First Eigenvector Second Eigenvector Third Eigenvector

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1 Non-euclidean structure of spectral color space Reiner Lenz a and Peter Meer b a) Dept. Science and Engineering, Campus Norrkoping, Linkoping University,SE-674 Norrkoping, Sweden,reile@itn.liu.se b) ECE Department,Rutgers University,94 Brett Road, Piscataway, NJ , 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 dicult to interpret. In this paper we show that for a large family of spectra the space of eigenvector coecients has a natural cone structure. Thus we can dene 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. Dening 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. 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 based 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, ). These systems have the advantage that many oftheirfeatures 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 dierent types of detectors. All colors are thus described by only three dierent 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 dierent reectance spectra produce the same sensor output under one illumination but dierent sensor outputs under a dierent light source. This is not only a problem for systems based on human color vision but also for similar technical systems such asrgb 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.,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 coecients describe the space of reectance spectra which are relevant for human color vision. These methods are eective as long as these coecients are only used for relatively simple numerical calculations. Results obtained in this way mayhowever be dicult 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:

2 the Lab-values. An illustration of the problems encountered is the following: assume you want to describe the color distributions of two dierent 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 coecients can be computed but these values are dicult to interpret since the raw coecients have no perceptual correlates. For Lab-based descriptions this is no problem since dierent 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 dicult to dene 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 rst show that the space of eigenvector coecients has a natural cone structure. This implies that there is a natural coordinate system in which these coecients can be described. It is then shown that the rst 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 coecient 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 diculties. The nature of the data allows investigations of higher order approximations along the same lines described below.. THE HYPERBOLIC STRUCTURE OF THE SPACE OF REFLECTION SPECTRA It is well known that the reectance spectra measured from the Munsell chips are all linear combinations of a few basis vectors.,{9 Usually the eigenvectors, computed from the set of measured reectance spectra, are taken as these basis vectors but other selections are also possible and have been used earlier.,7{ In the following we use as the variable describing wavelength and s() is a vector denoting an arbitrary spectral distribution. The eigenvectors (shown in Figure (a)) are written as b k () and for the coecients in the eigenvector expansion of the spectrum s() weuse k. Thus we have the following approximation: s() = KX k= k b k () () In the following we use mainly three basis functions K = and we thus choose to represent the spectrum s() by the coordinate vector =( ) in the coordinate system given by b () b () b (). In all earlier investigations (known to us) the vector is treated as a general elementinr. No further assumptions are made about other properties of these vectors. In our experiments we used a database consisting of reectance spectra of 78 color chips, 69 from the Munsell system and the rest from the NCS system. For each of the 69 chips of the Munsell System their spectra was measured from 8nm to 8nm at nm steps, while the samples from the NCS system were measured from 8nm to 78nm at nm intervals. These measurements were combined in one set consisting of 78 spectra (sampled in nm steps from 8nm to 78nm). 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 = ( ): ; ; () The only exception is the NCS chip 7-B (marked by the big dot in the Figure) a very dark blue color. For this spectrum the value of ; ; is equal P to -., i.e. is lying on the border of the cone. Furthermore we found that for all other spectra the values of K ; l= l 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 (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 coecients whereas is plotted along the z-axis. In the cone it is natural to consider three dierent types of natural curves:

3 Distribution of coefficients 7. Eigenvectors. Range 8nm 78nm. 78 NCS and Munsell spectra Ev. 4 Reflectance Wavelength First Eigenvector Second Eigenvector Third Eigenvector. Ev.. Ev. (a) The rst three eigenvectors (b) Distribution of coecient vectors Figure. Eigenvector expansion of color chips. The axis. hyperbola in planes containing the axis and. circles in planes perpendicular to the axis. This leads to the introduction of the hyperbolic coordinate system dened A =e For ' introduce the matrices D A and K: D() A cosh sinh cos ' A = h( ') () sinh sin ' cosh sinh sinh cosh A K(') cos ' ; sin ' A (4) sin ' cos ' The matrices A() and K(') and all possible products of them dene the group SO( ). It can also be shown that every matrix in SO( ) can be written as a product K( )A()K(') = T( ') which is similar to the Euler-angle representation of a three-dimensional rotation. We will therefore call ' the Euler angles of that transformation. The coordinate vector h( ') isgiven by: The axis, the hyperbola and the circles can now be described as: 7! A =( -axis,) h( ') A () 7! A =(hyperbola), ' 7! K(') We note also that these matrices form groups parametrized by one parameter since we A = (circle) (6) D( + )=D( )D( ) A( + )=A( )A( ) K(' + ' )=K(' )K(' ) (7)

