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Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1553
Spectral sharpening: sensor transformationsfor improved color
constancy
Graham D. Finlayson, Mark S. Drew, and Brian V. Funt
School of Computing Science, Simon Fraser University, Vancouver,
B.C., Canada V5A 1S6
Received March 8, 1993; revised manuscript accepted October 28,
1993; accepted October 28, 1993
We develop sensor transformations, collectively called spectral
sharpening, that convert a given set of sensorsensitivity functions
into a new set that will improve the performance of any
color-constancy algorithmthat is based on an independent adjustment
of the sensor response channels. Independent adjustment
ofmultiplicative coefficients corresponds to the application of a
diagonal-matrix transform (DMT) to the sensorresponse vector and is
a common feature of many theories of color constancy, Land's
retinex and von Kriesadaptation in particular. We set forth three
techniques for spectral sharpening. Sensor-based sharpeningfocuses
on the production of new sensors as linear combinations of the
given ones such that each new sensorhas its spectral sensitivity
concentrated as much as possible within a narrow band of
wavelengths. Data-based sharpening, on the other hand, extracts new
sensors by optimizing the ability of a DMT to accountfor a given
illumination change by examining the sensor response vectors
obtained from a set of surfacesunder two different illuminants.
Finally in perfect sharpening we demonstrate that, if illumination
andsurface reflectance are described by two- and three-parameter
finite-dimensional models, there exists a uniqueoptimal sharpening
transform. All three sharpening methods yield similar results. When
sharpened conesensitivities are used as sensors, a DMT models
illumination change extremely well. We present simulationresults
suggesting that in general nondiagonal transforms can do only
marginally better. Our sharpeningresults correlate well with the
psychophysical evidence of spectral sharpening in the human visual
system.
Key words: spectral sharpening, color constancy, color
balancing, lightness, von Kries adaptation.
1. INTRODUCTION
The performance of any color-constancy algorithm,whether
implemented biologically or mechanically, willbe strongly affected
by the spectral sensitivities of thesensors providing its input.
Although in humans thecone sensitivities obviously cannot be
changed, we neednot assume that they form the only possible input
to thecolor-constancy process. New sensor sensitivities can
beconstructed as linear combinations of the original
sen-sitivities, and in this paper we explore what the
mostadvantageous such linear transformations might be.
We call the sensor response vector for a surface viewedunder an
arbitrary test illuminant an observation. Theresponse vector for a
surface viewed under a fixed canoni-cal light is called a
descriptor. We take as the goal ofcolor constancy the mapping of
observations to descrip-tors. Since a descriptor is independent of
illumination,it encapsulates surface reflectance properties.'
In the discussion of a color-constancy algorithm thereare two
separate issues: the type of mechanism or ve-hicle supporting the
transformation from observations todescriptors in general and the
method used to calculatethe specific transformation that is
applicable under a par-ticular illumination. In this paper we
address only theformer and therefore are not proposing a completely
newtheory of color constancy.
A diagonal-matrix transformation (DMT) has beenthe
transformation vehicle for many color-constancy al-gorithms, in
particular von Kries adaptation,2 all theretinex/lightness
algorithms,3 -5 and, more recently,Forsyth's gamut-mapping
approach.6 All these algo-rithms respond to changing illumination
by adjusting the
response of each sensor channel independently, althoughthe
strategies that they use to decide on the actual ad-justments
differ.
DMT support of color constancy is expressed mathe-matically in
relation (1). Here pe denotes an observa-tion (a 3-vector of sensor
responses), where i and e areindex surface reflectances and
illumination, respectively.The vector pic represents a descriptor
and depends onthe single canonical illuminant. The diagonal
transformDe best maps observations onto descriptors. Through-out,
the superscript e denotes dependence on a variableilluminant and
the superscript c denotes dependence onthe fixed canonical
illuminant. Boldface indicates vectorquantities.
In general, there may be significant error in this
ap-proximation. Indeed West and Brill2 and D'Zmura andLennie7 have
shown that a visual system equipped withsensors having the same
spectral sensitivity as the hu-man cones can achieve only
approximate color constancythrough a DMT.
