectrum s()) we use Uk . Thus
we have the following approximation:
the following we use mainly three basis functions K =
and we thus choose to represent the spectrum s())
by the coordinate vector a =
in the coordinate system given by b1 ()) ,
(..\) ,
all earlier investigations (known to us) the vector a is treated as a general element in R .
further assumptions
are made about other properties of these vectors. In our experiments we used a database consisting of reflectance
spectra of 2782 color chips, 1269 from the Munsell system and the rest from the NCS system. For each of the 1269
chips of the Munsell System their spectra was measured from 380nm to 800nm at mm steps, while the 1513 samples
from the NCS system were measured from 380nm to 780nm at 5nm intervals. These measurements were combined
in one set consisting of 2782 spectra (sampled in 5nm steps from 380nm to 780nm). In the following we refer to this
set of spectra as the spectral database. From these spectra the eigenvectors were computed and the spectra were
approximated by linear combinations of these eigenvectors. An inspection of the properties of the coordinate vectors
showed that nearly all color chips in the spectral database are described by coordinate vectors which lie in the cone
only exception is the NCS chip 2070-B (marked by the big dot in the Figure) a very dark blue color. For this
spectrum the value of a —
4 — a
is equal to -0.01, i.e. is lying on the border of the cone. Furthermore we found
that for all other spectra the values of a —
[i2 ? was
positive for approximation orders K up to ten. We could
therefore also consider cones in spaces of dimensions less or equal ten. For the three-dimensional approximation the
distribution of the spectra is shown in Figure 1(b) in which each point corresponds to a chip in the spectral database.
The x- and y-coordinates in the plot correspond to the second and the third eigenvector coefficients u2, cr3
plotted along the z-axis.
the cone it is natural to consider three different types of natural curves:
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Fico Eigenvector
- - -
Second Eigenvecto
Third Eigenvector
Dsmbuton of coefhcents
1. The a axis
Figure 1. Eigenvector expansion of color chips
2. hyperbola in planes containing the o
axis and
3. circles in planes perpendicular to the o
This leads to the introduction of the hyperbolic coordinate system defined by:
For p, c, introduce the matrices D, A and K:
/ coshc
(sinh a cos
a, ip)
D(p) =
cosh a
sinh a
sinh a
cosh a
0 J
cos p
The matrices A(a) and K(ço) and all possible products of them define the group SO(2,
It can also be shown
that every matrix in SO(2,
can be written as a product K(i')A(a)K(p) =
a, ) which is similar to the
Euler-angle representation of a three-dimensional rotation. We will therefore call
a, ço the Euler angles of that
transformation. The coordinate vector h(p, a, ) is given by:
h(p,a,ço) =
The a axis, the hyperbola and the circles can now be described as:
p -+
= (hyperbola),
j =
We note also that these matrices form groups parametrized by one parameter since we have:
+ P2) = D(pi)D(p2),
A(ai + a2)
A(ai)A(a2), K(pi + P2) = K(çoi)K(p2)
Eoooncs. Rcno38oo - 780cc. 2182 NCS&odMocs.Hspnco
450 500 55.0 000 650 700
( a) The first three eigenvectors
2. Ev.
3. En.
Distribution of coefficient vectors
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These parameterizations of the curves is natural since a composition of mappings are described by additions of the
A first relation between the geometric and the perceptual concepts is obtained by an inspection of the first three
eigenvectors bk shown in Figure 1 (a) and the definition of the coefficients cr
scalar products:
Since b1 is almost a constant and since the spectral distributions have only non-negative entries we see that a is
roughly equal to the L1 norm, and thus the energy of the spectrum s(A).
call the origin of the space, the vector h(O, 0, 0) the "white Point" and h(p, 0, 0) the "achromatic axis".
Color spectra are traditionally described by their XYZ-coordinates (Chapter 3.3 in1 )
which are defined as the scalar
products between the spectrum and the tristimulus functions x(A), y(A), z(A)
= (s(A),x(A)),
Z =
Writing this as a matrix multiplication we get for the transformation from the hyperbolic to the XYZ-system the
following relation:
The elements of the matrix H are computed as:
where we used x1
x(A), x2(A) = y(A)
and x3(A) =
The matrix H is computed as:
H =
( 2.34
—3.00 )
\2.55 —2.84
Further understanding of role of the hyperbolic coordinates and can be obtained by relating them to the tradi-
tional chromaticity description in the Lab-system defined as1 (Sec. 3.3.9, we often write L, a,
b instead of L* ,
a* b*):
x 1/3
y 1/3
Y 1/3
116 ()
- 16;
500 [()
() ] ;
200 [()
- () ]
Here XN ,
describes the (XYZ)-coordinates of the white point (which should not be confused with the "white
point" mentioned above) .