4 These parameterizations of the curves is natural since a composition of mappings are described by additions of the parameter. A rst relation between the geometric and the perceptual concepts is obtained by an inspection of the rst three eigenvectors b k shown in Figure (a) and the denition of the coecients i as scalar products: k = hs() b k ()i (8) Since b is almost a constant and since the spectral distributions have only non-negative entries we see that is roughly equal to the L norm, and thus the energy of the spectrum s(). We call the origin of the space, the vector h( ) the \white Point" and h( ) the \achromatic axis".. RELATION BETWEEN THE LAB- AND THE HYPERBOLIC COORDINATE SYSTEMS Color spectra are traditionally described by their XYZ-coordinates (Chapter. in ) which are dened as the scalar products between the spectrum and the tristimulus functions x() y() z() : X = hs() x()i Y = hs() y()i Z = hs() z()i (9) 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 X Y A = Hh( '): () Z h kl = hx k () b l ()i () where we usedx () =x() x () =y() and x () =z(). The matrix H is computed as: H : :76 ;:6 :4 : ;:A () : ;:84 :9 Further understanding of role of the hyperbolic coordinates and ' can be obtained by relating them to the traditional chromaticity description in the Lab-system dened as (Sec...9, we often write L a b instead of L a b ): = " = # = " = # = L Y =6 ; 6 a X Y = ; b Y Z = ; () Y N X N Y N Y N Z N Here X N Y N Z N describes the (XY Z)-coordinates of the white point (which should not be confused with the \white point" mentioned above). Usually we dene the white point through the rst eigenvector, i.e. X N Y N Z N are the XYZ-coordinates of the rst 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: p b C ab = a + b (chroma) h ab = arctan (hue) (4) a Figure shows the relation between the hyperbolic coordinates ' and and the chromaticity coordinates in abspace. Figure (a) shows the function C ab ( ') and Figure (b) the function h ab ( ') (here a polar coordinate system in the ( ');space is used, i.e. the radial coordinate is given by and the angular coordinate by '. Tracing a circle around the origin in ( ');space varies the hue-angle ' while keeping the value of xed. Going on a straight line from the origin outward varies but leaves ' xed. Figure (b) shows that h ab is (up to a constant

5 ab-radius ab-angle Figure. Relation to Lab coordinates (a) Lab-Chroma (b) Lab-hue shift parameter) almost identical to ' for all values of : Figure (a) on the other hand illustrates that the relation between C ab and depends on the hue-angle (circles in ( ');space 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 ' coordinates of the spectrum. For all surface points the value of the variable is constant (and equal to the mean value of computed over all spectra). Figure shows the same information as Figure but now for the spectra in the spectral database. In these gures each point corresponds to a spectrum in the database. Figure (b) shows the dependence between the angular variable ' and the Lab-hue value h ab. It shows that there is a almost linear relation between ' and h ab except for the gray colors (Note that both variables are angles, i.e. periodic). For the relation between the hyperbolic coordinate and the Lab-chroma value C ab the situation is not as simple as can be seen from Figure (a). 7 Hyperbolic and Lab chroma for Munsell chips 6 Lab chroma (a) Lab-Chroma for chips (b) Lab-hue for chips Figure. Relation to Lab coordinates However selecting only a certain hue-range and plotting the C ab 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 demonstrates this relation for the hue ranges ;: <h ab < ;:77, ;:4 <h ab < ;:7, ;:7 <h ab < ;:78 and and :78 < h ab < :66: Here the spectra in the database were ordered according to their Lab-hue value h ab and divided into six sets with an approximately equal number of colors in each set. For each of these six sets the

