Color matching and characterization of surface coatings

A method of characterization of a coating on a surface such as a paint coating having metallic flakes distributed within such coating. Single and multi-angle models are utilized. The coating is characterized by measurements having components of light reflected by the metal flakes and attenuated on its exit route from the coating, of light scattered in the entering path of the beam as it travels through the coating, of light scattered by the reflected beam on its exit from the coating, of light scattered in the entering path then reflected by the metallic flakes and attenuated in the coating on exiting, and of light scattering in the exit path, both to the metallic flake and reflected by the flake and attenuated on its exit from the coating. Methods of color matching a coating having a metallic flake distribution within the coating are included using reflectance factors and tristimulus values and minimizing the sum of the squares of the deviation between samples.

BACKGROUND OF THE INVENTION 
The Kubelka-Munk Mode 
A two flux Kubelka-Munk model is widely used in the coatings industry for 
the color matching and batch correction of solid color coatings. This 
theory in its two constant version gives good color matches and batch 
corrections for solid color coatings in the pastel and midtone ranges. 
With dark colors the results are poorer but still usable. Coatings 
containing metallic flake have found wide acceptance in the automotive 
industry and are finding growing acceptance in the extrusion and 
industrial areas. These coatings exhibit a strong variation in appearance 
as the angle of view and illumination are changed. Attempts to color match 
coatings containing metal flakes using conventional color 
spectrophotometers and solid color matching software have been very 
disappointing. As a result, these coatings are often matched and batches 
corrected by visual tinting procedures. 
Despite many efforts to extend the two flux Kubelka-Munk theory to metallic 
coatings, the poor agreement between predicted reflectance values, coating 
composition, and visual appearance has been disappointing. Part of the 
difficulty lies in the limitations of conventional 45/0 and 
diffuse/near-normal measuring geometry spectrophotometers to adequately 
characterize the appearance of coatings containing metallic flake. In 
addition, part of the poor agreement seems to be associated with the 
nature of the color matching algorithm itself. 
The Kubelka-Munk model for a solid color considers a pigmented film of 
thickness x.sub.0. Light falls on the film from above. The two flux 
Kubelka-Munk model considers a downward flux of light, I, and an upward 
flux, J. The downward flux is reduced by absorption and scattering and 
increased by the light scattered from the upward flux into the downward 
directed flux according to the equation 
##EQU1## 
where k is the absorption coefficient and s is the scattering coefficient 
of the film. The minus sign in the derivative is the result of taking the 
downward direction along the negative x axis. 
The upward flux is reduced by absorption and scattering and increased by 
the light scattered from the downward flux into the upward directed flux 
according to the equation 
##EQU2## 
The film reflectance is defined as the ratio of the upward to the downward 
directed flux. 
EQU R=J/I 
The rate of change of the reflectance with film thickness is obtained by 
differentiating the reflectance equation with respect to x. 
##EQU3## 
Substituting the expressions for dJ/dx, dI/dx, and making use of the 
definition of R, we obtain 
##EQU4## 
The reflectance R can be obtained by solving the differential equation for 
R. Alternatively the two simultaneous differential equations for I and J 
can be solved and R follows from the ratio of the upward to the downward 
flux. The complete solution is required for optically thin films and is of 
interest for hiding power and transparency calculations. At complete film 
hiding (infinite optical thickness) the solution simplifies to the form 
used for color matching and batch correction as used in commercial color 
matching software. 
The solution of the differential equations for the fluxes I and J and the 
calculation of R for the Kubelka-Munk model are given in P. Kubelka, "New 
Contributions to the Optics of Intensely Light-Scattering Materials, Part 
I," Journal of Optical Society of America, 38, 448-457, 1948, along with a 
number of other useful relations. The final result for the reflectance of 
a film of thickness X over a substrate of reflectance R.sub.G is 
##EQU5## 
where a=1+k/s; 
b=(a.sup.2 -1).sup.1/2 ; 
k is the absorption coefficient of the film; 
s is the scattering coefficient of the film; 
Coth(b*s*X) is the hyperbolic cotangent of b*s*X. 
Taking the limit of R as X tends to .infin. gives 
EQU R.sub..infin. =a-b 
It is shown in the prior cited Kubelka article that 
EQU a=1/2* (1/R.sub..infin. +R.sub..infin.) 
EQU b=1/2* (1/R.sub..infin. -R.sub..infin.) 
The Kubelka-Munk equation relating to k/s to R.sub..infin. can easily be 
derived from 
EQU k/s=a-1 
giving 
EQU k/s=(1-R.sub..infin.).sup.2 /(2*R.sub..infin.) 
This last relationship provides the basis for relating sample composition 
to R.sub..infin. (the film reflectance at complete hiding or an infinite 
optical film thickness) and is used in many applications of the 
Kubelka-Munk theory to industrial color matching. This equation is well 
known in the coatings and plastics industries and is used as the basis for 
color matching, batch correction and strength calculations. 
An important, additional assumption in the theory is the additive character 
of k and s in relation to a film's composition. For a film containing N 
colorants, it is assumed that 
##EQU6## 
where k.sub.i is the absorption coefficient, s.sub.i is the scattering 
coefficient, and c.sub.i is the concentration of the ith colorant in the 
film. The summation is taken over the N colorants in the film. 
If a film's composition is known along with the absorption and scattering 
coefficients of each colorant, then the k and s for that film can be 
calculated. The reflectance of the film then follows from the relationship 
EQU R.sub..infin. =1+(k/s)-{(k/s).sup.2 +2*(k/s)}.sup.1/2 
which is the inverse of the equation previously given relating k/s to 
R.sub..infin.. 
Metallic Modifications to the Kubelka-Munk Model 
Various attempts have been made to use the Kubelka-Munk model to color 
match metallic flake containing coatings. The earliest attempts involved 
preparing mixtures of aluminum flake and colored pigments and calibrating 
the aluminum flake pigment as if it were a white pigment. The predicted 
color matches and batch corrections using this procedure have been so 
unreliable that these operations are now routinely carried out by visual 
matching and tinting procedures. 
J. G. Davidson (in a Ph.D. thesis, Rensselaer Polytechnic Institute in 
Troy, N.Y., February, 1971) proposed an alternate treatment of the 
metallic problem. He considered each aluminum flake as a single reflector 
with reflectance, r, and reflection cross section, .sigma.. He proposed 
that the changes to the downward and upward fluxes are given by the 
equations 
##EQU7## 
These equations differ from the previous Kubelka-Munk equations in that the 
aluminum flake pigment is assumed to remove light from the directed flux 
in proportion to its cross section, .sigma., and add light to the 
complimentary flux in amount r*.sigma. times the directed flux. The other 
pigments are assumed to contribute to the fluxes in the same manner as the 
preceding model for solid colors. 
Substituting these equations into the equation for dR/dx we obtain 
##EQU8## 
This differential equation can be solved for R. The solution is identical 
to the above Kubelka-Munk solution if s is replaced by (s+R*.sigma.) and 
(k+s) is replaced by (k+s+.sigma.). 
The solution for a complete hiding layer can be derived directly from the 
condition dR/dx=0. This yields the result 
##EQU9## 
E. D. Campbell (in a Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, 
N.Y., August, 1972) applied the Davidson modification to mixtures of 
metallic flake and colored pigments. Predictions for matching and batch 
corrections were shown to be unreliable. 
