Learning Center
Plans & pricing Sign in
Sign Out

ument Template


									                                                                    Axiphos GmbH
                                                                 Marketing, Trading and

                                                                                                  May 2002
                                    On whiteness formulas
After the recognition of whiteness being a                  increase in whiteness though. The situation
special color attribute o objects much work has             turns more complicated with the fact that a
been invested in defining whiteness as a                    slight blue shade is also interpreted by the eye
unique number that takes into account the                   as an increase in whiteness and leads to a
colorimetric perception of the object under                 wider definition of whiteness as having a finite
observation and assessment. Quantification of               amount of hue and therefore emphasizing the
perceived whiteness has been intimately                     importance of a preferred white.
related to lightness levels and the absence of              The use of fluorescence as a method to
any hue. White pigments show however in                     increment both lightness and blue hue
general a yellow shade originating from                     introduces a formidable task to elaborate a
impurities (mainly Iron ions in different                   formula that takes into account a colorimetric
oxidation stages) that has led to the terms                 compensation based on additive and sub-
“near white” and “off-white”; these colors are              ractive color mixing.
really perceived under comparison with a                    The original idea of characterizing whiteness
“preferred” white though and this is a result of            through a unique number is valid today only
the chromatic adaptation of the eye.                        after the definition of a preferred white has
As a consequence first attempt to quantify                  been defined that is highly dependent of the
whiteness have been related to the                          cultural group of the observer and the
quantification of yellowness, lower yellowness              application of the white object. From a
values are considered as showing higher                     technical point of view formulas based on
whiteness. Almost all pigments are quantified               colorimetric quantities give more information
in this way, remarkably also pulps and natural              since they are based on psychometric
raw fibers that are treated by a chemical                   quantities; nowadays efforts are invested in the
bleaching process to lower yellowness values.               definition of a whiteness subspace that
The colorimetric compensation of yellowness                 localizes the perceived color within the color
by adding a blue (or violet) dye is known as                solid, though the definition of preferred white
“bluing” and was quite widespread in the textile            axis numbers can be then transformed into
area, specially hand washing where chemical                 unique perceived whiteness that are strictly
bleaching does not apply. The addition of a                 part of a color appearance whiteness model.
dye leads irreversible to a loss of lightness, the          Literally hundreds of whiteness formulas exist
object appears grayish or duller as compared                and have been applied in the past, only a
with the not treated one, the compensation of               selected number of them a presented and
the yellow hue is interpreted by the eye as an              discussed in the next sections.

Primitive formulas
The first attempts of describing whiteness are              у() to describe the luminance factor under a
based on just lightness, yellowness or                      given observer and illuminant. This is purely a
blueness. The formula:                                      luminance value and does not report if the
                                                            observed object is bluish or yellowish (or by
W Y                                                        the same token having any other hue). The
tries to quantify whiteness just as related to
lightness, normally as a relative quantity to a             W B
preferred white defined by a Magnesium oxide
or Barium sulfate tablet, using the CIE function
                                               Axiphos GmbH
                                     Arend-Braye Str. 42, D-79540 Loerrach

                 Tel: (+49-7621) 426693 - Fax: (+49-7621) 426693 - Email:

                                                                                                   Page 1 of 8
tries to relate whiteness to a blue reflectance        Worth mentioning is the whiteness measured
defined either by the CIE function z() or to          by the Leukometer of VEB Carl Zeiss, Jena,
some ad hoc defined one as for example the             GDR:
paper brightness function B(); the relation to a
MgO or BaSO4 preferred white is implicitly             WI Leukometer  2  R459  R614
contained in the definition of the function B. It
is clear that the formula gives always positive        The use of not standardized band pass filters
numbers regardless of the real color of the            contributed to a loss of popularity of this type
observed substrate, furthermore numbers are            of formulas, specially after the introduction of
not corrected by the relative amount of                filter colorimeters that are based on the filters
absorbed yellow light and as such it does not          G (green), B (blue) and A (amber) that are
take into account the action of bluing                 related to the CIE y(), z() and the red-portion
techniques. The latter point can be corrected          of the x() color-matching functions weighted
by using some yellowness index that takes into         by a CIE standard illuminant (normally of type
account the relative amounts of blue and               A). In general following relationships apply:
yellow in the reflected light; first attempts to
describe yellowness were based on the use of              Amber filter          Rx   A
special band pass filters as for example:                 Green filter          RY   G
                                                          Blue filter           Rz   B
      R700  R450
W 
         R700                                          X  a  R X  b  RZ
                                                       Y  RY
where R is the reflectance value at the               Z  c  RZ
wavelength . Further examples are the
Stephansen formula:
                                                              1       b
                                                        RX     X      Z
WI Stephansen  2  R430  R670                               a      ac
                                                        RY  Y
and the Harrison formula:                                    1
                                                        RZ   Z
WI Harrison  100  R670  R430