A DMT will work better with some sensor sensitivitiesthan with
others, as one can see by considering how theillumination, surface
reflectance, and sensor sensitivitiescombine in the formation of an
observation. An observa-tion corresponds to
p = f E(A)S(A)R(A)dA, (2)where E(A), S(A), and R(A) denote
illumination, surfacereflectance, and sensor sensitivities,
respectively, and the
0740-3232/94/051553-11$06.00 © 1994 Optical Society of
America
Finlayson et al.
Pi C Depie. (1)
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1554 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994
integral is taken over the visible spectrum . For a DMTto
suffice in modeling illumination change,2 it must bethe case, given
a reference reflectance Sr, a secondaryarbitrary reflectance S, and
illuminants Ei and Ei, that,
f Ei(A)S(A)R(A)dA f EJ(A)S(A)R(A)dA@ = @ * ~~~~~~~~~~~(3)
f Ei(A)Sr(A)R(A)dA fEi(A)S(A)R(A)dAAs others have observed, one
way to ensure that this
condition holds is to use extremely narrow-band sensors,which in
the limit leads to sensors that are sensitive to asingle wavelength
(Dirac delta functions).6 Our intuitionwhen we began this work was
that if we could find a lin-ear combination of sensor sensitivities
such that the newsensors would be sharper (more narrow band), then
theperformance of DMT color-constancy algorithms shouldimprove and
the error of relation (1) would be reduced.It should be noted,
however, that Eq. (3) can be satisfiedin other ways, such as by the
placement of constraints onthe space of illuminants or
reflectances.2 With the addi-tion of sharpening, relation (1)
becomes
Tpi - Tpi, (4)
where T denotes the sharpening transform of the originalsensor
sensitivities. It is important to note that applyinga linear
transformation to response vectors has the sameeffect as
application of the transformation to the sensorsensitivity
functions.
The sharpening transform effectively generalizesdiagonal-matrix
theories of color constancy. Otherauthors2 36 also have discussed
this concept of an inter-mediate (or sharpening) transform.
However, our workappears to be the first to consider the precise
form of thistransform.
The sharpening transform is a mechanism throughwhich the
inherent simplicity of many color-constancyalgorithms can be
maintained. For example, Land'sretinex algorithm requires color
ratios to be illuminationindependent (and hence implicitly assumes
a diagonal-matrix model of color constancy), which, as can be
seenfrom Eq. (3), they generally will not be. It seems dif-ficult
to improve the accuracy of retinex ratioing di-rectly without
making the overall algorithm much morecomplicated8 ; however, if a
simple, fixed sharpeningtransformation of the sensors as a
preprocessing stageis applied, the rest of the retinex process can
remainuntouched. Similar arguments apply to Forsyth's stan-dards
for coefficient rule (CRULE)6 and Brill's9 volumet-ric theory.
We initially present two methods for calculating T:sensor-based
and data-based sharpening. Sensor-basedsharpening is a general
technique for determining thelinear combination of a given sensor
set that is maxi-mally sensitive to subintervals of the visible
spectrum.This method is founded on the intuition that narrow-band
sensors will improve the performance of DMTtheories of color
constancy. We apply sensor-basedsharpening over three different
ranges in order to gen-erate three new sharpened sensors that are
maximallysensitive in the long-wave, medium-wave, and
shortwavebands. Figure 1 below contrasts the cone fundamentals
derived by Vos and Walraven10 (VW) before and aftersharpening.
Although the new sensitivity functions aresharper, they are far
from meeting the intuitive goal ofbeing very narrow band (i.e.,
with strictly zero response inall but a small spectral region);
nonetheless, we performsimulations that show that they in fact work
much betterthan the unsharpened cone fundamentals. In Section 2we
present the details of sensor-based sharpening.