Radiative Transfer Theory 
S. Chandrasekhar in "Radiative Transfer Theory" (Oxford, Clarenden Press, 
1950) describes the foundations of radiative transfer theory and applies 
this theory to a variety of astrophysical problems concerning steller 
atmospheres and planetary atmospheres. Of particular interest are the 
exposition of the foundations of radiative transfer theory and its 
application of this theory to the planetary atmosphere problem. 
Chandrasekhar has investigated the angular scattering of planetary and 
stellar atmospheres and formulated the equations and methods to solve 
angular scattering problems. The basic equation of transfer for the 
angular intensity as a function of angle and optical depth is 
##EQU10## 
where I is the intensity of the light incident on the coating; 
.tau. is the optical depth of the coating; 
.mu. is the cosine of the observation angle, .theta..sub.2 ; 
.phi. is the azimuthal angle of the observation; 
.mu.' is the cosine of an illumination angle, .theta.'; 
.phi.' is an azimuthal angle; and 
p is the scattering phase function. 
Let the flux incident on surface of the coating be designated by F, the 
cosine of the angle of illumination by .mu..sub.o, and the azimuthal angle 
of illumination by .phi..sub.o. 
It is sometimes convenient to distinguish between the reduced incident 
radiation 
EQU .pi.Fe.sup.-.tau./.mu..sbsp.o 
which penetrates to the level .tau. and the diffuse radiation field that 
arises from one or more scattering or reflection processes. The equation 
of transfer for the diffuse radiation field is 
##EQU11## 
For a diffuse isotropic radiation field with an albedo .OMEGA..sub.o, the 
equation of transfer is 
##EQU12## 
This equation has been solved by Chandrasekhar who showed that the exact 
solution is 
##EQU13## 
where the H(.mu.) functions are defined by the nonlinear integral equation 
##EQU14## 
and can be numerically estimated by an iteration procedure described by 
Chandrasekhar. These functions have been tabulated by several 
investigators and can be computer generated on a personal computer. 
Approximate Methods Based on Averaging Intensities 
Prior to and since Chandrasekhar's exact solution to the problem many 
efforts have been made to solve the radiative transfer equation by 
approximate methods for various isotropic and anisotropic scattering 
models of interest in atmospheric science and technology. 
One such method for isotropic scatterers was proposed by Schuster in 
"Radiation Through Foggy Atmospheres," Astrophys. Journal, 21, 1, 1905. 
This method involves the directional averaging of intensities. 
Starting from the basic equation of transfer for isotropic scatterers 
##EQU15## 
where the reduced incident radiation is included with the scattered 
radiation, we define two average fluxes by the equations 
##EQU16## 
The flux J is an upward directed flux and the flux I is a downward 
directed flux. 
Integrating the equation of transfer over .mu. from 0 to 1 and then over 
.mu. from 0 to -1 we obtain the following two equations 
##EQU17## 
The integrals on the left sides of these equations can be approximated as 
follows. 
##EQU18## 
Combining these results, a set of simultaneous linear differential 
equations is obtained for the upward and downward fluxes. 
##EQU19## 
Equations of this type have been used by many investigators to solve a 
variety of technical problems involving turbid materials. Kubelka and Munk 
in "Ein Beitrag zur Optik der Farbanstriche" (Z. Tech. Phys., 12, 593-601, 
1931) derived similar equations which Kubelka ("New Contributions to the 
Optics of Intensely Light-Scattering Materials, Part I," Journal Opt. 
Society of Am., 38, 448-457, 1948) cast into the form that is widely used 
today for color matching coatings, plastics, inks, and related materials. 
A Radiative Transfer Derivation of the Kubelka-Munk Equation 
Kubelka and Munk derived their turbid media theory without consideration of 
the more rigorous radiative transfer equation. The following line of 
thought establishes the relation between the average flux equations 
derived from the radiative transfer equation and the Kubelka-Munk 
equations. 
The albedo of the material, .omega..sub.o, is related to the absorption 
coefficient, K, and a scattering coefficient, S, of the material by the 
relation 
EQU .omega..sub.o =S/(K+S) 
The optical depth .tau. is related to K, S, and the distance x inside the 
material by the relation 
EQU .tau.=(K+S) x 
Introducing these relationships and rearranging the differential equations 
for the averaged intensities result in the Kubelka-Munk equations 
##EQU20## 
In these equations x is the distance from the unilluminated surface 
according to Kubelka-Munk rather than the distance from the illuminated 
surface which is the convention used for the radiative transfer equations. 
The Kubelka-Munk absorption coefficient, k, and scattering coefficient, s, 
are related to the radiative transfer absorption coefficient, K, and 
scattering coefficient, S, by 
EQU s=S/2 
EQU k=2 K 
These equations are strictly valid for diffuse illumination and diffuse 
viewing conditions. These conditions are seldom encountered in practice. 
Despite this limitation the results obtained using 45/0 and diffuse near 
normal instruments has been very satisfactory for pastel and midtone solid 
color materials. The results for deeptone or dark materials have been less 
satisfactory but usable for industrial color matching. 
In the case of materials showing strong directional appearance 
characteristics such as metal flake pigmented coatings, pearlescent 
pigmented coatings, etc., the theory has been inadequate to such an extent 
that these materials are matched by visual matching methods. 
The inability of average flux approximations to predict the angular 
variation of coating appearance is the main reason for the failure efforts 
to modify the Kubelka-Munk equations to handle metal flake containing 
coatings. These and other efforts have treated the metal flakes as another 
pigment perhaps with anisotropic scattering characteristics along with the 
color pigments. The color pigments are typically a fraction of a 
wavelength of the incident light in size while the metal flakes range in 
size from 60 to 200 times the wavelength of light. Further the metal 
flakes are opaque to and are moderate to good reflectors of incident 
light. 
An Approximate Radiative Transfer Solution for Isotropic Scattering 
Approximate solutions to the radiative transfer equations based upon 
averaging fluxes lead to useful formulas for determining the constants for 
the colored pigments used in metallic coatings and for color matching 
solid colors. 
The diffuse radiative transfer equation for isotropic scatterers having an 
albedo of .omega..sub.o is 
##EQU21## 
which can be rewritten as 
##EQU22## 
where 
##EQU23## 
The intensity of scattered light in the direction .mu. to an observer 
outside the coating film for incident collimated light coming from a 
direction .mu..sub.o is 
##EQU24## 
If we can obtain an approximate expression for S(.tau.), we can perform the 
integration over .tau. and calculate the reflected light intensity for the 
coating film. We assume here a film of infinite optical thickness which 
corresponds to the important case of a completely hiding coating layer. 
The approximate diffuse flux equations for isotropic scatterers can be 
solved to give estimates of the upward and downward diffuse fluxes. 
##EQU25## 
These equations are for the diffuse fluxes since we have separated out the 
direct intensity from the diffuse intensity in the radiative transfer 
equation given above. 
Solution of these equations for I and J gives 
##EQU26## 
where 
EQU .lambda.=2 (1-.omega..sub.o).sup.1/2 
The source function S(.tau.) then is estimated by 
##EQU27## 
Introducing this result into the equation for I(O,.mu.,.mu..sub.o) and 
performing the integration over .tau., we obtain after some algebra 
##EQU28## 
This equation can be related to the exact result obtained by Chandrasekhar, 
if we set the H(.mu.) function equal to 
##EQU29## 
This simple relation is a good estimate for the H(.mu.) functions and also 
gives a good estimate of the exact value of the radiance factor, 
.beta.(.mu.,.mu..sub.o). 
