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Noneuclidean structure of spectral color space
Reiner Lenz a and Peter Meer
a) Dept. Science and Engineering, Campus Norrköping,
Linköping University,SE601 74 Norrköping, Sweden,reile©itn.liu.se
b) ECE Department,Rutgers University,94 Brett Road
Piscataway, NJ 088548058, USA,meer©caip.rutgers.edu
ABSTRACT
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, noneuclidean geometry, MehlerFok transform
1. MOTIVATION
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 CIELab system, for example, the Lcomponent describes the
lightness, and the length and the orientation of the vector in the abplane describe the saturation and the hue of
the spectrum. The euclidean distance between two Labvectors 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 bestknown 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 multichannel
cameras which allow a better reconstruction of the underlying color spectra than the usual RGBcameras. 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 lowdimensional 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
A
color version of the paper is available under: http:
//www. itn. liu.
se/reile/prints/noneucol .ps
Part
of the EUROPTO Conference on Polarization and Color Techniques in Industrial Inspection
Munich, Germany . June 1999 SPIE Vol. 3826. 0277786X/99/$1 0.00
101
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the Labvalues. 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 ndimensional 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 Labbased descriptions this is no problem since different
mean values for the Lcomponents have the obvious interpretation of being lighter/darker on average. Since the
rawmeasurements lack simple interpretations it is also difficult to define which statistics are best suited for a given
application.
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.
2. THE HYPERBOLIC STRUCTURE OF THE SPACE OF REFLECTION SPECTRA
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:
s(A)=kbk(A)
(1)
In
the following we use mainly three basis functions K =
3
and we thus choose to represent the spectrum s())
by the coordinate vector a =
(ai
,
a2
,
a3)
in the coordinate system given by b1 ()) ,
b2
(..\) ,
b3
()).
In
all earlier investigations (known to us) the vector a is treated as a general element in R .
No
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
C={(ai,a2,a3):a?—a—aO}
(2)
The
only exception is the NCS chip 2070B (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 threedimensional 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 ycoordinates in the plot correspond to the second and the third eigenvector coefficients u2, cr3
whereas
ai
is
plotted along the zaxis.
In
the cone it is natural to consider three different types of natural curves:
102
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Fico Eigenvector
  
Second Eigenvecto
Third Eigenvector
Dsmbuton of coefhcents
3
c
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
axis.
This leads to the introduction of the hyperbolic coordinate system defined by:
For p, c, introduce the matrices D, A and K:
fcTi\
/ coshc
=
e
(sinh a cos
=
h(p,
a, ip)
\sinhasinp)
(1
D(p) =
e
(0
\0
0
1
0
0\
(
0
J
,
A(a)
=
(
1)
\
cosh a
sinh a
0
sinh a
cosh a
0
(1
0 J
,
K(p)
=
(0
1)
\0
0
cos p
sinp
0
—
sin
cp
cos
The matrices A(a) and K(ço) and all possible products of them define the group SO(2,
1).
It can also be shown
that every matrix in SO(2,
1)
can be written as a product K(i')A(a)K(p) =
T(,
a, ) which is similar to the
Eulerangle representation of a threedimensional rotation. We will therefore call
a, ço the Euler angles of that
transformation. The coordinate vector h(p, a, ) is given by:
fi\
h(p,a,ço) =
D(p)K(ço)A(a)
(
0
)
(5)
\01
The a axis, the hyperbola and the circles can now be described as:
(1\
('\
fi\
p +
D(p)
(
0
=
(oaxis,)
a
A(a)
(
0
)
= (hyperbola),
K(p)
(
0
j =
(circle)
(6)
\\01
\0J
\0J
We note also that these matrices form groups parametrized by one parameter since we have:
D(pi
+ P2) = D(pi)D(p2),
A(ai + a2)
=
A(ai)A(a2), K(pi + P2) = K(çoi)K(p2)
(7)
Eoooncs. Rcno38oo  780cc. 2182 NCS&odMocs.Hspnco
0.01
400
450 500 55.0 000 650 700
750
Wo&ch
( a) The first three eigenvectors
2. Ev.
3. En.
(b)
Distribution of coefficient vectors
(3)
(4)
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These parameterizations of the curves is natural since a composition of mappings are described by additions of the
parameter.
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
as
scalar products:
ak
(s(A),bk(A))
(8)
Since b1 is almost a constant and since the spectral distributions have only nonnegative entries we see that a is
roughly equal to the L1 norm, and thus the energy of the spectrum s(A).
We
call the origin of the space, the vector h(O, 0, 0) the "white Point" and h(p, 0, 0) the "achromatic axis".