Sensor-based sharpening does not take into account
thecharacteristics of the possible illuminants and reflect-ances
but considers only the sharpness of the resultingsensor. Our second
sharpening technique, data-basedsharpening, is a tool for
validating the sensor-basedsharpening method. Given observations of
real surfacereflectances viewed under a test illuminant and
theircorresponding descriptors, data-based sharpening findsthe
best, subject to a least-squares criterion, sharpen-ing transform
T'f. Interestingly, data-based sharpeningyields stable results for
all the test illuminations thatwe tried, and in all cases the
data-based-derived sensorsare similar to the fixed sensor-based
sharpened sensors.Data-based sharpening is presented in Section
3.
In Section 4 we present simulations evaluatingdiagonal-matrix
color constancy for sharpened andunsharpened sensor sets. Over a
wide range of illu-minations, sensor-based sharpened sensors
provide asignificant increase in color-constancy performance.
Data-based sharpening is related to Brill's9 volumetrictheory of
color constancy. This relationship is exploredin Section 5. Through
spectral sharpening, the volumet-ric theory is shown to be
informationally equivalent toLand'sl white-patch retinex.
The data-based sharpening technique finds the opti-mal
sharpening transform for a single test illuminant.In Section 6 we
investigate the problem of finding a goodsharpening transform
relative to multiple illuminants.We show that if surface
reflectances are three dimen-sional and illuminants two dimensional
there exists asharpening transform with respect to which a
diagonalmatrix supports perfect color constancy. This
analysisconstitutes a third technique for deriving the
sharpeningtransform.
In Section 7 we relate our work specifically to theoriesof human
color vision. Sharpened spectral sensitivitieshave been measured in
humans.12- 6 We advance thehypothesis that sharpened sensor
sensitivities arise as anatural consequence of optimization of the
visual system'scolor-constancy abilities through an initial linear
trans-formation of the cone outputs.
2. SENSOR-BASED SHARPENING
Sensor-based sharpening is a method of determining thesharpest
sensor, given an s-dimensional (s is usually 3)sensor basis R(A)
and wavelength interval [Al, A2]. Thesensor Rt(A)c, where c is a
coefficient vector, is maximallysensitive in [Al, A2] if the
percentage of its norm lying inthis interval is maximal relative to
all other sensors. Wecan solve for c by minimizing:
I = L [R(A)tc]2dA + p, [R(A)tc]2dA - ,
Finlayson et al.
I
(5)
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Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1555
where (o is the visible spectrum, 0 denotes wavelengthsoutside
the sharpening interval, and jz is a Lagrange, mul-tiplier. The
Lagrange multiplier guarantees a nontrivialsolution for Eq. (5):
the norm of the sharpened sensor isequal to 1. Moreover, this
constraint ensures that thesame sharpened sensor is recovered
independent of theinitial norms of the basis set R(A).
By differentiating with respect to c and equating to thezero
vector, we find the stationary values of Eq. (5):
2 aI = f R(A)R(A)tcdA + pff R(A)R(A)tcdA =0.(6)
Differentiating with respect to /,t simply yields the
con-straint equation f, [R(A)tc]2 dA = 1. The solution ofEq. (6)
can thus be carried out, assuming that the con-straint holds.
Define the s x s matrix A(a) = It R(A)R(A)t dA so thatEq. (6)
becomes
A()c = -A(w)c. (7)
Assuming a nontrivial solution c 0, , 0 0 -and rearrang-ing Eq.
(7), we see that solving for c (and consequently forthe sharpened
sensor) is an eigenvector problem:
[A(co)]-'A(5)c = -nc. (8)
There are s solutions of Eq. (8), each solution corre-sponding
to a stationary value, so we choose the eigenvec-tor that minimizes
f¢ [R(A)tc]2 dA. It is important thatc be a real-valued vector, as
it implies that our sharpenedsensor is a real-valued function. That
c is real valued fol-lows from the facts that the matrices [A())]-'
and A(+)are positive definite and that eigenvalues of the product
oftwo positive-definite matrices are real and nonnegative.' 7
Solving for c for each of three wavelength intervals yieldsa
matrix '2' for use in relation (4). The matrix C isnot dependent on
any illuminant and denotes the sensor-based sharpening
transform.