##EQU30## 
It can be used as a starting point to calculate the exact H(.mu.) functions 
and is useful for the estimation of calibration constants for colored 
pigments used in metallic coatings. 
This equation is also sufficiently accurate for color matching calculations 
for solid colors. 
Bridgeman in "Two-flux Formulae for the Total and Directional Reflectance 
of a Semi-infinite Diffuser" (Die Farbe, 35/36, 41-49, 1988/1989) has 
proposed similar approximations to the H(.mu.) functions and has arrived 
at the same formula for the H(.mu.) function given above by a process of 
empirical fitting his two flux solutions to exact computer generated 
H(.mu.) function data. 
A useful insight into the physical significance of J and I, the upward and 
downward averaged fluxes, can be obtained by evaluating the upward flux 
for .tau.=0 
##EQU31## 
After rearrangement J becomes 
##EQU32## 
This last expression is the well-known result for the total reflected flux 
from a surface illuminated by a collimated beam of light of intensity 
.pi.F from a direction .mu..sub.o (see R. G. Giovanelli, "Reflection by 
Semi-infinite Diffusers," Optical Acta, 2, 153-62, 1955). 
Giovanelli shows that the radiance factor for diffuse illumination and 
viewing from a direction .mu. is given by 
EQU .beta.=1-H(.mu.) (1-.omega..sub.o).sup.1/2 
This important result applies to the diffuse/near normal type of 
measurement geometry in widespread use in commercial color measuring 
equipment. 
SUMMARY OF THE INVENTION 
Characterization of coatings, such as metallic paints, is made by 
representing the reflected and scattered light as having components from 
the following: light from the incident beam which is reflected by the 
metal flake towards the observer; light scattered by the colored pigments 
which is reflected by the metal flake towards the observer; and light 
scattered by the colored pigments towards the observer. 
The radiance factor of a metal flake containing coating is given by the sum 
of the direct radiance contribution of the incident beam reflected by the 
metal flake and the scattered light contribution to the radiance 
##EQU33## 
in which 
##EQU34## 
A good first approximation is to assume the metal flakes are perfect 
mirrors reflecting light with no loss or attenuation. The scattered light 
contribution to the radiance then becomes 
##EQU35## 
If the metal flakes are assumed to be specular ("mirror") reflectors that 
have some loss or attenuation, the scattered reflected light contribution 
is 
##EQU36## 
This result differs from previous methods in that each metal flake is 
treated as a reflecting substrate rather than a pigment and then the total 
optical effect is found by integrating over the metal flake distribution 
in the film. 
Color matching and batch correction can be made using the above 
characteristics by wavelength matching to the spectrophotometric 
reflectance curve simultaneously at each illuminating and viewing geometry 
of interest. A wavelength match may be refined by using tristimulus 
iteration procedures. The tristimulus procedure includes requiring that 
the sums of the squares of the deviations of the match tristimulus values 
from the standard for three illuminants to be a minimum at each geometry. 
This procedure has the advantage that it produces the least metameric 
match to the standard using the three X, Y, Z values for the primary 
illuminant, but requires highly repeatable and reproducible instrumental 
measurements.

DESCRIPTION OF PREFERRED EMBODIMENTS 
Radiative Transfer Theory for Metallic Coatings 
We model a colored metallic coating as a collection of metallic reflecting 
plates imbedded in a medium containing isotropic scattering colored 
pigments. The metal flakes are oriented more or less parallel to the film 
surface. 
Consider a collimated beam, F, of light incident as shown in FIG. 1 at an 
angle -.theta..sub.o to the outwardly drawn normal on a metal flake 
containing coating, 2. The flake, 1, is embedded in a medium pigmented 
with absorbing and scattering colored pigments. The flake, 1, is oriented 
generally parallel to the surface 3 of the coating, 2, at an optical depth 
.tau. from the air/coating interface. 
Neglecting for the present the refraction effects that occur at the 
air/coating boundary, the incident flux F enters the medium and is 
attenuated by absorption and scattering as it travels through the film. 
The collimated attenuated beam will reach the metal flake and be reflected 
at various angles with reflection coefficient r(.mu.,.mu..sub.o) where 
.mu. is the direction cosine of the angle of reflection .theta. and 
.mu..sub.o is the direction cosine of the angle of the incident beam 
.theta..sub.o. 
The intense incident beam will be specularly reflected at the specular 
angle .theta..sub.o with direction cosine .mu..sub.o, pass through the 
film again, and be attenuated by absorption and scattering as it travels 
through the layer of optical thickness .tau. and leaves the film. 
An observer viewing the film at an angle .theta. (direction cosine .mu.) 
will see all the light scattered and reflected in the direction .mu.. In 
all future references to direction the reference will be to the direction 
cosines since they play a fundamental role in radiative transfer 
calculations. 
The scattered and reflected light in the direction .mu. seen by the 
observer will consist of the following components: 
a. light from the incident beam reflected by the metal flake in the 
direction .mu. and attenuated as it leaves the film in the direction .mu.. 
b. light scattered by the primary beam in the direction .mu. as it travels 
through the layer of optical thickness .tau.. 
c. light scattered in the direction .mu. by the reflected primary beam as 
it leaves the film in the direction .mu..sub.o through the layer of 
optical thickness .tau.. 
d. light scattered in the layer .tau. by the incident beam in the direction 
of the flake, reflected by the flake in the direction .mu., and attenuated 
as it travels out of the film through the layer of thickness .tau. to the 
observer. 
e. light back scattered by the exiting primary beam toward the metal flake 
in the direction -.mu., reflected by the flake in the direction .mu., and 
attenuated on its way to the observer in the layer of optical thickness 
.tau.. 
These contributions of scattered and reflected light seen by the observer 4 
will be calculated and summed to give the intensity of light viewed by an 
observer from a direction .mu. for a single metal flake 1 embedded in a 
medium coating 2 containing absorbing and scattering colored pigments. 
We shall first calculate the optical effect of a single aluminum flake 
imbedded in the film containing isotropic scattering color pigments and 
then sum this optical effect over the aluminum flake distribution to 
obtain the total intensity of scattered and reflected light for the 
coating. 
To accomplish our calculation we will rely on results and methods used by 
astronomers who have studied the planetary atmosphere problem. 
The coating film containing the colored pigments will correspond to the 
planetary atmosphere and the aluminum flake will represent the ground with 
arbitrary reflecting characteristics. The combined effect of the colored 
pigmented layer and flake is represented by R. 
Using the operators and the adding or doubling method described by H. C. 
van de Hulst in Multiple Light Scattering, Tables, Formulas, and 
Applications, Volumes 1 and 2 (Academic Press, New York and London, 1980), 
we obtain the following result 
EQU R=R.sub.f +T.sub.f R.sub.a [1+R.sub.f R.sub.a +(R.sub.f R.sub.a).sup.2 + . 
. . ] T.sub.f 
where 
R.sub.f is the reflection function for the pigmented layer; 
T.sub.f is the transmission function for the pigmented layer; 
R.sub.a is the reflection function for the aluminum flake. 
For an arbitrary flake reflection function no further simplification is 
possible and the expression for R must be evaluated term by term by 
successively applying these operator functions in the order indicated. 
Once R is known, the reflected intensity, I, can be calculated from the 
incident intensity, I.sub.o, from the relation 
EQU I=R I.sub.o 
where the above operator equation involves the evaluation of the following 
integral. 