6 distribution in the ( h ab ) space were plotted. Summarizing we see that the rst eigenvector coecient measures the intensity of the spectrum and that the norm of the vector ; measures the colorfulness of the spectrum, i.e. the distance to the gray color with the same intensity. In the coecient space we can thus introduce the metric ; ; () which measures the \whiteness" of a spectrum. In this interpretation the transformations in the group SO( ) are the whiteness preserving linear transformations of spectral space. 4. FOURIER-TRANSFORM IN COLOR SPACE In the previous sections we described a color spectrum with three coordinate values: '. It can be shown that the eect of can be treated independent of the other two variables. We will thus in the following assume that it has a xed 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 ; = ;cosh sinh cos ' sinh sin ' which will be denoted by ; '. Since we use ' to denote coordinates we will often use! to describe the parameters of the linear transformations. 9 Hyperbolic vs. ab radius. Angle Range: Hyperbolic vs. ab radius. Angle Range: Lab chroma 4 Lab chroma (a) ;: <h ab < ;:77 (b) ;:4 <h ab < ;:7 9 Hyperbolic vs. ab radius. Angle Range: Hyperbolic vs. ab radius. Angle Range: Lab chroma 4 Lab chroma (c) ;:7 <h ab < ;:78 (d) :78 <h ab < :66 Figure 4. The relation between ' and C ab

7 In the previous section we showed that the linear transformations in the group SO( ) preserved the \whiteness" of color spectra. These transformations can all be parametrized by their Euler angles and it is therefore sucient to consider transformations K(!) and A(). The rotations K(!) dene also rotations in space. More complicated is the operation of the A() operations. Applying A() to the point ; ' and denoting the coordinates of the resulting point by ; b b' we get for the new coordinates the relations: cosh b = cosh cosh +sinhsinh cos ' e b' = sinh cosh +coshsinh cos ' + i(sinh sin ') sinh b For jcos 'j = this is a simple addition/subtraction of the hyperbolic parameters and thus a shift along the hyperbola. For other values of ' this is a much more complicated transformation. For a general transformation with Eulerangles!! we write such a transformation as T(!! ). Making the following discussion more concrete we assume that we have functions p( ') which could describe the probability distributions of the color points in images. Applying a whiteness-preserving transformation T(!! ) to color space will map the color distribution p( ') to the distribution p(b b'). For the transformation A() = T( ) the coordinate transformation is given by Equation(6). This transformation on the functions p is denoted by a superscript describing he transformation of the underlying color space: (6) p(b b') =p(t ; ( ')) = p T ( ') (7) General pattern recognition problems that can be formulated in this framework are:. Invariant feature extraction: For all probability distributions construct features F such thatf (p) =F (p T ) for all transformations T.. Given two probability distributions p 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 7! p(' ;!). In this case the functions p can be developed into a Fourier series in the variable ' : p(') = n a ne in' and the Fourier coecients a n will undergo a simple multiplication a n 7! a n e in!. Invariant features are in this case the absolute values of the Fourier coecients and the transformation parameter! can be estimated from the phase factor. We also note that these features form a complete set since the Fourier coecients dene the function p completely. The general problem is now to nd 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 see { ). 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. For integers or half- Our outline of the appropriate Fourier transform starts with the Legendre functions. integers m n and a complex constant we introduce the functions (see page 4 in 4 ) P mn(cosh t) = Z For the special case n =we obtain (see page in 4 ): P m(cosh t) = and for m = n =: cosh t +sinh t ei +n cosh t +sinh t e;i ;n e i(m;n) d (8) ;( +) ;( ; m +) P;m (cosh t) (m ) P ;( +) m(cosh t) = ;( + m +) Pm (cosh t) (m ) (9) P (cosh t) =P (cosh t) =P (cosh t) ()