We sharpened two sets of sensor sensitivities: thecone
absorptance functions measured by Bowmaker andDartnell' 5 (BOW) and
the cone fundamentals derived byVos and Walraven' 0 (VW), which
take into account thespectral absorptions of the eye's lens and
macular pig-ment. The BOW functions were sharpened with respectto
the wavelength intervals (in nanometers) [400,480],[510,550], and
[580,650] and the VW fundamentalswith respect to the intervals
[400,480], [520,560], and[580,650]. (All spectra used in this paper
are in therange 400-650 nm measured at 10-nm intervals.)
Theseintervals were chosen to ensure that the whole visiblespectrum
would be sampled and that the peak sensitivi-ties of the resulting
sensors would roughly correlate withthose of the cones.
The results for the VW sensors are presented in Fig. 1(those for
the BOW sensors are similar), where it can beseen that the
sharpened curves contain negative sensitivi-ties. These need not
cause concern in that they do notrepresent negative physical
sensitivities but merely repre-sent negative coefficients in a
computational mechanism.Clearly, the sharpening intervals are
somewhat arbitrary.
They were chosen simply because the resulting sharpenedsensors
appeared, to the human eye, significantly sharperthan the
unsharpened sensors. Nevertheless, the inter-vals used are
sensible, and their suitability is verified bythe fact that they
provide sharpened sensors that are inclose agreement with those
derived by data-based sharp-ening as described in Section 3. The
actual values of thec's in Eq. (8) are given in Section 6
below.
Figure 1 contrasts the sharpened VW sensor set withthe
corresponding unsharpened set; the degree of sharp-ening is quite
significant. The peak sensitivities ofthe new sensors are shifted
with respect to the initialsensitivities; this shift is due to both
the choice of sharp-ening intervals and the shape of the VW
sensitivities.The sharpened long-wavelength mechanisfn is
pushedfurther toward the long-wavelength end of the spectrum;in
contrast, the medium-wavelength mechanism isshifted toward the
shorter wavelengths; and the short-wavelength mechanism remains
essentially the same.Intriguingly, field sensitivities of the human
eye mea-sured under white-light conditioning with long
testflashes13 are sharpened in an analogous manner.
Table 1 contrasts the percentage-squared norm lying inthe
sharpening intervals for the original versus the sharp-ened curves.
For both the VW and the BOW sensors thedegree of sharpening is
significant. Furthermore, from
0.4
' > 0.3
. - 0.2E Ct (o 0.1
0.0
-0.1
. .
EN :U)E
' a)Z to
z.N>
0.4
0.3
0.2
0.1
0.0
0.4
0.3
0.2
0.1
0.0I
/ / \
400 450 500 550 600 650Wavelength
400 450 500 550 600 650Wavelength
. , ;_,--,----- ,- ;-/
400 450 500 550 600 650Wavelength
Fig. 1. VW fundamentals (solid curves) are contrasted withthe
sharpened sensitivities derived through sensor-based(dotted curves)
and data-based (dashed curves) sharpening.Top, long-wavelength
mechanism; middle, medium-wavelengthmechanism; bottom,
short-wavelength mechanism.
Finlayson et~ al.
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1556 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994 Fnasne l
Table 1. Percentage of Total Squared Normin the Sharpening
Intervals
% Squared NormSensors [400,480] [510,550] [580,650]
BOW, original 98.9 51.4 30.1BOW, sharpened 99.4 66.3 76.2
[520,560]VW, original 97.5 67.6 40.3VW, sharpened 97.8 74.7
89.1
Ire~ is unique also, always exists, and equals [e]-l,
fordiagonal [De].
It is interesting to compare Eq. (12) with the relationfor the
problem of finding the best general transform G that maps
observations obtained under a test illuminantto their corresponding
descriptors:
Wc,: G e~e. (13)
Fig. it is clear that the sharpening effect is not limitedto the
sharpened interval.