##EQU37## 
A similar operator equation gives the transmitted light intensity, I, from 
the incident light intensity, I.sub.o. 
EQU I=T I.sub.o 
##EQU38## 
The evaluation of expressions for R and T for various physical models and 
materials is the subject of many papers in astrophysics and meteorology 
according to van de Hulst. 
Because the pigments used in metallic coating films are dispersed to a high 
degree of transparency in order to give the maximum appearance change with 
varying angles of view and illumination, the partially transparent 
atmosphere model is a good physical approximation to describe metallic 
coatings. 
TIALLY TRANSENT ATMOSPHERE MODEL 
Because single scattering and reflection events are physically most 
significant for a partially transparent atmosphere it is possible to 
obtain relatively simple mathematical expressions for the reflected and 
scattered light intensities. 
Our development of the calculation of the various contributions to the 
reflected intensity will consist of two parts. 
The major contribution to the reflected intensity comes from the 
attenuation of the primary incident beam as it travels through the 
pigmented layer, is reflected by the metal flake toward the observer, and 
is attenuated again by the pigmented layer before reaching the observer. 
A smaller but significant contribution to the reflected intensity comes 
from the scattering of light by colored pigments and reflection of this 
scattered light by the metal flake in the direction of the observer. 
These contributions will be calculated separately for a single metal flake 
imbedded in a film containing colored pigments, added together, and then 
summed over the metal flake distribution to obtain the total optical 
effect. 
a. Direct Reflection by the Attenuated Incident Beam 
Consider a single plate located inside the film at an optical depth .tau. 
from the surface that is illuminated by a collimated beam of light of 
intensity .pi.F from a direction -.theta..sub.o from the outwardly drawn 
normal to the surface. 
As the incident beam travels through the film it will be attenuated by 
absorption and scattering. At the metal flake surface it will have the 
intensity 
EQU .pi.F e.sup.-.tau./.mu. o 
After reflection by the metal flake in the direction (.mu.,.phi.) the 
intensity will be 
EQU 1/2.sigma..sub.o F r(.mu.,.phi.;.mu..sub.o,.phi..sub.o).mu..sub.o 
e.sup.-.tau./.mu. o 
where .sigma..sub.o is the area of the metal flake which is the order of a 
1000 to 10,000 times larger than the colored pigment particles. 
The intensity contribution from a single metal flake viewed by an observer 
from the direction .mu., .phi. outside the film will be 
EQU 1/2.sigma..sub.o F r(.mu.,.phi.;.mu..sub.o .phi..sub.o)(.mu..sub.o 
/.mu.)e.sup.-.tau./.mu. e.sup.-.tau./.mu. o 
The total intensity viewed by the observer (.mu.,.phi.) is the sum of the 
contributions from all the flakes in the film distributed at different 
optical depths. 
Various metal flake distributions can be postulated. For a uniform 
distribution of metal flakes, the cumulative fractional film area covered 
by metal flakes, P(.tau.), can be shown to be 
EQU P(.tau.)=1-e.sup.-.tau./.tau. o 
where .tau..sub.o is an optical depth parameter that depends on 
.sigma..sub.o, the area of a metal flake, c, the concentration of metal 
flakes per unit volume, and (K+S) the optical constants of the colored 
pigments in the film according to the equation 
EQU .tau..sub.o =(K+S)/(.sigma..sub.o c)=(K+S)/.sigma. 
The derivative of P(.tau.) with respect to .tau. gives the fraction of 
flake coverage in the layer .tau., .tau.+d.tau.. 
EQU dP(.tau.)/d.tau.=(1/.tau..sub.o) e.sup.-.tau./.tau. o 
The total intensity of light viewed by the observer (.tau.,.phi.) is equal 
to 
##EQU39## 
where the integral or summation of flake reflections is taken over the 
depth of the film which is considered here to be an infinite half plane or 
a completely covering optical layer. 
Performing the integration and introducing the expression for .tau..sub.o 
##EQU40## 
defining 
EQU .beta.(.mu.,.phi.;.mu..sub.o,.phi..sub.o)=I(0,.mu.,.phi.;.mu..sub.o,.phi..s 
ub.o)/F.mu..sub.o 
and 
##EQU41## 
we obtain the result 
##EQU42## 
which accounts for the optical properties of the metallic film except for 
the weak scattering of the transparent color pigments used to color the 
metal flake coating. 
b. Scattered Light Reflections 
The estimation of the weak scattering is a critical aspect of the theory 
since this weak scattering is responsible for the difference in flop 
between different coatings that have the same face appearance. The 
reflection and transmission functions for first order scattering in an 
optical layer of thickness .tau. are 
##EQU43## 
Because of the strongly directional nature of the metal flake scattering, 
we can approximately model the flake particles in the correction term as 
being specular reflectors. 
i. Ideal Specular Reflector 
In the case of an ideal or perfect specular reflector, the incident and 
scattered light will be reflected at the flake surface like a mirror with 
no loss or attenuation. The net optical effect can be obtained by 
reflecting the layer into its mirror image. Following Van de Hulst, the 
reflection function for the scattered light will then consist of the sum 
of the reflection and transmission functions for a single layer of 
thickness 2.tau.. 
EQU R.sub.s 
(.mu.,.mu..sub.o,.phi.,.tau.)=R(.mu.,.mu..sub.o,.phi.,2.tau.)+T(.mu.,.mu.. 
sub.o,.phi.,2.tau.) 
For collimated incident radiation 
EQU I.sub.o =.pi.F.delta.(.mu.-.mu..sub.o).delta.(.phi.-.phi..sub.o) 
the intensity is given by 
EQU I=R.sub.s (.mu.,.mu..sub.o,.phi..sub.o,.tau.) I.sub.o 
which is equal to 
##EQU44## 
This result for the scattered light intensity for a single metal flake must 
be summed over all the metal flakes in the film and added to the 
contribution from the direct component to give the total intensity seen by 
an observer outside the film. 
Integrating this result over the metal flake distribution we obtain 
##EQU45## 
ii. Partial Specular Reflector 
In the case of a specular reflector with a reflection function 
EQU R.sub.a =r(.mu.)[.delta.(.mu.-.mu..sub.o)/2.mu.]2.pi. 
.delta.(.phi.-.phi..sub.o) 
where r(.mu.) is the reflection coefficient of the flake for light incident 
from a direction .mu., a more involved calculation gives the following 
result for the scattered light reflected intensity 
##EQU46## 
This result reduces to our previous expression for a perfect specular 
reflector if both r(.mu.) and r(.mu..sub.o) are set equal to one. 
Integrating these equations for a single metal flake over the assumed 
uniform aluminum flake distribution in the coating, we obtain the result 
##EQU47## 
This result reduces to the simpler perfect specular reflector result if 
both r(.mu.) and r(.mu..sub.o) are set equal to one. 
c. Combined Direct and Scattered Reflected Light 
The reflectance of a metal flake containing coating is given by the sum of 
the direct radiance contribution and the scattered light contribution to 
the radiance. 
##EQU48## 
In the case of bright aluminum flake coatings, the perfect specular 
approximation gives very good estimates of the observed reflectance. For 
metal flakes with lower reflectances, the more complicated result for 
partial specular reflectors is required. 