we define the white point through the first eigenvector, i.e. XN ,YN
are the
XYZ-coordinates of the first eigenvector. The variable L describes the lightness properties of a spectrum whereas
the ab part characterizes its chromaticity. The polar coordinates in the ab-plane have perceptual correlates in the
sense that the angular part corresponds to the hue and the radial part to the chroma of the spectrum:
Cab a2
arctan ()
Figure 2 shows the relation between the hyperbolic coordinates and a and the chromaticity coordinates in ab-
space. Figure 2(a) shows the function Cab(a, )
Figure 2(b) the function hab(a, cp) (here a polar coordinate
system in the (a, )
is used, i.e. the radial coordinate is given by a and the angular coordinate by p. Tracing
a circle around the origin in (a, p)—space varies the hue-angle p while keeping the value of a fixed. Going on a
straight line from the origin outward varies a but leaves
fixed. Figure 2(b) shows that hab is (up to a constant
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Figure 2. Relation to Lab coordinates
(a) Lab-Chroma
(b) Lab-hue
shift parameter) almost identical to ço for all values of c. Figure 2(a) on the other hand illustrates that the relation
between Cab and a depends on the hue-angle (circles in (ct,
are mapped to ellipses in ab-space).
The color of the surfaces in these plots is an indication of the color of the underlying spectrum. It is computed
from the a, coordinates of the spectrum. For all surface points the value of the variable p is constant (and equal
to the mean value of p computed over all spectra).
Figure 3 shows the same information as Figure 2 but now for the spectra in the spectral database. In these
figures each point corresponds to a spectrum in the database. Figure 3(b) shows the dependence between the
angular variable p and the Lab-hue value hab. It shows that there is a almost linear relation between p and hab
except for the gray colors (Note that both variables are angles, i.e. 2ir periodic). For the relation between the
hyperbolic a coordinate and the Lab-chroma value Cab the situation is not as simple as can be seen from Figure 3(a).
(a) Lab-Chroma for chips
(b) Lab-hue for chips
Figure 3. Relation to Lab coordinates
However selecting only a certain hue-range and plotting the a, Cab coordinates for all chips in this hue-segment
shows that there is locally a linear relation between these variables but that this relation changes over hue. Figure 3
demonstrates this relation for the hue ranges 3.13 <
< 1.77, —1.42 < hab < 1.07,
1.07 < hab < 0.78
and and 0.78 < hab < 1.66. Here the spectra in the database were ordered according to their Lab-hue value hab
and divided into six sets with an approximately equal number of colors in each set. For each of these six sets the
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distribution in the (cr,
space were plotted. Summarizing we see that the first eigenvector coefficient a1 measures
the intensity of the spectrum and that the norm of the vector (a2a3)
the colorfulness of the spectrum, i.e.
the distance to the gray color with the same intensity. In the coefficient space we can thus introduce the metric
— a
which measures the "whiteness" of a spectrum. In this interpretation the transformations in the group 80(2, 1)
the whiteness preserving linear transformations of spectral space.
In the previous sections we described a color spectrum with three coordinate values: p, c,
. It
can be shown
that the effect of p can be treated independent of the other two variables. We will thus in the following assume
that it has a fixed value and we will ignore it in this section. The space of color spectra is thus a hyperboloid
in three-dimensional space. The corresponding point in three-dimensional space has coordinates (a1, a2, a3) =
(cosh ,
p) which will be denoted by (cl,
Since we use c, to denote coordinates we will
often use $,
w to
describe the parameters of the linear transformations.
0 0.2
04 00 00
(d) 0.78 < hab < 1.66
Figure 4. The relation between c, and Cab
(a) 3.13 < hab
< 1.77
1 2/22 . 0
(b) —1.42 < hab < 1.07
14y24b010 *
A03i0200-44 0 /04< .. I (001
:* :-
'. *
4 a
00 00
2 10
14. 10 2
(c) —1.07 < hab < 0.78
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In the previous section we showed that the linear transformations in the group SO(2,
preserved the "whiteness"
of color spectra. These transformations can all be parametrized by their Euler angles and it is therefore sufficient
to consider transformations K(w) and A($). The rotations K(w) define also rotations in space. More complicated
is the operation of the A(3) operations. Applying A(3) to the point (,
and denoting the coordinates of the
resulting point by (,
) we
get for the new coordinates the relations:
3 cosh c + sinh $
@ —
cosh fi
a cos p + i(sinh a sin )
For Icos cI
= 1
this is a simple addition/subtraction of the hyperbolic parameters and thus a shift along the hyperbola.