3. RELATION BETWEEN THE LAB AND THE HYPERBOLIC COORDINATE
SYSTEMS
Color spectra are traditionally described by their XYZcoordinates (Chapter 3.3 in1 )
which are defined as the scalar
products between the spectrum and the tristimulus functions x(A), y(A), z(A)
x
= (s(A),x(A)),
1"
=
(s(A),y(A)),
Z =
(s(A),z(A))
(9)
Writing this as a matrix multiplication we get for the transformation from the hyperbolic to the XYZsystem the
following relation:
fx\
I Y
)
=Hh(p,,).
(10)
\Z)
The elements of the matrix H are computed as:
hkl
(Xk()t),bl(/\))
(11)
where we used x1
(A)
=
x(A), x2(A) = y(A)
and x3(A) =
z(A).
The matrix H is computed as:
/2.35
.76
—1.26\
H =
( 2.34
.01
—3.00 )
(12)
\2.55 —2.84
1.39)
Further understanding of role of the hyperbolic coordinates and can be obtained by relating them to the tradi
tional chromaticity description in the Labsystem defined as1 (Sec. 3.3.9, we often write L, a,
b instead of L* ,
a* b*):
Y
1/3
x 1/3
y 1/3
Y 1/3
1/3
L*
116 ()
 16;
a*
500 [()
() ] ;
b*
200 [()
 () ]
(13)
Here XN ,
N
,
ZN
describes the (XYZ)coordinates of the white point (which should not be confused with the "white
point" mentioned above) .
Usually
we define the white point through the first eigenvector, i.e. XN ,YN
,
ZN
are the
XYZcoordinates 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 abplane 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
+
b2
(chroma);
hab
arctan ()
(hue)
(14)
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, )
and
Figure 2(b) the function hab(a, cp) (here a polar coordinate
system in the (a, )
—space
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 hueangle 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
104
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Figure 2. Relation to Lab coordinates
(a) LabChroma
(b) Labhue
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 hueangle (circles in (ct,
p)—space
are mapped to ellipses in abspace).
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 Labhue 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 Labchroma value Cab the situation is not as simple as can be seen from Figure 3(a).
(a) LabChroma for chips
(b) Labhue for chips
Figure 3. Relation to Lab coordinates
However selecting only a certain huerange and plotting the a, Cab coordinates for all chips in this huesegment
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 <
hab
< 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 Labhue value hab
and divided into six sets with an approximately equal number of colors in each set. For each of these six sets the
105
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distribution in the (cr,
hab)
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)
measures
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
—
cr
(15)
which measures the "whiteness" of a spectrum. In this interpretation the transformations in the group 80(2, 1)
are
the whiteness preserving linear transformations of spectral space.
4. FOURIERTRANSFORM IN COLOR 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 threedimensional space. The corresponding point in threedimensional space has coordinates (a1, a2, a3) =
(cosh ,
sinh
ccos
,
sinh£1sin
p) which will be denoted by (cl,
p).
Since we use c, to denote coordinates we will
often use $,
w to
describe the parameters of the linear transformations.
D'.d%
—'
(Ø
bGc.
.4'4:
*
2O
0 0.2
04 00 00
4
0
2
106
(d) 0.78 < hab < 1.66
Figure 4. The relation between c, and Cab
(a) 3.13 < hab
< 1.77
1 2/22 . 0
(b) —1.42 < hab < 1.07
(
14y24b010 *
A03i020044 0 /04< .. I (001
(0
.
:* :
0,
40
'. *
4 a
20
0
02
0.0
00 00
2 10
14. 10 2
(c) —1.07 < hab < 0.78
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In the previous section we showed that the linear transformations in the group SO(2,
1)
preserved the "whiteness"
of color spectra. These transformations can all be parametrized by their Euler angles and it is therefore sufficient
to consider transformations K(w) and A($). 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 (,
ço)
and denoting the coordinates of the
resulting point by (,
) we
get for the new coordinates the relations:
cosh
=
cosh
3 cosh c + sinh $
sinh
c
cos
@ —
sinh
/3
cosh
c
+
cosh fi
sinh
a cos p + i(sinh a sin )
(16)
e
sinh
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 ,
3,
W2).
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 whitenesspreserving transformation T(wi ,
3,
W2)
to color space will map the color distribution p(c, )
to
the distribution p(&, ).
For the transformation A($) =
T(O,
/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(&,)
=p(T'(,ço)) pT(a,)
(17)
General
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) =
F(pT)
for
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 hueshift operators and the functions are
independent of
: p(p)
p(ço
—
w).