As in the Kubelka-Munk theory, it is assumed that the nonmetallic colorants 
in the film contribute additively to K and S. Thus 
EQU K=.SIGMA..sub.i K.sub.i * c.sub.i 
EQU S=.SIGMA..sub.i S.sub.i * c.sub.i 
where K.sub.i is the absorption coefficient, S.sub.i is the scattering 
coefficient, and c.sub.i is the concentration of the ith nonmetallic 
colorant. The summations are taken over the number of nonmetallic 
colorants in the film. 
If more than one aluminum pigment is present in the film, then .sigma. is 
given by 
##EQU49## 
where .sigma..sub.i is the cross section of the ith aluminum pigment. The 
summation is taken over the number of aluminum pigments in the film. 
FIRST SURFACE CORRECTIONS 
The above reflectance is derived without regard for the external and 
internal reflections that take place at the air coating interface. Better 
color matching predictions may be obtained for coatings if a correction is 
made for these reflections. The rigorous radiative transfer equations for 
making surface corrections are formulated by V. V. Sobolev (in A Treatise 
on Radiative Transfer, D. Van Nostrand Company, Inc., Princeton, N.J., 
1963), R. G. Giovanelli ("Reflection by Semi-infinite Diffusers," Optical 
Acta, Vol. 2, No. 4, December, 1955, pp. 153-162), and S. E. Orchard 
("Reflection and Transmission of Light by Diffusing Suspensions," Journal 
Optical Soc. of Am., 59, 1584-1597). The approximations to the rigorous 
radiative transfer equations proposed by J. L. Saunderson in "Calculation 
of the Color of Pigmented Plastics," Journal Optical Soc. of Am., 32, 
727-736, 1942, are the most widely used equations for diffuse/near-normal 
measurements. 
The scattering and reflecting pigments in a metallic film are dispersed in 
a resin matrix while the film reflectance is measured in air. The 
refractive index of the resin matrix is generally about 1.5 while the 
refractive index of air is 1. 
An incident collimated beam of light of flux .pi.F(m) per unit area falling 
on the coating film will be refracted at the plane parallel film surface 
as it crosses from a region of lower to a region of higher refractive 
index. 
If m is the cosine of the external angle of incidence, then the cosine of 
the internal angle of incidence, .mu..sub.o, is given by Snell's law of 
refraction 
EQU 1-m.sup.2 =n.sup.2 (1-.mu..sub.o).sup.2 
where n is the ratio of the refractive index of the resin matrix to that of 
air. 
Assuming the upper surface of the coating film is a specular reflector, its 
directional reflection characteristics are described by Fresnel's 
equations. 
The incident illumination is .pi.F(m)m. The direct flux immediately below 
the film surface, measured on a plane parallel to the beam, is 
##EQU50## 
where 
EQU t(m)=1-r(m) 
and 
##EQU51## 
The observed radiance factor, .beta..sub.obs, is related to the radiance 
factor inside the film just below the surface, .beta.*, by the relation 
EQU .beta..sub.obs ={1-r(.mu..sub.o)}{[1-r(.mu.)]/n.sup.2 }.beta.* 
This in turn is related to the radiance factor calculated by radiative 
transfer theory in the absence of the refractive index change between the 
coating film and air, .beta.(.mu.,.phi.;.mu..sub.o,.phi..sub.o), by the 
equation 
##EQU52## 
Continuing the single scattering assumption and introducing 
EQU r(.mu.,.phi.)=r(.mu.)[.delta.(.mu.-.mu..sub.o)/2.mu.]2.pi. 
.delta.(.phi.-.phi..sub.o) 
for the specular boundary reflection, we obtain the following relation 
between .beta.* and .beta. 
EQU .beta.*(.mu.,.phi.;.mu..sub.o,.phi..sub.o)=.beta.(.mu.,.phi.;.mu..sub.o,.ph 
i..sub.o) / {1-r(.mu.).beta.(.mu.,.phi.;.mu.,.phi.)} 
The observed or experimental radiance factor, .beta..sub.obs, is 
##EQU53## 
Since r(.mu.) and r(.mu..sub.o) are in the range 0.04 to 0.06 for typical 
angles of incidence and view the corrected reflectance and measured 
reflectance are close enough to ignore these corrections as a first 
approximation for metallic color matching. 
This finding is in contrast to that for solid colors where the diffuse 
scattering predominates leading to much larger corrections because of the 
total internal reflection of the diffuse light incident on the film air 
interface at angles greater than the critical angle. 
COLOR MATCHING 
Matching is the same whether for formulation or batch control. The measured 
parameters are compared with combinations of likely pigments to yield 
combinations that best approximate the desired color coating under various 
conditions. 
Wavelength matching attempts to match the spectrophotometric reflectance 
curve as closely as possible simultaneously at each illuminating and 
viewing geometry of interest. This procedure works best when the pigments 
chosen for the match are the same as those present in the sample 
composition. 
The following scheme can be used to estimate the least squares wavelength 
match to the reflectance curve. Weighting factors, W, may be used to 
emphasize the data at one or more geometries. At each geometry, l, for 
each wavelength, j, we have 
##EQU54## 
We can rewrite this equation as 
EQU .delta..sub.1 ={(.beta..sub.lj -.beta..sub.lj.sup.*)*(.mu..sub.1 
+.mu..sub.ol)*(K.sub.lj +S.sub.ij)}+{(.beta..sub.lj -.beta.*.sub.lj 
-.beta.s.sub.lj)*.sigma..sub.1 *.mu..sub.1 *.mu..sub.ol }=0 
Define Q as 
##EQU55## 
It is now required that the partial derivative of Q with respect to 
c.sub.m, the concentration of the mth pigment, be zero for each 
nonmetallic and aluminum flake pigment. Thus 
##EQU56## 
where N is the total number of pigments in the sample. 
Introducing 
##EQU57## 
since concentrations are expressed in weight percent, the additional 
relationship is given 
##EQU58## 
Eliminating the Nth pigment, the set of equations which determine N-1 
pigment concentrations 
##EQU59## 
The remaining or Nth pigment concentration is obtained by difference of 
the others from 100. 
A wavelength match may be refined by using the resultant pigment 
concentrations as the starting point for a tristimulus match. This may 
lead to a more perceptually accurate match particularly when the pigments 
used to match the standard are different than those that were used to make 
the standard. Tristimulus matching is more sensitive to variations in 
panel preparation and measurement. 
Various known tristimulus matching techniques can be used. A new method 
using a least squares tristimulus iteration procedure is particularly 
effective when matching metallic finishes simultaneously at multiple 
illuminating and viewing geometries. The three tristimulus values (X,Y,Z) 
at each geometry, l, are expanded in a Taylor series about a set of trial 
values of colorant concentrations, c.sub.i.sup.0. These equations are then 
solved for a set of .DELTA.c.sub.i values. The calculation is repeated 
with a new set of concentrations c.sub.i =c.sub.i.sup.0 +.DELTA.c.sub.i 
until no improvement in the color match is obtained. To improve the 
method, the natural logarithm of the tristimulus value is expanded in a 
Taylor series as follows: 
##EQU60## 
where the summation is taken over the N colorants in the coating. Thus the 
.DELTA.c.sub.i satisfy the following equations 
##EQU61## 
where the tristimulus values are now subscripted with k to denote the kth 
illuminant choice. In general the values of X, Y, Z are calculated for 
three illuminants to detect the degree of metamerism between the standard 
and match. As with wavelength matching, weighting factors, W, can be 
introduced to emphasize one or more geometries. 
Now let 
##EQU62## 
and define 
##EQU63## 
where the summation is made over the K illuminants chosen for matching. 