3. DATA-BASED SHARPENINGIt could be the case that the best
sensors for DMT al-gorithms might vary substantially with the type
of -lumination change being modeled. If so, sensor-basedsharpening,
which does not take into account any of thestatistical properties
of collections of surfaces and illumi-nants, might perform well in
some cases and poorly inothers. To test whether radically different
sharpeningtransformations are required in different
circumstances,we explore a data-based approach to deriving
sharpenedsensitivities, in which we optimize the sensors for DMT
al-gorithms by examining the relationship between observa-tions,
obtained under different test illuminants, and theircorresponding
descriptors.
Let WI be a 3 n matrix of descriptors generated froma set n of
surfaces observed under a canonical illuminantEc. Similarly, let We
be the matrix of observations of nsurfaces imaged under another
test illuminant El. Tothe extent that DMT-based algorithms suffice
for colorconstancy, WC and We should be approximately equiva-lent
under a DMT:
WC =::DeWe. (9)
The diagonal transform is assumed to be an approxi-mate mapping
and will have a certaln degree of error.The idea of spectral
sharpening is that this approximationerror can be reduced if We and
We are first transformedby a matrix rfe;
q'eWc =:De'eWe. (10)
D e will in fact be optimal in the least-squares sense if it
is defined by the Moore-Penrose inverse:
De= rf~ewc[9-ewe]+,(1
where + denotes the Moore-Penrose inverse. (TheMoore-Penrose
inverse of the matrix A is defined as
A+- At[AAt]-1.) Now [Te] must be chosen to ensurethat [De] is
diagonal. To see how to do this, carry outsome matrix manipulation
to obtain
[T'e]-lveje = Wc[We]+. (12)
Since the eigenvector decomposition of the matrix onthe
right-hand side of Eq. (12), Wc[We]+ = UleDe[(Je]-l,is unique, its
similarity to the left-hand side implies that
Such optimal fitting effectively bounds the possible
per-formance within a linear model of color constancy. Whenthe
approximation of relation (13) is to be optimized in
theleast-squares sense, G I is simply
(14)
Equation (12) can be interpreted, therefore, as simplythe
eigenvector decomposition of the optimal generaltransform G .Of
course, it is obvious that the opti-mal transform could be
diagonalized; what it is impor-tant is that if one knows a
sharpening transform Te~],then the best least-squares solution
relating ffeWc and
IWe is precisely the diagonal transformation De; thatis,
TeWc[cyeWe]+ = De. In other words, when one isusing the sharpened
sensors, the optimal transform isguaranteed to be diagonal, so
finding the best diagonaltransform after sharpening is equivalent
to finding theoptimal general transform. Therefore sharpening
allowsus to replace the problem of determining the nine pa-rameters
of GeI with that of determining only the threeparameters of De.
Data-based sharpening raises two main questions: (1)Will the
resulting sensors be similar to those obtained bysensor-based
sharpening, and (2) will the derived sensorsvary substantially with
the illuminant? To answer thesequestions requires the application
of data-based sharpen-ing to response vectors obtained under a
single canonicalilluminant and several test illuminants. For
illuminantswe used five Judd daylight spectra' 9 and CIE standard
il-luminant A,10 and as reflectances we used the set of 462Munsell
spectra 20 We arbitrarily chose Judd's D55 (55stands for 5500 K) as
the canonical illuminant; descrip-tors are the response vectors for
surfaces viewed underD55. For each of the other illumninants El, we
derivedthe data-based sharpening transform Tje in accordancewith
Eq. (12).
Figure 2 shows for the VW sensors the range of the fivesets of
data-based sharpened sensors obtained for map-ping between each of
the five illuminants and D55. Forthese five illuminants, the
results are remarkably stableand hence relatively independent of
the particular illumi-nant, so the mean of these sharpened sensors
character-izes the set of them quite well. Referring once again
toFig. 1, we can see that these mean sensors are very simi-lar to
those derived through sensor-based sharpening.
The stability of the results for the five cases and
thesimilarity of these results to the sensor-based resultsis
reassuring; nonetheless, it would be nice to find thesharpening
transformation that optimizes over all theilluminants
simultaneously. This issue is addressed inSection 6, where we show
that, given illuminant and re-flectance spectra that are two and
three dimensional,
Finlayson et al.
Ge = Wc[Wel+.