               Observer       Illuminant       a          b              c
              2°              A            1.044623   0.053849       0.355824
                              C            0.783185   0.197520       1.182246
                              D65          0.770180   0.180251       1.088814
              10°             A             1.05719    0.05417        0.35202
                              C             0.77718    0.19566        1.16144
                              D65          0.768417   0.179707       1.073241

Within this formalism yellowness formulas take         The formula of Taube:
the form:
                                                       WTaube  G  4  G  B 
   A B
    G                                                  was develop by subtracting the amount of
                                                       yellowness (second term) from the index of
and subsequently the Stephansen formula is:            lightness and can also be expressed as:

WI Stephansen  2  RZ   R X                        WTaube  4  B  3  G

and the Harrison formula is:                           Closely related is also the whiteness index of
                                                       the ASTM:
WI Harrison  RZ  R X  100
                                                       WI  3.388  Z  3  Y

                                                                                             Page 2 of 8
Formulas based on a uniform color system (UCS)
It was quite early recognized that yellowness
formulas or those based on the relative
                                                                           WMacAdam  Y  k  pc2
differences of blue and yellow light were not
sufficient to describe whiteness, specially of
those objects whitened through bluing                                      where pe is the colorimetric purity and k is a
techniques. The importance of lightness was                                constant that depends on the application, and
recognized as an important contribution to                                 the Judd formula given by:
whiteness perception as with the Hunter
formula:                                                                   WI ( Judd ,1936)  Y  6700  S 

WHunter  L  3  b
                                                                           where S is the distance between the sample
                                                                           and the preferred white in the Judd’s UCS
where L and b are Hunter coordinates defined                               triangle. The factor 6700 is optimized for
as:                                                                        grading laundry white goods and may assume
            Y                                                              a different value for other applications.
L  100                                                                   The closely related formula:

          0.0102  X n             X                                     WI (Coppock )  10  Y  2  p e2
a  175                         
                                  X Y                
             Y                     n  n               
               Yn                                                          is due to W.A. Coppock and known as the
                                                                           Chemstrand Whiteness Scale.
                                                                           Further formulas based on the principle of
            0.00847  Z n         Y  Z                
b  70                           
                                  Y                   
                                                                           colorimetric purity are the Vaeck formula:
                Y                  n Zn               
                                                                           WVaeck  Y  k  E u, v 

and (Xn,Yn,Zn) are the coordinates of the                                  where the equivalent luminescence E(u,v) is
achromatic point. The simplicity of the Hunter                             defined in the MacAdam UCS diagram and its
formula is remarkable and takes clearly into                               value for a particular (u,v) must be looked up in
account the importance of having high                                      a nomogram and it defines the dominant
lightness and neutral blue b values.                                       wavelength of 472 nm as preferred whiteness
Close relatives of this formula are the                                    hue, and the formula of Anders and Daul:
MacAdam formula given by:

                                                                0.5  x     0.5  x n               
                                                                 y   arctg  y
                         W Anders Daul  2  Y  1520  arctg                                        65
                                                                                n                   

where (xn,yn) is the coordinate of achromatic                              Further developments are the first Selling
point for D65.                                                             formula:
The fact that deviations from the neutral blue                                                                    2
                                                                           WSelling  100  100   Y 2   k  s 
                                                                                                        
yellow axis may contribute to perceived
whiteness leads to the Hunter-Judd formula:                                                             

                         30                      
                                                            1  Y 

WHunter Judd  1                    2
                                          b   2
                                                                 
                                                             2            Y 2   YMgO  Ysample
                                                                                 
                                                                                 
where a and b are Hunter coordinates as
defined above.                                                             and
In this respect the original Hunter formula can
be generalized as:
                                                                           s     u 2  v2
W  100      L   p
                        L  a 2  b 2           2
                                                                           being the distance between the sample and
                                                                           the preferred white on the MacAdam’s UCS
where Lp is the lightness of the preferred white,                          diagram and k is a constant.
in case of MgO it assumes the value 100.                                   A simplification of the latter is the second
                                                                           Selling formula:

                                                                                                                      Page 3 of 8
                                                     A last formula worth mentioning is the Friele
WSelling  100  100  Y   k 's 
                               2             2       formula given by:
                                                                                    2           2
                                                                          M  S
                                                     WFriele      A L      

with                                                                       b  c

Y  YMgO  Ysample                                  where (A,b,c) are constants and (L,M,S) are
                                                     the length of long, medium and short axis of
and s defined as above and k’ is a constant         color discrimination ellipsoid centered on the
with the typical value of 9.5 10 .                   preferred white. This formula is remarkable
The Croes formula is given as:                       since it recognizes fully the importance of
                                                     deviations from the neutral blue axis as
                                                     contributions     to     perceived    whiteness,
WI Croes  Y  13.2  Y   u  u n 2  v  vn 2   furthermore it attempts to compensate for the
                                                     different sensitivities from lightness, hue and
where (u,v) are coordinates in MacAdam UCS           chroma contributions.
diagram and (un,vn) are the coordinates of the
preferred white.

Formulas considering fluorescence
The extensive use of Fluorescent Whitening           The question of preferred white was however
Agents (FWA) to increase perceived whiteness         delegated to second importance since FWA
achieves the compensation of substrate               manufacturers tried to develop whiteness
yellowness through an additive color mixing          formulas tailored to the characteristics of their
process; a considerable amount of blueness           products.
can be introduced without loosing luminance,         The formula of Stensby:
on the contrary a modest lightness increase
results in objects showing dazzling whites.          WStensby  L  3  a  3  b
Depending on their chemical structure,
fluorescence produced by FWAs can lead to
                                                     derives from the Hunter formula and shows
neutral, or to red- or green-shaded whiteness;
                                                     clearly a preference for redder whites, while
the existence of shade preferences is
                                                     the formula of Berger:
illustrated by the formula:
                                                     W Berger  Y  a  Z  b  X
                      220  (G  B)   100  G 

WI (C 429)    100                          
                      G  0.242  B   2          with

originally due to Hunter but modified to give a                                           a           b
neutral white preference, or the formula                        2° observer             3.400       3.895
                                                                10° observer            3.448       3.904
               L  3  b  10  Y  21 Y  Z 
WI (CDML)                                           shows a preference for green whites as well
                               Y                     the formula of Croes:
that shows a blue white preference.
Due to the additive nature of the process it was     WI Croes  RY  RZ  R X
readily recognized that linear formulas could
be built for measuring perceived whiteness in
                                                     Much of the development of linear whiteness
any of the colorimetric spaces:
                                                     formulas was done by Ganz, who formulated a
                                                     general formula as:
W    B    G    A  k1
W    L   b    a  k2                        WGanz  Y  P  x0  x   Q   y 0  y 
W    Y    x    y  k3
                                                     where the values of the parameters determine
since they can be regarded as variations of the      the hue preference as seen from the table:
same theme; this represents also a first
rationalization of existing formulas, since most                          hue preference
of them can fit in one of the listed formulas,                        red     neutral   green
allowing a classification of origins and                    P        -800      +800     +1700
preferences.                                                Q       +3000     +1700      +900

                                                                                                       Page 4 of 8
                                                       the sample, daylight conditions were chosen
where (x0,y0) is the coordinate of the                 as reference for work on whiteness
achromatic point for the D65 illuminant.               determination and modern formulas are strictly
At this stage it was clearly recognized the            valid for D65.
importance of the amount of UV acting onto

Modern formulas: general linear forms
Starting point is the Roesch color solid as            The isoleukai for Y=100 and d= 470 nm
depicted in the figure, under following                consist of fairly parallel and equally spaced flat
conditions:                                            curves that can be approximated by straight
     Illuminant D65                                   lines; this is the case for example for the
     Dominant wavelength for neutral                  formulas of Berger and Stensby but the lines
        whites 470 nm                                  have different slope because of their different
                                                       preference for greenish or reddish whites and
The plane with the said dominant wavelength            is controlled by the angle  in the figure.
describes (blue) colors with the same spectral         To set up the whiteness formula following
purity at different levels of color saturation S,      parameters must be determined numerically:
the colors perceived as white will lie within a              The gain of whiteness with increasing
limited region on this plane; perpendicular to                  saturation ∂W/∂S
this plane are the ones corresponding to hues.               The impact of lightness on whiteness
Curves with same whiteness are called                           ∂W/∂Y
isoleukai, these are defined by the whiteness                The impact of the hue on whiteness
formula (regardless of its general form), it must               ∂W/∂H
be remarked that an isoleuke contains the
same values for whiteness W but different
values for hue (or shade deviation).