The method requires that the .DELTA.c.sub.i be determined to satisfy the 
following conditions 
##EQU64## 
This leads to the following equations for the .DELTA.c.sub.i 
##EQU65## 
where 
##EQU66## 
and k now runs from 1 to 3K. 
These equations are solved for the first (N-1) .DELTA.c.sub.i values. Those 
values are then used to determine a new set of trial c.sub.i values and 
the process is continued by iteration until no further improvement in 
color difference between the standard and match is obtained. 
This method appears to be superior to the conventional Allen procedure for 
reproducing the composition of metallic color panels. While the residual 
color differences at match may tend to be larger than those obtained by 
the Allen procedure, the match seems to be less metameric. 
COLORANT CALIBRATION 
Before color matches can be made or batches corrected, the absorption and 
scattering coefficients for the colored pigments must be determined. The 
cross section of the metal pigments must also be determined. The process 
of determining these constants is called pigment calibration. Whereas in 
color matching, the optical constants of the pigments are known and the 
concentrations required to match a standard color are determined, in 
pigment calibration, mixtures of known concentrations are used to 
determine the optical constants. 
The optical constants are considered properties of the pigments as they are 
dispersed in a given resin system. Optical constants are wavelength 
dependent and must be calculated at each wavelength of interest. There are 
several techniques which may be used in determining the optical constants. 
In a typical pigment calibration a specific set of calibration panels is 
prepared at complete hiding, i.e., infinite optical thickness, and 
measured with a reflectance spectrophotometer at several illuminating and 
viewing geometries and at a number of wavelengths. The examples presented 
in this patent were done with a Datacolor MMK111 goniospectrophotometer 
which illuminates the sample at -45 degrees from the normal (to the left 
of the sample normal) and measures at 20, 0 and -25 degrees from the 
normal. Many researchers prefer describing the illuminating and viewing 
geometry in terms of the observation angle from the specular (mirror or 
gloss) angle. This convention will be followed in this patent. For the 
Datacolor MMK 111, the specular angle would be 45 degrees from the normal 
and the three geometries would be respectively described as 25 degrees 
from the specular angle, 45 degrees from the specular, and 70 degrees from 
the specular. Although the instrument measures a sample at 1 nm wavelength 
increments, the data was mathematically reduced to 20 nm increments from 
400 to 700 nm, i.e., 400, 420, . . . , 680, 700 nm. Another instrument 
that has been used successfully is the Macbeth 5010 goniospectrophotometer 
at 20, 45, and 75 degrees from the specular angle. The equations are not 
limited to those instruments, geometries, or wavelength ranges. 
I. CALCULATING THE ALBEDO 
In calibrating colorants it is sometimes necessary to know the albedo, 
.omega., that produced an observed reflectance in a panel containing no 
metal flake pigment. The albedo can be calculated by using the solid color 
radiative transfer theory and an iterative technique. 
After correction for first surface affects (if desired), the internal 
radiance factor of a solid color can be found from 
##EQU67## 
The value of .omega. is determined by using the secant method to find the 
root of the equation 
##EQU68## 
The H function at a given angle, .mu., can be estimated by 
##EQU69## 
where 
EQU .lambda.=2 * (1-.omega.).sup.1/2 
or calculated with a computer by the methods previously described. 
II. CALIBRATIONS USING A REFERENCE ALUMINUM PIGMENT 
One technique of pigment calibration that has been used with success is to 
calibrate a reference aluminum (or other metal flake pigment), calibrate a 
black using a masstone of the black and a mixture of the black with 
aluminum, and to calibrate all other colorants using a mixture of the 
colorant in aluminum and either the colorant in black or a masstone of the 
colorant. In this technique the reflection coefficient for the reference 
aluminum flake, r.sub.al, is equal to the reflectance of the masstone 
panel which may be corrected for the first surface affects, and the cross 
section for that reference flake, .sigma..sub.al, is set equal 1.0. 
Computations of the absorption and scattering coefficient are made using 
the downhill simplex method developed by Nelder and Mead and described by 
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling in 
"Numerical Recipes" (Cambridge University Press, Cambridge, N.Y., and 
Melbourne, 1986). To use this technique, an initial estimate must be made 
for the absorption and scattering coefficients. The Nelder-Mead algorithm 
will adjust either just the absorption coefficient or both the absorption 
and scattering coefficients for each wavelength at each geometry until the 
sum of the squares of the difference between the experimental reflectance 
and the calculated reflectance is minimized. The minimized constants are 
then stored for later use in color matching and batch correction. 
A. CALIBRATING A BLACK COLORANT OR OTHER OPAQUE COLORANT USING A MASSTONE 
AND A MIXTURE OF THE COLORANT IN THE REFERENCE ALUMINUM 
Two panels are required to calibrate a black or other opaque colorant, a 
masstone of the colorant and a mixture of the colorant with the reference 
aluminum (or other metallic flake pigment). The reflection coefficient of 
the aluminum, r.sub.al, the cross section of the aluminum, .sigma..sub.al, 
the concentration of the reference aluminum, c.sub.al, and the 
concentration of the black or other opaque colorant, c.sub.b, in the 
colorant/aluminum mixture are known, and the reflectance of the 
colorant/aluminum mixture, r.sub.mix, is measured and may be corrected for 
first surface affects. 
Estimates of (K+S) and S are made at each wavelength for each geometry 
using the equations 
##EQU70## 
in which .omega..sub.b is the albedo of the masstone. If (K+S) is less 
than S, a new estimate of (K+S) is made using 
EQU (K+S)=S/.omega..sub.b 
An estimate of the absorption coefficient is then found by subtraction 
EQU K=(K+S)-S 
The estimates of K and S are processed with the Nelder-Mead algorithm and 
an optimized value for the absorption coefficient of the black or opaque 
colorant, K.sub.b, is returned. 
The scattering coefficient for the colorant, S.sub.b, is then calculated 
from 
##EQU71## 
B. CALIBRATING COLORANTS USING MIXTURES IN A REFERENCE ALUMINUM AND IN A 
CALIBRATED BLACK 
Two panels are required to calibrate colorants using this technique, a 
mixture of the colorant with the reference aluminum (or other metallic 
flake pigment) and a mixture of the colorant with a calibrated black. The 
reflection coefficient of the aluminum, r.sub.al, the cross section of the 
aluminum, .sigma..sub.al, the concentration of the reference aluminum in 
the colorant/aluminum mixture, c.sub.al, the concentration of colorant in 
the colorant/aluminum mixture, c.sub.ca, the concentration of the 
calibrated black in the colorant/black mixture, c.sub.b, the concentration 
of the colorant in the colorant/black mixture, c.sub.cb, the absorption 
coefficient of the black, K.sub.b, and the scattering coefficient of the 
black, S.sub.b, are known. The reflectances of the colorant/aluminum 
mixture, r.sub.ca, and the colorant/black mixture, r.sub.cb, are measured 
and may be corrected for first surface affects. 