For other values of p this is a much more complicated transformation. For a general transformation with Euler-
angles Wi ,
3, w2 we
write such a transformation as T(wi ,
Making the following discussion more concrete we assume that we have functions p(a, ço) which could describe the
probability distributions of the color points in images. Applying a whiteness-preserving transformation T(wi ,
to color space will map the color distribution p(c, )
the distribution p(&, ).
For the transformation A($) =
/3, 0) the coordinate transformation is given by Equation(16). This transformation on the functions p is denoted
by a superscript describing he transformation of the underlying color space:
=p(T'(,ço)) pT(a,)
pattern recognition problems that can be formulated in this framework are:
1. Invariant feature extraction: For all probability distributions construct features F such that F(p) =
all transformations T.
2. Given two probability distributions p, T which are related by an unknown transformation T compute the
transformation T.
Let us consider the special case in which the transformations are the hue-shift operators and the functions are
independent of
: p(p)
In this case the functions p can be developed into a Fourier series in the
: p() =
>:: ae'°
and the Fourier coefficients a will undergo a simple multiplication a —+ ae'.
Invariant features are in this case the absolute values of the Fourier coefficients and the transformation parameter w
can be estimated from the phase factor. We also note that these features form a complete set since the Fourier
coefficients define the function p completely. The general problem is now to find a feature extraction process, a Fourier
transform, for the whole group of transformations and general functions on the hyperboloid, which corresponds to the
ordinary Fourier transform when restricted to hue-shifts. The full description of the underlying theory of harmonic
analysis on this group is beyond the scope of this paper (for a complete description see1115). Here will only
introduce the Legendre functions as the functions which correspond to the exponential function and we will describe
the Mehler-Fok transform as the corresponding Fourier transform.
Our outline of the appropriate Fourier transform starts with the Legendre functions. For integers or half-
integers m, n and a complex constant T we introduce the functions (see page 314 in14)
t) =
(cosh + sinh e°)
(cosh + sinh e_) ei(m_ dO
For the special case n =
we obtain (see page 322 in'4):
0(cosht) =
1)m(cosht) (m  0);
0(cosht) =
F(r+rn 1)
cosht) (m <0) (19)
and for m =
q30(cosht) =
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(b) '43_112(cosht), (n = 0.4)
Figure 5. A few Legendre Functions
Here 3' and
are the (associated) Legendre functions. A few typical examples of (associated) Legendre functions
are shown in Figure 5.
These functions satisfy the following addition formula:
e_i(m(cosh &) =
ek(cosh $)(cosh ) (21)
k= —
k runs over the integers m, n are integers and runs over the half-integers m, n are half-integers. The relation
between the coordinates is the same as in Equation (16). Note that the left-hand side of the equation contains a
phase factor eintI) where the angle t' is a function of a, ,
defined by:
e2 —
i-e"2 + sinh
sinh cose'2
For the special case m =
0 this becomes:
r(cosh cosh $ + sinh sinh fi cos ) =
e(cosh )k(cosh $) (23)
This describes how the value of the Legendre function at the point (cr, ) transforms under a hyperbolic transformation
with parameter 3. We see that the transformed Legendre function is a series of associated Legendre functions with
the same parameter r. The simple multiplicative transformation property of the exponential function under shifts is
thus replaced by an infinite sum over related functions.
This gives the required transformation formula under a hyperbolic transformation (Equation (16)) and the as-
sociated Legendre functions are thus the appropriate generalizations of the exponential functions for hyperbolic
transformations. This holds for complex values of the parameter r which has not yet been specified. Special cases
will be discussed later.