                whiteness axis

                  W                                                                        point


                                                    dominant wavelength
                                                          420 nm

In general the following relationship holds:            W        W   
                                                                       tan  
                                                        H        S   

                                                                                                      Page 5 of 8
and each whiteness point is characterized by             W   cos    
                                                   P      
                                                                            
                                                         S   cos  
the slope of the isoleukai:

      W S                                            W   sin     
      W Y                                      Q      
                                                         S   cos  

and their angle  with respect to the line with    where η is the angle between the line with d=
d= 470 nm:                                        470 nm and the x axis, and (x0.y0) is the

  45  arctg
               W S  W H                   achromatic point for D65.
                                                   The next step is the evaluation of the shade
               W S  W H                   deviation or tint by the formula:

                                                   TGanzGriesser  m  x0  x   n   y 0  y 
It must be remarked that the numerical value
of (∂W/∂S) sets the extension of the whiteness
scale and is closely related to the amount of      where
                                                          cos 
UV present in the illuminant; this is a direct     m
consequence of the presence of fluorescence                BW
resulting from excitation of the FWA and
inherent to the nature of the formula.             and
With these definitions following linear formula
can be written down:                                     sin  
WGanz  D  Y  P  x0  x   Q   y 0  y 
                                                   and  is the angle of the perpendicular to the
where                                              line with d= 470 nm and the bandwidth BW is
                                                   a constant related to the sensitivity of the eye
         W                                        to distinguish different shades of white.

The Ganz and Ganz-Griesser formulas
The formulas are expressed by:                              (x0.y0) = (0.313795,0.330972)

WGanz  Y  P  x0  x   Q   y 0  y         The numerical scale sets up following
                                                   threshold values:
TGanzGriesser  m  x0  x   n   y 0  y         threshold       for   undistinguishable
                                                           whiteness sample pairs: 5 Ganz
                                                           whiteness points
and describe whiteness and shade deviation              threshold for undistinguishable shade
(tint) for daylight D65. The formula parameters            deviation in sample pairs: 0.5 Ganz-
adopt following numerical values:                          Griesser points
         D= ∂W/∂Y = 1                             The      instrument   for   conducting    the
         ∂W/∂S= 4000                              measurements must be equipped with a
         dominant wavelength is 470 nm (η=        device to regulate the amount of UV falling
          48.18154° and α=41.81852°)               onto the samples, this amount must be set
         φ= 15° (light preference for greenish    (and maintained) to an amount similar to that
          whites)                                  encountered in daylight in order to obtain
         BW= 0.0008                               reliable whiteness data. Some problems arise
                                                   because        small  unavoidable    physical
Under these conditions following parameters        differences among instruments result in large
are calculated:                                    discrepancies      in  measured    whiteness
                                                   numbers; for this reason the Ganz-Griesser
         P = -1868.322                            formulas are applied with instrument-specific
         Q = -3695.690                            parameters calculated with the aid of proper
         m = -931.576                             calibrated samples. The formulas are
         n = 833.467                              expressed as:

                                                                                                      Page 6 of 8
WGanz  Y  P  x  Q  y  C                                       calibrated instrument; this procedure leads to
                                                                    satisfactory results when inter-instrumental
                                                                    comparison (specially shade deviation values)
TGanzGriesser  m  x  n  y  k                                  are mandatory, though to the price of non-
                                                                    transferable parameters.
where the values of (P,Q,C) and (m,n,k) are
not universal and apply only for the specific