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Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1557
0.4
~0 0.3,N .> 0.2a _E n; ' 0.1
0.0
-0.1
0.4
0.3
0.2
0.1
0.0
0.4
0.3
0.2
0.1
0.0
400 450 500 550 600 650Wavelength
400 450 500 550 600 650Wavelength
40Q 450 500 550 600 650Wavelength
Fig. 2. Data-based sharpening generates different
sharpenedsensors for each illuminant. The range of sharpened
curvesover all the test illuminants (CIE A, D48, D65, D75, andD100)
mapped to D55 is shown for the VW cone mechanisms.Top,
long-wavelength mechanism; middle, medium-wavelengthmechanism;
bottom, short-wavelength mechanism.
respectively, there exists a unique optimal
sharpeningtransform.
4. EVALUATING SHARPENED SENSORS
Since the sensor-based and data-based sharpened sensorsare
similar, we restrict our further attention to the evalu-ation of
sensor-based sharpened sensors alone. The re-sults for VW and BOW
sensors are similar as well, so weinclude figures only for the VW
case.
For each illuminant we generated sensor responsesfor our 462
test surface reflectances, using both thesharpened and the
unsharpened VW sensors so that wecould determine how much
sharpening improves DMTperformance.
To measure the performance of a DMT in mapping testobservations
to canonical descriptors, we compare fit-ted observations
(observations mapped to the canonicalilluminant by means of a
diagonal matrix) with corre-sponding (canonical) descriptors. The
Euclidean dis-tance between a fitted observation qe and its
descriptorpC, normalized with respect to the descriptor's
length,provides a good error metric, given the definition of
colorconstancy that we are using. This percent normalized
fitted distance (NFD) metric is defined as
NFD = 100 * IIp - qej1IOpCI (15)
Let WI be a 3 X 462 matrix of descriptors correspond-ing to the
462 surfaces viewed under the canonical illu-minant. Similarly, let
WI denote the 3 X 462 matrix ofobservations for the 462 surfaces
viewed under a test illu-minant. Relation (9) can then be solved to
yield the bestdiagonal transformation in the least-squares sense;
thisprocedure will be called simple diagonal fitting. SinceD e is a
diagonal matrix, each row of We is fitted indepen-dently. The
components of De are derived as follows:
D,, = Wi [We]+ = Wvc[We,]t/Wie[We]t , (16)
where the single subscript i denotes the ith matrix rowand the
double subscript ii denotes the matrix element atrow i column
i.
Given a fixed set of sensor functions, Eq. (16) yieldsthe best
diagonal transformation that takes observationsunder the test
illuminant onto their corresponding de-scriptors. Simple diagonal
fitting, therefore, does not in-clude sharpening but rather for a
fixed set of sensors findsthe best diagonal matrix DI for that set
of sensors.
For performance comparisons we are also interested inthe NFD
that results under transformed diagonal fitting.Transformed
diagonal fitting proceeds in two stages:
1. 'tWc D"ec'We, where ' is the fixed sharpeningtransform and D
I is calculated by means of Eq. (16);
2. q1`19jWc r1-lDeTwe.
Application of T`1 transforms the fitted observationsback to the
original (unsharpened) sensor set so that anappropriate comparison
can be made between the fittingerrors: the performance of sharpened
diagonal-matrixconstancy is measured relative to the original
unsharp-ened sensors.
Figure 3 shows NFD cumulative histograms for diago-nal fitting
of VW observations (solid curves) and diago-nal fitting of
sharpened VW observations (dotted curves).For each illuminant the
sharpened sensors show bet-ter performance than the unsharpened
ones, as indi-cated by the fact that in the cumulative NFD
histogramsthe sharpened sensor values are always above those forthe
unsharpened ones. In general the performance dif-ference increases
as the illuminant color becomes moreextreme: from D55 to D100 the
illuminants become pro-gressively bluer and toward CIE A,
redder.
Figure 4 shows the cumulative NFD histograms for di-agonal
fitting of VW observations (solid curves) and trans-formed diagonal
fitting of sharpened VW observations(dashed curves). Once again it
is clear that the sharp-ened sensors perform better; however, the
performancedifference is greater, which shows that the question
ofsensor performance is linked to the axes in which colorspace is
described.