For an application in feature extraction we will in general need a Hubert space which contains both the function p
which should be analyzed and the Legendre functions. The scalar product in this Hubert space is furthermore
invariant under the operation of the transformation group in the sense that
(fT,gT) =
for all elements f, g in the Hilbert space and all transformations T in the group. We can then compute the feature
extracted from the transformed function T using:
(pTl,3) =
and use the transformation rule in Equation (23) to compute the transformed Legendre function
(a) 3k.1_l/2(cosht), (k =
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and investigate only the components c,
. For
each of the images we estimated the joint probability distributions
of (a, cc) and the densities of c and
separately. These probability distributions were estimated using a kernel
density estimation method.
The two two-dimensional probability distributions for two flower images in the same position under the illuminants
Philips Ultralume fluorescent (ph-ulm) and Macbeth 5000K fluorescent in conjunction with the Rosolux 3202 Full
Blue filter (mb5000+3202) are shown in Figure 7.
The distributions of the c and
q variables are shown in 8. These
distributions show that the blue illuminant mainly effects the distribution of the 0
variable: it is more concentrated
and shifted. The distributions of the a values are very similar in both cases but the second peak in the distributions
is clearly shifted to the right for the blue illuminant. Translated into the perceptual framework developed above
these results state that the colors in the blue image are more saturated and shifted towards blue.
In the last experiment we computed for all 110 images in the database the Mehler-Fok transform at the posi-
tions r =
(k =
.4). A few of the results obtained are shown in Figure 9(a-c) In these figures we see
R 5OOO.32O2 N
A...- —
Figure 7. Density distributions for (a, ) computed
from flower images
(a) Distribution of a
Figure 8. Probability densities computed from flower images
that for low values of the parameter r (which correspond to the "low frequency" responses of the ordinary Fourier
(a) mb5000+3202
(b) ph-ulm
b5OOO+32O2- k* ,g.s
- - -
(b) Distribution of
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( d) O.5i—1/2 aligned vs. non-
aligned objects
Figure 9. Features computed with the Mehier-Fok transform
transform) there is a consistent connection between the spectral properties of the light source and the value of the
extracted features: The images taken under the blue illuminant (mb5000+3202) have in all cases the highest feature
value. These features are stable against variation of the placement of the objects in the scene as can be seen from
Figure 9(d). In this Figure we show the distributions of the features computed with the Legendre function O.5.i1/2.
The values computed from the first 55 images (showing the eleven objects in the fixed position) and the values corn-
puted from the last 55 image (with random orientation of the same objects). The solid line represents the aligned
objects, the dashed line the non-aligned objects. We see that the distributions are almost identical for the same
object in different positions but different for different objects and different illuminants. For some of the objects (such
as the book) there is no great variation between the color distributions in the different images since only the position
of the object points in the image vary. For other objects, like the balls or the flowers, there is however a significant
change since different parts of the object are visible in the different images.
In the space of color spectra we introduced a coordinate system based on the eigenvector expansion of the color spectra
measured from a database of Munsell and NCS system. The coordinates of the visible spectra lie all in a cone. In
the cone we identified a special coordinate system in which the components correspond roughly to intensity, hue and
saturation. We then illustrated the relation between these coordinates and the Lab system. For this coordinate there
is a transform which corresponds to ordinary Fourier transform in euclidean geometry. For functions depending only
on the saturation variable this Fourier transform is the classical Mehier-Fok transform. Finally we illustrated how
this transform can be used to construct pattern recognition algorithms which can be useful in analyzing distributions
(a) ¶43—1/2
(b) O.5.i—1/2
(c) I34i—1/2
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of color spectra. We did not discuss sampling, numerical and implementation problems which need to be solved in
a successful application.
The Munsell spectra is available from the Information Technology Dept., Lappeenranta University of Technology,
Lappeenranta, Finland under the address: http : //www .lut
html The NCS spectra was obtained from the Scandinavian Color Institute in Stockholm courtesy to B. Kruse.
The images are available under the address http : //www .cs.
The kernel density toolbox
is available under R. Lenz was supported by grant 98-530
from the Swedish Research Council for Engineering Sciences and grant 98.7 from CENIIT, the Center for Industrial
Information Technology at Linköping University.
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... They analyzed "a database consisting of reflectance spectra of 2782 color chips, 1269 from the Munsell system and the rest from the NCS system." 2 The Munsell spectra are the same ones we analyze below. They find that the spectral data are described by coordinate vectors which lie in a cone and they therefore define a hyperbolic coordinate system to represent the data. ...