8 (a) P ki;= (cosh t) (k =::4) (b) P n i;= (cosh t) (n =::4) Figure. A few Legendre Functions Here P m and P are the (associated) Legendre functions. A few typical examples of (associated) Legendre functions are shown in Figure. These functions satisfy the following addition formula: e ;i(m b'+n ) P mn(cosh b) = X k=; e ;ik' P mk(cosh )P kn(cosh ) () where k runs over the integers if m n are integers and runs over the half-integers if m n are half-integers. The relation between the coordinates is the same as in Equation (6). Note that the left-hand side of the equation contains a phase factor e ;in where the angle is a function of ' dened by: e i('+ )= = cosh t cosh t ei'= + sinh t sinh t cosh t cos e;i'= () For the special case m = n = this becomes: P (cosh cosh +sinh sinh cos ') = X k=; e ;ik' P k (cosh )P ;k (cosh ) () This describes how the value of the Legendre function at the point( ') transforms under a hyperbolic transformation with parameter. We see that the transformed Legendre function is a series of associated Legendre functions with the same parameter. The simple multiplicative transformation property of the exponential function under shifts is thus replaced by an innite sum over related functions. This gives the required transformation formula under a hyperbolic transformation (Equation (6)) and the associated Legendre functions are thus the appropriate generalizations of the exponential functions for hyperbolic transformations. This holds for complex values of the parameter which has not yet been specied. Special cases will be discussed later. For an application in feature extraction we will in general need a Hilbert space which contains both the function p which should be analyzed and the Legendre functions. The scalar product in this Hilbert space is furthermore invariant under the operation of the transformation group in the sense that hf T g T i = hf gi (4) 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 p T using: hp T; P i = hp P T i () and use the transformation rule in Equation () to compute the transformed Legendre function P T :

9 4.. The Mehler-Fok transform The Legendre functions provide a family of functions which 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 (in 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 which will be described next (see page 7 in ). Here we will only consider functions of the parameter and hyperbolic transformations. The variables and ' are kept xed and will be ignored to simplify notations. The Mehler-Fok transform states now that the Legendre functions with parameter = i ; = form a complete function system (see page 48 in 4 ): then Put: F (x) = c() = Z Z F (x)p ;i;= (x) dx (6) c()p i;= (x) tanh() d (7) This shows that the system P i;= (x) : R forms a complete set in which functionsinthevariable can be expanded. This can be generalized to functions which are not only functions of but of the angular variable ' as well. We refer the interested reader to chapter. in. A more traditional approach can be found in chapter 7 of 6 and also in chapters 7 and 8 in. 7. EXPERIMENTS In the previous sections we sketched how the geometry of spectral space implies the existence of a kind of Fourier transform compatible with this geometry. Numerical and practical problems, such as the computation of the Legendre functions, the selection of a nite number of lter parameters and the sampling of the functions involved have to be solved before the theoretical results can be applied to solve practical problems. We will not go into that here but rather illustrate the performance of the method with an example. In our experiment we used two databases consisting of images each. The images were captured under carefully controlled conditions. 8 The images show objects under ve dierent illuminants. In the rst database the objects are all in 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 ve images of the same object under dierent illuminants in dierent 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 images are rather dark. We converted RGB-vectors to vectors of eigenvector coecients by using a (a) syl-cwf (b) mb- (c) mb-+ (d) ph-ulm (e) halogen Figure 6. Flower images matrix multiplication. From them the hyperbolic coordinates ' were computed. In the following we ignore

10 and investigate only the components '. For each of the images we estimated the joint probability distributions of ( ') and the densities of and ' separately. These probability distributions were estimated using a kernel density estimation method. The two two-dimensional probability distributions for two ower images in the same position under the illuminants Philips Ultralume uorescent (ph-ulm) and Macbeth K uorescent in conjunction with the Rosolux Full Blue lter (mb+) are shown in Figure 7. The distributions of the and variables are shown in 8. These distributions show that the blue illuminant mainly eects the distribution of the variable: it is more concentrated and shifted. The distributions of the 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 images in the database the Mehler-Fok transform at the positions = k= (k = :::4). A few of the results obtained are shown in Figure 9(a-c) In these gures we see Flower, mb+ Illuminant Flower, ph ulm Illuminant Blue Density.. ULM Density.. φ φ... (a) mb+ (b) ph-ulm Figure 7. Density distributions for ( ) computed from ower images Densities of for the ph ulm and mb+ flower images Ph Ulm mb Densities of φ for the ph ulm and mb+ flower images Ph Ulm mb Density 6 Density φ (a) Distribution of (b) Distribution of Figure 8. Probability densities computed from ower images that for low values of the parameter (which correspond to the \low frequency" responses of the ordinary Fourier