Estimates of (K+S) and S for the colorant are then made at each wavelength 
for each geometry using the equations 
##EQU72## 
If (K+S) is greater than or equal to S, then the estimate of the 
absorption coefficient is calculated from 
EQU K=(K+S)-S 
If (K+S) is less than S, then the estimate of the absorption coefficient is 
calculated from 
EQU K=S-(K+S) 
The estimates of K and S are processed with the Nelder-Mead algorithm and 
an optimized value of absorption coefficient for the colorant, K.sub.c, is 
returned. The scattering coefficient for the colorant, S.sub.c, is then 
calculated from 
##EQU73## 
C. CALIBRATING OTHER ALUMINUM PIGMENTS 
Two panels are required to calibrate other aluminum pigments, a masstone of 
the other aluminum and a mixture of the other aluminum with a calibrated 
black. The concentration of the other aluminum in the black/aluminum 
mixture, c.sub.b, and the concentration of the calibrated black in the 
black/aluminum mixture, c.sub.al, the absorption coefficient of the 
calibrated black, K.sub.b, and the scattering coefficient of the 
calibrated black, S.sub.b, are known. The reflectance of the masstone 
aluminum and the calibrated black with aluminum mixture, r.sub.ba, are 
measured and may be corrected for first surface affects. 
The reflection coefficient for the other aluminum, r.sub.al, is equal to 
the reflectance of the masstone panel which may be corrected for the first 
surface affects. 
For other aluminum pigments, the Nelder-Mead algorithm is used to optimize 
the aluminum cross section in the black/aluminum mixture. The initial 
estimate of the cross section is 
##EQU74## 
After optimizing, the other aluminum's cross section is calculated from 
EQU .sigma..sub.al =.sigma. * c.sub.al /c.sub.b 
III. CALIBRATIONS USING A REFERENCE WHITE PIGMENT 
A second technique of colorant calibration uses calibration panels 
consisting of a reference white masstone, a mixture of the white with a 
black colorant, a masstone aluminum panel, a mixture of aluminum with the 
calibrated black colorant, and, for each additional colorant, a mixture of 
the colorant with the calibrated white and a mixture of the colorant with 
the calibrated black. Except for the aluminum panel, these equations 
presented with this technique can also be used to match solid colors using 
approximations to the full radiative transfer equation other than the 
Kubelka-Munk approximations. In the following descriptions it is 
understood that a different set of optical constants are being calculated 
for at each geometry for each wavelength measured. For this technique an 
iterative solution is shown. It is possible to use the Nelder-Mead 
algorithm to solve these equations. 
A. CALIBRATING THE REFERENCE WHITE 
To calibrate a reference white, a masstone panel containing the reference 
white colorant is required. For each geometry at each wavelength the 
scattering coefficient, S, is set equal to 1.0. For each reflectance value 
which may be corrected for the first surface reflections, the albedo, 
.omega., is computed. The absorption coefficient at each wavelength, K, is 
then 
EQU K=(1.0/.omega.)-1.0 
B. CALIBRATING BLACK AND OTHER OPAQUE COLORANTS 
Two panels are required to calibrate a black colorant, a mixture of the 
black with the reference white and a black masstone. The concentration of 
the black, c.sub.b, and the concentration of the reference white, c.sub.w, 
in the black and white mixture are known as are the absorption, K.sub.w, 
and scattering, S.sub.w, coefficients for the reference white. For each 
reflectance value which may be corrected for the first surface 
corrections, the albedo for the black and white mixture, w.sub.bw, and the 
albedo for the black masstone, .omega..sub.b, are computed. From these 
quantities, the sum of the absorption and scattering coefficients, 
(K+S).sub.bw, for the black and white mixture can be calculated from 
##EQU75## 
The scattering coefficient of the black is calculated from 
EQU S.sub.b =(K+S).sub.bw * .omega..sub.b 
and the absorption coefficient of the black is calculated from 
EQU K.sub.b =(K+S).sub.bw -S.sub.b 
Other colorants can be calibrated using these equations by preparing a 
mixture of the colorant in white and a masstone of the colorant and by 
calculating the appropriate concentrations and albedos. This technique 
should only be used when the masstone panel of the colorant can be 
prepared at complete hiding. 
C. CALIBRATING OTHER NON-METALLIC COLORANTS 
Two panels are required to calibrate other non-metallic colorants, a 
mixture of the colorant with the reference white and a mixture of the 
colorant with a calibrated black. The concentration of the colorant, 
c.sub.cw, and the concentration of the reference white, c.sub.w, in the 
colorant and white mixture are known as are the absorption, K.sub.w, and 
scattering, S.sub.w, coefficients of the reference white. The 
concentration of the colorant, c.sub.cb, and the concentration of the 
calibrated black, c.sub.b, in the colorant and black mixture are also 
known as are the absorption, K.sub.b, and scattering, S.sub.b, 
coefficients of the calibrated black. For each reflectance value which may 
be corrected for the first surface corrections, the albedo for the 
colorant and white mixture, .omega..sub.cw, and the albedo for the 
colorant and black mixture, .omega..sub.cb, are computed. From these 
quantities the scattering coefficient, S.sub.c, of the colorant can be 
calculated as follows. 
##EQU76## 
The sum of the absorption and scattering coefficients for the colorant can 
also be calculated from these quantities and thus the absorption 
coefficient of the colorant itself as follows. 
##EQU77## 
D. CALIBRATING METAL (ALUMINUM) PIGMENTS 
With this technique two panels are required to calibrate a metal pigment, a 
masstone of the metal pigment and a mixture of the metal pigment and a 
calibrated black. The metal flake is usually aluminum but the equations 
are not restrictive to other metallic flake pigments. Aluminum will be 
assumed for the remainder of this discussion. The reflection coefficient 
of the aluminum flake, r.sub.al, is equal to the reflectance of the 
masstone panel which may be corrected for the first surface affects. The 
cross section is found with an iterative procedure using the reflection 
coefficient and the radiance factors of the black and aluminum mixture, 
.beta..sub.bm, which may be corrected for the first surface affects. The 
concentration of the aluminum flake, c.sub.al, and the concentration of 
the calibrated black, c.sub.b, are known as are the absorption, K.sub.b, 
and scattering, S.sub.b, coefficients of the calibrated black. 
First an approximation of the aluminum cross section, .sigma..sub.al, is 
made from the following equations. 
##EQU78## 
The following iterative loop is performed until the difference between the 
variables R and RO is sufficiently low, 0.00001 being a good ending point. 
##EQU79## 
E. CALIBRATING OTHER WHITE COLORANTS 
Two panels are required to calibrate additional white colorants, a masstone 
of the white and a mixture of the white with a calibrated black. The 
concentration of the white, c.sub.w, and the concentration of the 
calibrated black, c.sub.b, in the white and black mixture are known as are 
the absorption, K.sub.b, and scattering, S.sub.b, coefficients for the 
calibrated white. For each radiance factor which may be corrected for the 
first surface affects, the albedo for the white masstone, .omega..sub.w, 
and the albedo for the white and black mixture, .omega..sub.wb, are 
computed. From these quantities, the sum of the absorption and scattering 
coefficients, (K+S).sub.wb, for the white and black mixture can be 
calculated from 
##EQU80## 
The scattering coefficient of the new white is calculated from 
EQU S.sub.w =(K+S).sub.wb * .omega..sub.w 
and the absorption coefficient of the new white is calculated from 
EQU K.sub.w =(K+S).sub.wb * s.sub.w 
MATCHING 
A strategy for matching solid colors is to make an initial match 
calculation using wavelength matching procedures followed by a least 
squares tristimulus refinement of the solution to the final match. The 
presently preferred matching method for metallic panels involves making a 
least squares wavelength match at all illuminating and viewing geometries 
of interest. The current state of the art techniques of panel preparation 
and measurement are neither as uniform nor as reproducible as with solid 
colors. Wavelength matching predicts panel compositions that are not as 
sensitive to experimental variations as is tristimulus matching. If the 
experimental variability can be lowered, the use of a tristimulus 
refinement might improve the final predictions. 