... They find that the spectral data are described by coordinate vectors which lie in a cone and they therefore define a hyperbolic coordinate system to represent the data. They present a three-dimensional figure of the distribution of the chips ( Fig. 1(b) in Ref. 2) that is somewhat like the upper left panel of our Fig. 2. The structure we describe in this article could be characterized as a set of nested cone-like structures, each made up of a single Munsell Chroma with lower Chroma values being inside higher values. The narrow tips of each cone-like structure are found at the lowest Value levels. ...
... The curve from 430 to 660 nm in Fig. 4 shows W ␣ ϭ (w ␣ ), each representing the overall weights of 231 wavelength values in the respective basis factors ␣ (␣ ϭ 2 and 3). The configuration of 1269 colors in this plot is based on P2 and P3 of SVD analysis obtained by procedure (2) in Appendix I. For comparison we also calculated the Euclidean distances among all wavelengths and , D 231ϫ231 ϭ (d ), and used metric scaling to embed them in E 3 by the procedure (3) in Appendix I. ...
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In this article we describe the results of an investigation into the extent to which the reflectance spectra of 1269 matt Munsell color chips are well represented in low dimensional Euclidean space. We find that a three dimensional Euclidean representation accounts for most of the variation in the Euclidean distances among the 1269 Munsell color spectra. We interpret the three dimensional Euclidean representation of the spectral data in terms of the Munsell color space. In addition, we analyzed a data set with a large number of natural objects and found that the spectral profiles required four basis factors for adequate representation in Euclidean space. We conclude that four basis factors are required in general but that in special cases, like the Munsell system, three basis factors are adequate for precise characterization. © 2003 Wiley Periodicals, Inc. Col Res Appl, 28, 182–196, 2003; Published online in Wiley InterScience ( DOI 10.1002/col.10144
... (52) is the Rao-Siegel metric encountered in section 2.2 when we have discussed the Resnikoff model, with the difference that Resnikoff applied it on the whole cone H + (2, R) and not on the level set H + 1 (2, R). This metric has been used also in [32] and in [14] in the context of CIE (Commission Internationale de l'Éclairage) colorimetry. ...
... Under the condition that b 0 (λ) has only positive entries, and with an appropriate scaling of basis functions, we showed in 14 that the vectors σ are located in a cone. In the following we will only use three basis vectors (K = 2), and thus concentrate on coordinate vectors located in the cone: ...
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Understanding the properties of time-varying illumination spectra is of importance in all applications where dynamical color changes due to changes in illumination characteristics have to be analyzed or synthesized. Examples are (dynamical) color constancy and the creation of realistic animations. We show how group theoretical methods can be used to describe sequences of time changing illumination spectra with only few parameters. From the description we can also derive a differential equation that describes the illumination changes. We illustrate the method with investigations of black-body radiation and measured sequences of daylight spectra
... This conical structure will be verified for a number of spectral databases and for multi-spectral images of real-world scenes. We showed earlier that this conical structure has profound consequences for further processing in, for example, computer vision and pattern recognition [6, 8, 9] . In contrast to these earlier investigations were we motivated the conical structure by the non-negativity of color spectra we give here a more precise description of the origin and the structure of the cone. ...
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The physical properties of color are usually described by their spectra, eigenvector expansions or low-dimensional descriptors such as RGB or CIE-Lab. In the first part of the paper we show that many of the traditional methods can be unified in a framework where color spectra are elements of an infinite-dimensional Hilbert space that are described by projections onto low-dimensional spaces. We derive some fundamental geometrical properties of the subset of the Hilbert space formed by all color spectra. We describe the projection operators that map the elements of the Hilbert space to elements in a finite dimensional vector space. This leads to a generalization of the concepts of spectral locus and purple line. It will be shown that for geometrical rea- sons the color space is topologically equivalent to a cone. In the second part of the paper we illustrate the theoretical concepts with four large databases of spectra from color systems and a series of multi-spectral images of natural scenes. We verify the conical property of color space for these databases and compute their, geometrically defined, spectral locus and chromaticity properties. In the last sec- tion we relate the natural co-ordinate system in the conical color space to the traditional polar co-ordinates in CIE- Lab. We show that there is a good agreement between the geometrically defined hue-variable and the angular part of the polar co-ordinate system in CIE-Lab. There is also a clear correlation between the geometrical and the CIE-Lab saturation descriptors.