11 τ = *i / τ =.*i / hal mb+ mb ulm cwf hal mb+ mb ulm cwf Feature Value Feature Value Image Number Image Number (a) P ;= (b) P :i;= τ = 4*i / Comparing aligned and non aligned objects hal mb+ mb ulm cwf 4 Aligned Objects Random Positions Feature Value. Feature Value Image Number ball ball book coff crun flow jave maca rope sham tide Object Number (c) P 4i;= (d) P :i;= aligned vs. nonaligned objects Figure 9. Features computed with the Mehler-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 (mb+) 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 P :i;=. The values computed from the rst images (showing the eleven objects in the xed position) and the values computed from the last 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 dierent positions but dierent for dierent objects and dierent illuminants. For some of the objects (such as the book) there is no great variation between the color distributions in the dierent images since only the position of the object points in the image vary. For other objects, like the balls or the owers, there is however a signicant change since dierent parts of the object are visible in the dierent 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 identied 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 Mehler-Fok transform. Finally we illustrated how this transform can be used to construct pattern recognition algorithms which can be useful in analyzing distributions

12 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: html The NCS spectra was obtained from the Scandinavian Color Institute in Stockholm courtesy to B. Kruse. The images are available under the address The kernel density toolbox is available under R. Lenz was supported by grant 98- from the Swedish Research Council for Engineering Sciences and grant 98.7 from CENIIT, the Center for Industrial Information Technology at Linkoping University. REFERENCES. G. Wyszecki and W. Stiles, Color Science, Wiley & Sons, London, England, ed., 98.. M. D. Fairchild, Color Appearance Models, Addison-Wesley, C. A. Parraga, G. Brelsta, T. Troscianko, and I. R. Moorehead, \Color and luminance information in natural scenes," J. Opt. Soc. Am. A, pp. 6{69, March D. L. Ruderman, T. W. Cronin, and C.-C. Chiao, \Statistics of cone responses to natural images: implications for visual coding," Journal of the Optical Society of America A, pp. 6{4, Aug J. Cohen, \Dependency of the spectral reectance curves of the Munsell color chips," Psychon Science, pp. 69{7, J. Parkkinen, J. Hallikainen, and T. Jaaskelainen, \Characteristic spectra of Munsell colors," Journal of the Optical Society of America A 6(), pp. 8{, S. Usui, S. Nakauchi, and M. Nakano, \Reconstruction of Munsell color space by a ve-layer neural network," Journal of the Optical Society of America A 9, pp. 6{, April 99. Color. 8. L. T. Maloney, \Evaluation of linear models of surface spectral reectance with small numbers of parameters," Journal of the Optical Society of America A, pp. 67{68, October M. D'Zmura and G. Iverson, \Color constancy. I. Basic theory of two- stage linear recovery of spectral descriptions for lights and surfaces," Journal of the Optical Society of America A, pp. 48{6, 99.. R. Lenz, M. Osterberg, J. Hiltunen, T. Jaaskelainen, and J. Parkkinen, \Unsupervised ltering of color spectra," Journal of the Optical Society of America A (7), pp. {4, S. Lang, SL(,R), Graduate Texts in Mathematics (Vol. ), Springer Verlag, New York, Berlin, Heidelberg, 98.. M. Sugiura, Unitary Representations and Harmonic Analysis, Kodansha Ltd., John Wiley and Sons, 97.. A. Wawrzynczyk, Group Representations and Special Functions, D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, N. Vilenkin and A. Klimyk, Representation of Lie groups and special functions, Mathematics and its applications : 7, Kluwer Academic, D. Gurarie, Symmetries and Laplacians, Introduction to Harmonic Analysis, Group Representations and Applications, vol. 74 of North Holland Mathematics Studies, North Holland, Amsterdam, London, New York and Tokyo, I. N. Sneddon, The Use of Integral Transforms, McGraw Hill Book Company, N. N. Lebedev, Special Functions and their Applications, Dover Publications, New York, B. Funt, K. Barnard, and L. Martin, \Is machine colour constancy good enough," in Proc. ECCV-98, pp. 44{ 49, Springer Verlag, 998.