For the examples following, three illuminating and viewing geometries are 
considered. This should not be considered as a limitation, however, since 
matches can be predicted for a single geometry. The maximum number of 
geometries is limited only by the measuring instruments commercially 
available and the size of the computer doing the calculations. All 
pigments used in the example were calibrated individually at each 
geometry. Following currently accepted practice, for the instrument used 
to generate the examples, each geometry is specified by the angle from the 
specular reflectance angle at which the specimen is viewed. The 
illumination was at 45 degrees to the specimen's normal, and viewing was 
at 25, 45, and 70 degrees from the specular reflectance angle. Other 
instrumental designs used to generate reflectances at multiple angles can 
be used equally as well. 
EXAMPLES: TWO PIGMENT METALLIC MIXTURES 
An example of a match of a mixture containing a single aluminum pigment and 
a single colored pigment is shown in the following table. The table lists 
the colorants, the parts of colorant in the sample, the calculated 
composition of the panel, and the color differences between the predicted 
match and the measured standard panel calculated for three selected 
illuminants. 
______________________________________ 
Colorant Parts Calc 
______________________________________ 
Trans Red Oxide 
50 50.04 
Coarse Aluminum 
50 49.96 
DED 25 = 0.08 DED 45 = 0.52 
DED 70 = 1.04 
______________________________________ 
EXAMPLES: FOUR COLORANT METALLIC MIXTURES 
A set of mixtures was prepared using four pigment blends made from one 
aluminum and three colored pigments. The match results are shown in the 
following tables. Pigments were calibrated individually, matches made at 
each geometry of interest (25, 45, and 70 degrees from the specular 
angle), and an overall least squares (LS) match made. Each table contains 
the panel identification, a listing of the pigments, the actual panel 
composition, the calculated panel compositions, and the color differences 
between the predicted matches and the measured standard panel calculated 
for daylight at each geometry of interest. 
______________________________________ 
25 45 70 LS 
Colorant Parts Calc Calc Calc Calc 
______________________________________ 
Mixture 1A 
Phthalo Blue RS 
50 50.8 50.2 50.0 51.3 
Indo Yellow RS 
20 14.6 15.3 13.3 16.1 
Carbon Black S 
2 2.9 1.7 2.1 1.9 
Coarse Aluminum 
28 31.7 32.7 34.6 30.8 
Color Diff DED 25 0.50 -- -- 2.02 
Color Diff DED 45 -- 0.35 -- 1.31 
Color Diff DED 70 -- -- 0.43 2.20 
Mixture 2A 
Phthalo Blue RS 
40 37.8 38.0 39.6 39.7 
Phthalo Green YS 
40 42.6 41.7 37.8 41.5 
Carbon Black S 
2 1.8 0.6 1.0 1.3 
Coarse Aluminum 
18 17.7 19.7 21.6 17.5 
Color Diff DED 25 0.31 -- -- 1.94 
Color Diff DED 45 -- 0.35 -- 4.32 
Color Diff DED 70 -- -- 0.70 3.38 
Mixture 3A 
Phthalo Green YS 
50 57.0 52.5 51.5 54.2 
Indo Yellow GS 
20 14.7 18.0 18.6 19.0 
Carbon Black S 
2 2.7 1.3 0.9 1.8 
Coarse Aluminum 
28 25.7 28.2 29.0 25.0 
Color Diff DED 25 0.08 -- -- 3.56 
Color Diff DED 45 -- 0.27 -- 3.01 
Color Diff DED 70 -- -- 0.25 4.84 
______________________________________ 
The least square wavelength match provides a better overall prediction of 
the panel composition than the single geometry matches, but the predicted 
color difference is not as low. This is due to the variations in 
measurements and panel preparation. 
The color matching and characterizations for coatings as described herein 
will usually be practiced in conjunction with computer software because of 
the lengthy mathematical calculations. Such computers can perform such 
calculations much faster than the traditional manual methods. Programs may 
be generated in many user languages which lend themselves to mathematical 
calculations; presently preferred software includes turbo pascal. Software 
lends itself to the practice of the invention especially in the color 
matching and batch correction processes where many reiterations may be 
required. 
The flow chart shown in FIG. 2 is one presently preferred series of steps 
using multiple angle measurements for color matching a metallic coating, 
specifically a metallic paint coating. Referring to FIG. 2, the first 
step, 5, is a standard measurement of the reflectances at multiple angles. 
It is to be understood that while multiple angles are used in this 
embodiment, a single angle may also be used, with limitations, in the 
practice of this invention. Multiple angles may have specific advantages 
when it is desired to match the extreme "flop" that may be desirable in 
some coatings. The measured reflectances may then be corrected, step 6, 
for the resin/air interface as previously described herein. The next step, 
7, includes selection of colorants to be used in the match. Past 
experience or manufacturing availability of the specific paint or pigments 
may strongly influence which colorants are used initially in the match. In 
step 8, the formulation defining the concentrations of the previously 
selected colorants is determined using a least squares wavelength matching 
method at each angle based upon the prior discussed characterizations of 
the light flux for both the measured sample and the selected colorants. In 
step 9, the reflectances of the formulation are then calculated using the 
prior discussed characterization method. In step 10, these reflectances 
may then be corrected for the air/resin interface similar to step 6. The 
color coordinates of the formulations are then calculated in step 11. The 
tristimulus values calculated in step 11 are then compared to the 
tristimulus values for the sample in step 12 and a color difference 
calculated. The desired match is displayed at step 14 or otherwise 
utilized. 
FIG. 3 shows a flow chart of a single angle metallic batch correction 
process using the teachings of the invention. The first step in such 
process, 14, is to measure the standard coatings reflectance at multiple 
angles. Single angle measurements could also be used within the scope of 
this invention with their inherent limitations. The measured reflectances 
are then recorded for the standard. Step 15 includes a similar measurement 
of the batch (trial) colors reflectances at multiple angles or at a single 
angle as step 14. The color differences between the standard and the batch 
are then calculated, 16. If the difference is acceptable in the decision, 
step 17, then the correction process can be stopped and the trial can be 
used as an acceptable duplication of the standard coating. If the color 
differences calculated in step 16 differ substantially from the standards, 
then the colorants used in the formulation of the trial enter the process, 
19, and will be used to compare the characteristics with the standard to 
arrive at a batch addition. The multiple angle metallic color matching 
characterization process previously described herein is used then to 
characterize the standard, 20. The same metallic color matching process is 
as previously described and its related equations are used to describe the 
trial sample 21. The difference between the calculated formulations of the 
trial batch and the standard are then determined. The necessary additions 
of the colorants as actually used in the trial batch to match the standard 
are then calculated, 22. This calculation is done so that the percentages 
of the colorants in the revised batch are equal to those of those 
calculated for the standard as determined in step 20. The changes in 
concentrations and/or the additions of the colorants used in the trial are 
then displayed, 23, or otherwise used so that the batch can be corrected 
with such information. After the batch has been corrected it may be 
desirable to repeat the process steps 15 through 23 until an acceptable 
match in step 17 is achieved. 
While certain presently preferred embodiments of the invention have been 
described hereinabove, it is to be understood that other methods of 
practicing the invention as will be apparent to those skilled in the art 
are included within the scope of the following claims.