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In this paper, we provide an overview on the foundation and first results of a very recent quantum theory of color perception, together with novel results about uncertainty relations for chromatic opposition. The major inspiration for this model is the 1974 remarkable work by H.L. Resnikoff, who had the idea to give up the analysis of the space of perceived colors through metameric classes of spectra in favor of the study of its algebraic properties. This strategy permitted to reveal the importance of hyperbolic geometry in colorimetry. Starting from these premises, we show how Resnikoff’s construction can be extended to a geometrically rich quantum framework, where the concepts of achromatic color, hue and saturation can be rigorously defined. Moreover, the analysis of pure and mixed quantum chromatic states leads to a deep understanding of chromatic opposition and its role in the encoding of visual signals. We complete our paper by proving the existence of uncertainty relations for the degree of chromatic opposition, thus providing a theoretical confirmation of the quantum nature of color perception.
Abstract Most descriptions of the colour rendering properties of light sources are based on the calculation of colour differences for a number of reflectance samples between the light source and a reference source. The CIE colour-rendering index CRI is a single number based on the average colour difference for 8 reflectance samples. Ever since its introduction the CRI has been discussed and many alternative suggestions have been made to improve the description of the colour rendering. These metrics comprise amongst others colour fidelity-, colour gamut- and colour preference based metrics and combinations of these. All these new proposals try to capture a certain colour rendering aspect in a single number index to be used instead of or together with the existing colour rendering index. The “traditional” CRI has been critized on the fact that a single number index does not provide sufficient information on the colour rendering properties of light sources. Creating a (second) single number metric seems to be a bit of a paradox, as it suffers from the same limitation by providing an average number to characterise certain colour aspects for all colours. In our view the best way to provide users of lighting in whatever application with accurate information on colour characteristics is to provide hue specific information on the most important colour characteristics such as hue, chroma and lightness. In this paper we will first describe our methods of supplying hue specific information via colour rendering vectors and colour rendering icons. We will illustrate why we believe it is important to add hue specific information and what the shortcomings are of a second hue-averaged metric. We realize that color appearance models have advanced since our first calculations and that it would be more appropriate to use CIECAM02. Also, recently several papers have appeared discussing improved reflectance sample sets. This, however, does not change the key message in this paper. Furthermore, the methodology we describe can easily be adapted to a different color space such as CIECAM02 and to a different set of reflectance samples.
Conference Paper
The spectra of color can represent a color in the most accurate way, but the dimension of the spectral data is too high to process. This paper aims to analyze the spectral reflectance curves of 1269 Munsell standard color samples with some influential algorithms in manifold learning. Experimental results reveal that the intrinsic dimension of the embedded manifold in the spectral Munsell color space is 3 and the 3-dimensional structure of this manifold looks like a cone, consistent with the development and structure of the Munsell color system.
A novel general transformation between reflectance spectra and the corresponding coordinates of the Munsell Color System is presented. The coefficient values of the transformation were experimentally determined by mapping the actual reflectance spectra of the chips in the Munsell Book of Color into the Munsell Color Order System and by minimizing the distance between calculated and actual coordinates. The experiment was repeated with a selected set of points of the Munsell Renotation System. Both the Smith–Pokorny functions and the CIE 1931 standard color-matching functions were used as a basis of the transformation. There is a good correspondence between calculated and actual coordinates of the Munsell Color System. It is also shown that the linear part of the same transformation applied to the basis functions results in one achromatic response function and two chromatic response functions in accordance with the opponent-colors theory. © 2005 Wiley Periodicals, Inc. Col Res Appl, 31, 57–66, 2006; Published online in Wiley InterScience ( DOI 10.1002/col.20173
Color scaling experiments have established that perceived colors are distributed on the surface of a hypersphere in spherical space. A formal mathematical model of the color space is defined. Color differences as estimated by an observer are equal to the chord distance between corresponding points on the surface in spherical space. In one mathematical model are united: brightness, saturation, and hue as expressed in the empirical Munsell and NCS systems; complementary colors; large color differences; and contrast effects that are not represented in other models. © 2008 Wiley Periodicals, Inc. Col Res Appl, 33, 113–124, 2008.
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The paper gives a short overview over some basic facts from the representation theory of groups and algebras. Then we describe iterative algorithms to normalize coefficient vectors computed by expanding functions on the unit sphere into a series of spherical harmonics. Typical applications of the normalization procedure are the matching of different three-dimensional images, orientation estimations in low-level image processing or robotics. The algorithm illustrates general methods from the representation theory of Lie-groups and Lie-algebras which can be used to linearize highly-non-linear problems. It can therefore also be adapted to applications involving groups different from the group of three-dimensional rotations. The performance of the algorithm is illustrated with a few experiments involving random coefficient vectors.