The CIE formulas
The formulas are expressed by:                                      As with the Ganz formula, the instrument for
                                                                    conducting the measurements must be
WCIE  Y  800  x0  x   1700   y 0  y                      equipped with a device to regulate the amount
                                                                    of UV falling onto the samples and this amount
                                                                    must be set (and maintained) to an amount
TCIE  900  x0  x   650   y 0  y                           similar to that encountered in daylight in order
                                                                    to obtain reliable whiteness data.
and describe whiteness and shade deviation                          The CIE equations are object of a norm issued
(tint) for daylight D65. The formula parameters                     by the CIE and adopted by many institutions
adopt following numerical values:                                   like ISO, Tappi, AATCC, DIN, ASTM, etc.
                                                                    Strictly speaking the CIE formulas are valid
         D= ∂W/∂Y = 1                                              only for illuminant D65 and for UV amounts
         ∂W/∂S= 1800.36                                            similar to daylight, however some institutions
         dominant wavelength is 470 nm (η=                         allow to use the CIE formulas in conjunction
          48.18154° and α=41.81852°)                                with illuminants different than D65. As shown
         φ= 16.6173° (light preference for                         above the value of ∂W/∂S determines the
          greenish whites)                                          scaling of the whiteness values and it is closely
         BW= 0.000901                                              related to the amount of fluorescence excited
         (x0.y0) = (0.313795,0.330972)                             from the FWA; there has been no study about
                                                                    the behavior of the isoleukai for other
The numerical scale               sets     up   following           illuminants. Recently the ISO has extended the
threshold values:                                                   CIE formulas to be used in conjunction with the
                                                                    illuminant C (introducing the term “indoor
     threshold    for     undistinguishable                        whiteness”), on the grounds that although the
      whiteness sample pairs: 2.3 CIE                               amount of UV differs notably from that of
      whiteness points                                              daylight, the coordinates of the achromatic
                                                                    point do not differ much. While probably the
    threshold for undistinguishable shade
                                                                    isoleukai are not too distorted compared with
      deviation in sample pairs: 0.2 CIE
                                                                    those of daylight, the assumption remains to
      shade points
                                                                    be proved true.
                                                                    In a later work Ganz gave the CIE equations in
                                                                    CIE-L*a*b* space as:

                                                                                    
                           WCIE  L*a*b*  2.41  L*  4.45  b *  1  0.0090  L*  96  141 .4

TCIE l *a*b*  1.58  a *  0.38  b *                            These are a generalization of the originally
                                                                    postulated linear formulas.

Performance of linear whiteness formulas
One must not forget that linear formulas were                       Another problem arising from the linear form is
developed for fluorescent whites, they perform                      that the formulas are “open”, any sample, even
fairly well for medium to high whiteness levels,                    a colored one will show certain degree of
but the linear approximation starts breaking                        whiteness; proper assessment requires
down for very high whiteness values or for                          assistance form the human observer. While
medium to strong shaded samples. For                                limits for whiteness have been postulated, for
samples with low content of fluorescence or                         example:
especially for just bleached materials they do                            samples are white if -20 < W Ganz <
not give reliable data and one should switch to                              (8*Y-490)
a yellowness formula to obtain proper                                     samples are white if 40 < W CIE < (5*Y-
assessment.                                                                  280)

                                                                                                          Page 7 of 8
the validity of these limits is highly dependent
on the observer and they fail specially with                  if W CIE > 5*Y-275
heavy shaded samples.
Recently Uchida has proposed corrective                       W  PCIE  2  P 
terms to the CIE formula as follows:

if 40 < W CIE< 5*Y-275                                        where

W  WCIE  2  TCIE 


             Pw  5  Y  275  800  U  V  100  Y   x
                                                                       1700  R  T  100  Y   y

                                                 2° observer       10° observer
                                      U          0.2761            0.2742
                                      V          0.00117           0.00127
                                      R          0.2727            0.2762
                                      T          0.0018            0.00176

Hundreds of whiteness formulas have been
published or applied in different fields over the
last 70 years and under a variety of conditions.
Certainly the field of whiteness has evolved
also during all those years, imposing new
additional challenges to the developed
formulas, the instrumental data has however
stayed back and failed to provide enough
reliable data to conduct quantitative and
conclusive studies.
Whiteness perception, although occupying a
small amount of the total color solid, is still a
psychochromatic phenomena attached to three
quantities as any other color, this is a hint that
a proper description must be based on three
quantities that describe fully the perceived
Still the quest for the “most beautiful white”
remains open, but it is a recognized fact that
its definition depends on cultural background
of the observer group and application of the
object, a specialization of whiteness formulas
based on absolute principles seems possible
and represents truly the goal of the next
developments in this area.

                                                                                                                 Page 8 of 8

To top