Finally, Fig. 5 contrasts the cumulative NFD histo-grams for
optimal fitting of VW observations [the unre-stricted least-squares
fit of relation (13) (solid curves)]and transformed diagonal
fitting of sensor-based sharp-
C) .N>X ,_
z n
x >a) N >
E Ci a)Z Un
Finlayson et al.
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1558 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994
100
80
_ 60
840
20
100
80
60
C 40
20
CIE A
0 2 4 6 8 10%error
D75
D48100
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C 60
40
20
0
100
80
n 600QC 40
20
0
D65
100
80
"I
60
. 40
20
0 2 4 6 8 10%error
D1 00100
80
_ 60
C 40
20
0
0 2 4 6 8 10%error
Average
0.2 4 6 810 0 2 4 6 810 0 2 4 6 8 10%error %error %error
Fig. 3. Cumulative NFD histogram obtained with each
testilluminant (CIE A, D48, D65, D75, and D100) for diagonalfitting
of VW observations (solid curves) and diagonal fittingof
sensor-based sharpened VW observations (dotted curves).The sixth
cumulative NFD histogram shows the average fittingperformance.
Finlayson et al.
ened VW observations (dashed curves). For these cases,with the
exception of CIE A, a DMT achieves almostthe same level of
performance as the best nondiagonaltransform.
5. DATA-BASED SHARPENINGAND VOLUMETRIC THEORY
Data-based sharpening is a useful tool for validatingour choice
of sensor-based sharpened sensors. However,more than this,
data-based sharpening can also be viewedas a generalization of
Brill's9 volumetric theory of colorconstancy. In that theory Brill
develops a method forgenerating illuminant-invariant descriptors
that is basedon two key assumptions:
1. Surface reflectances are well modeled by a three-dimensional
basis set and are thus defined by a surfaceweight vector oa. For
example, S(A) = 1 Si(A)o-1.
2. Each image contains three known reference re-flectances. In
the discussion that follows, Qe denotes the3 x 3 matrix of
observations for the reference patches seenunder Ee(A).
Given the first assumption, observations for surfacesviewed
under Ee(A) are generated by application of alighting matrix (a
term first used by Maloney2 1) to surfaceweight vectors:
pe = A eo, (17)
where the ijth entry of Ae is equal to f Rj(A)Ee(A)Sj(A)dA.
100
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2C 40
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0
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0 2 4 6 8 10%error
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C 40
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CC 60
c 40
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Average
CIE A
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C 40
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_60
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100
80
-C 60
C 40
20
0
100
80
C 40
20
0
0 2 4 6 8 10 0 2 4 6 810 0 2 4 6 810%error %error %error
Fig. 4. Cumulative NFD histogram obtained with each
testilluminant (CIE A, D48, D65, D75, and D100) for diagonal
fittingof VW observations (solid curves) and transformed diagonal
fit-ting of sensor-based sharpened VW observations (dotted
curves).The sixth cumulative NFD histogram shows the average
fittingperformance.
100
80
-60S
C 40
20
0
100
80
60
C 4021
20
0
0 2 4 6 810 0 2 4 6 810 0 2 4 6 810%error %error %error
Fig. 5. Cumulative NFD histogram obtained with each
testilluminant (CIE A, D48, D65, D75, and D100) for optimal
fittingof VW observations (solid curves) and transformed diagonal
fit-ting of sensor-based sharpened V' observations (dotted
curves).The sixth cumulative NFD histogram shows the average
fittingperformance.
-
Finlayson et al.
It follows immediately that Qe is a fixed linear transformof the
lighting matrix:
Qe Ae_4, (18)
where the columns of .A correspond to the surface weightvectors
of the reference reflectances and are independentof the illuminant.