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The 1257 reflectance spectra of the chips in the Munsell Book of Color—Matte Finish Collection (Munsell Color, Baltimore, Md., 1976) were measured with a rapid acousto-optic spectrophotometer. Measured spectra were sampled from 400 to 700 nm at 5-nm intervals. The correlation matrix of this sample set was formed, and the characteristic vectors of this matrix were computed. It is shown, contradictory to earlier recommendations [Psychon. Sci. 1, 369 (1964)], that as many as eight characteristic spectra are needed to achieve good representation for all spectra.
We have constructed a wine-glass-type five-layer neural network and generated an identity mapping of the surface spectral-reflectance data of 1280 Munsell color chips, using a backpropagation learning algorithm. To achieve an identity mapping, the same data set is used for the input and for the teacher. After the learning was completed, we analyzed the responses to individual chips of the three hidden units in the middle layer in order to obtain the internal representation of the color information. We found that each of the three hidden units corresponds to a psychological color attribute, that is, the Munsell value (luminance), red-green, and yellow-blue. We also examined the relationship between the internal representation and the number of hidden units and found that the network with three hidden units acquires optimum color representation. The five-layer neural network is shown to be an efficient method for reproducing the transformation of color information (or color coding) in the visual system.
We gathered hyperspectral images of natural, foliage-dominated scenes and converted them to human cone quantal catches to characterize the second-order redundancy present within the retinal photoreceptor array under natural conditions. The data are expressed most simply in a logarithmic response space, wherein an orthogonal decorrelation robustly produces three principal axes, one corresponding to simple changes in radiance and two that are reminiscent of the blue–yellow and red–green chromatic-opponent mechanisms found in the primate visual system. Further inclusion of spatial stimulus dimensions demonstrates a complete spatial decorrelation of these three cone-space axes in natural cone responses.
The spectral reflectance curves of 433 chips in the Munsell Book of Color have been found to depend on only 3 components which account for 99.18% of the variance. It is suggested that this 3-component dependency may be a characteristic of all organic pigments, including those in the retina, and thus explain the trichromatic nature of color vision. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Conference Paper
A wine-glass-type five-layered neural network (81-10-3-10-81) has been constructed, and identity mapping has been realized on the set of surface spectral reflectance data of Munsell color chips by a backpropagation learning algorithm. The network is divided into two parts: encoder (81-10-3) and decoder (3-10-81). Surface spectral reflectance data as physical attributes of color are transformed nonlinearly in each part. After identity mapping learning was completed, the response pattern of the three hidden units in the middle layer was analyzed to obtain the internal representation of color information acquired by self-learning. As a result, it was found that each hidden unit responds to psychological color attributes, that is, one for the value and the other two units for the constant value plane of the Munsell color system which consists of the hue and chroma. The nonlinear analysis method using five-layered neural networks is shown to be an efficient method for elucidating the color information coding mechanisms in the visual system
Recent computational models of color vision demonstrate that it is possible to achieve exact color constancy over a limited range of lights and surfaces described by linear models. The success of these computational models hinges on whether any sizable range of surface spectral reflectances can be described by a linear model with about three parameters. In the first part of this paper, I analyze two large sets of empirical surface spectral reflectances and examine three conjectures concerning constraints on surface reflectance: that empirical surface reflectances fall within a linear model with a small number of parameters, that empirical surface reflectances fall within a linear model composed of band-limited functions with a small number of parameters, and that the shape of the spectral-sensitivity curves of human vision enhance the fit between empirical surface reflectances and a linear model. I conclude that the first and second conjectures hold for the two sets of spectral reflectances analyzed but that the number of parameters required to model the spectral reflectances is five to seven, not three. A reanalysis of the empirical data that takes human visual sensitivity into account gives more promising results. The linear models derived provide excellent fits to the data with as few as three or four parameters, confirming the third conjecture. The results suggest that constraints on possible surface-reflectance functions and the "filtering" properties of the shapes of the spectral-sensitivity curves of photoreceptors can both contribute to color constancy. In the last part of the paper I derive the relation between the number of photoreceptor classes present in vision and the "filtering" properties of each class. The results of this analysis reverse a conclusion reached by Barlow: the "filtering" properties of human photoreceptors are consistent with a trichromatic visual system that is color constant.