In this case the functions p can be developed into a Fourier series in the
variable
: 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 hueshifts. 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 MehlerFok 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)
2ir
T+fl
Tfl
3(cosh
t) =
f
(cosh + sinh e°)
(cosh + sinh e_) ei(m_ dO
(18)
For the special case n =
0
we obtain (see page 322 in'4):
0(cosht) =
F(rrn
1)m(cosht) (m 0);
0(cosht) =
F(r+rn 1)
cosht) (m <0) (19)
and for m =
n
=
0:
q30(cosht) =
q3(cosht)
=
q3(cosht)
(20)
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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= —
oc
where
k runs over the integers m, n are integers and runs over the halfintegers m, n are halfintegers. The relation
between the coordinates is the same as in Equation (16). Note that the lefthand side of the equation contains a
phase factor eintI) where the angle t' is a function of a, ,
$
defined by:
e2 —
cosh

cosh
ie"2 + sinh
sinh cose'2
(22)
—
cosh
For the special case m =
n
0 this becomes:
r(cosh cosh $ + sinh sinh fi cos ) =
koc
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) =
(f,g)
(24)
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) =
(p,q3)
(25)
and use the transformation rule in Equation (23) to compute the transformed Legendre function
108
(a) 3k.1_l/2(cosht), (k =
0.4)
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4.1. The MehierFok transform
The Legendre functions provide a family of functions whicli have a similar transformation property under hyperbolic
transformations as the exponential function under shifts. Another aspect of Fourier transform is completeness. i.e. we
can express each function (iii a large function space) uniquely as linear combinations of the exponential functions. For
the Legendre functions and the hyperbolic transformations the corresponding transform is the Mehler—Fok transform
vIucli will he described next (see page 373 ill'3)
Here we will only consider functions of the parameter o and hyperbolic transformations. The variables p and
are kept fixed and will be ignored to simplify notations. The MehlerFok transform states now that the Legendre
functions with parameter
iv
—
1/2
form a complete function system (see page 485 in'4):
Put:
e(p) =
/ F(x)_1_,12(x)
d
(26)
then
F(z) =
f c(p),_l/2(I)ptanh(p)
dp
(27)
This shows that the system {ip1/9(I)
:
p E }
forms
a complete set in which functions in the variable Q can
be expanded. This can be generalized to functions which are not only functions of but of the angular variable
as well. \Ve refer the interested reader to chapter 10.5 in.3 A more traditional approach can be found in chapter 7
of'6 and also in chapters 7 and 8 in.17
5. EXPERIMENTS
In the previous sections we sketched how the geometry of spectral space implies the existence of a kind of Fourier
transform conipatible with this geometry. Numerical and practical problems, such as the computation of the Legendre
functions, the selection of a finite number of filter parameters T
and
the sampling of the functions involved have to
be solved before the theoretical results can he applied to solve practical problems. \Ve will not go into that here hut
rather illustrate the performance of the method with an example.
In our experiment we used two databases consisting of 55 images each. The images were captured under carefully
controlled conditions) The images show 11 objects under five different illuminants. In the first database the
objects are all imi the same position and only the illuminants vary. In the second database both the illumination
and the objects placement in the scene is changed. One series of five images of the same ohject under different
illmimninants in different positions is shown in Figure 6. The images in the databases were originally used to test color
constancy algorithms and great care was taken that there was no clipping under the brightest lighting conditions. As
a result the iluages are rather dark. We converted RGBvectors to vectors of eigelivect.or coefficients by using a 3 x 3
Figure 6. Flower images
matrix multiplication. From theni the hyperbolic coordinates p. (1.
'p were
computed. In the following we ignore f)
109
(a) sylewf
(h) mnh5000
(c) nib500fl+3202
(d) phuln,
(e) halogen
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110
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 twodimensional probability distributions for two flower images in the same position under the illuminants
Philips Ultralume fluorescent (phulm) 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 MehlerFok transform at the posi
tions r =
k/lO
(k =
0..
.4). A few of the results obtained are shown in Figure 9(ac) In these figures we see
R 5OOO.32O2 N
os

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) phulm
b5OOO+32O2 k* ,g.s
PhUtm
  
mb5OOO+32Oj
(b) Distribution of
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( d) O.5i—1/2 aligned vs. non
aligned objects
Figure 9. Features computed with the MehierFok 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 nonaligned 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.
6. CONCLUSIONS
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 MehierFok transform. Finally we illustrated how
this transform can be used to construct pattern recognition algorithms which can be useful in analyzing distributions
111
(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.
ACKNOWLEDGEMENTS
The Munsell spectra is available from the Information Technology Dept., Lappeenranta University of Technology,
Lappeenranta, Finland under the address: http : //www .lut
.
fi/ltkk/tite/research/color/lutcs_database.
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.
sfu
.
Ca/Co1our/imagedb.
The kernel density toolbox
is available under http://science.ntu.aC.uk/msor/cCb/densest.html. R. Lenz was supported by grant 98530
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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