Now, given any arbitrary responsevector pe, one can easily generate
an illuminant-invariantdescriptor by premultiplying with
[Qe]-l:
[Qe]l-pe = -q--[Aep]lAeu = -A' 0. (19)
The color-constancy performance of Brill's volumetrictheory is
linked directly to the dimensionality of surfacereflectances. Real
reflectance spectra are generally notthree dimensional (Maloney2 '
suggests that a basis set ofbetween three and six functions is
required), leading toinaccuracies in the calculated
descriptors.
Data-based sharpening, like volumetric theory, aimsto generate
illuminant-invariant descriptors by applyinga linear transform. If
we impose the extremely strongconstraint that all surface
reflectances appear in eachimage, then data-based sharpening can be
viewed as analgorithm for color constancy. As a color-constancy
al-gorithm, data-based sharpening has a distinct advantageover
volumetric theory in that it is optimal with respectto the
least-squares criterion and consequently is guar-anteed to
outperform the volumetric method. Unfortu-nately, this performance
increase is gained at the expenseof the extremely strong
requirement that all surface re-flectances appear in each
image.
In practice we can weaken this constraint and assumethat there
are k known reference patches per image,where k is small. Novak and
Shafer2 2 develop a simi-lar theory called supervised color
constancy, which isbased on the assumption that there are 24 known
refer-ence patches in each image; however, unlike
data-basedsharpening, their constancy transform is derived by
ex-amination of the relationship between measured re-sponses and
finite-dimensional models of reflectance andillumination. Certainly
for the reference patches them-selves the data-based sharpening
method will outper-form Novak's supervised color constancy since,
for thesepatches, data-based sharpening finds the optimal
least-squares transform. However, further study is requiredfor
comparing overall color-constancy performance. Inthe limiting case,
where k = 3, data-based sharpeningreduces to the volumetric
theory.
Volumetric theory requires three reference patchesin order to
recover the nine parameters of [e]-1 andthereby achieve color
constancy. As shown by the per-formance tests of the preceding
section, however, aftera fixed sharpening transformation of DMT
models illu-mination change almost as well as does a
nondiagonalmatrix. Since only three parameters instead of nineneed
to be determined for specification of the diagonalmatrix when
sharpened sensors are being used, only onereference patch is
required instead of three for achievingcolor constancy. This
follows because a single responsevector seen under a test
illuminant Ee(A) can be mapped
Vol. 11, No. 5/May 1994/J. Opt Soc. Am. A 1559
to its canonical appearance by a single diagonal matrix:
pC = Depe,
Pic
(20)
(21)
If we choose our reference patch to be a white re-flectance,
then, through sharpening, volumetric theory re-duces to Land's
white-patch retinex.1 Similarly, Westand Brill2 consider
white-patch normalization to be con-sistent with von Kries
adaptation.
We performed a simulation, called transformed white-patch
normalization, to evaluate the quality of color con-stancy
obtainable with use of a single reference patch.For each
illumination (CIE A, D48, D55, D65, D75, andD100) we
1. generated a matrix We of observations of Munsellpatches for
VW sensors;
2. transformed observations to the (sensor-based)sharpened
sensors, T'we;
3. calculated DeT'We, where Diie = l/pW (the re-ciprocal of the
ith sharpened sensor's response to thewhite patch; the Munsell
reflectance closest to the uni-form white, in the least-squares
sense, was chosen as thewhite reference patch);
4. transformed white-patch normalized observations-back to VW
sensors, 9`1DegWe.
Again D55 was the canonical light. Thus we evalu-ated constancy
by calculating the NFD between thewhite-patch-corrected
observations seen under D55 (thedescriptors) and the
white-patch-corrected observationsunder each other illuminant. In
Fig. 6 we contrast thecumulative NFD histograms for white-patch
normaliza-tion (dotted curves) with those for the optimal
fittingperformance (solid curves). White-patch normalizationyields
very good constancy results that are generallycomparable with the
optimal fitting performance.
6. SHARPENING RELATIVE TOMULTIPLE ILLUMINANTS
Data-based sharpening was introduced primarily to vali-date the
idea of sensor-based sharpening and to ensurethat our particular
choice of sharpening parameters ledto reasonable results. Figure 2
shows that the optimalsensors, as determined by means of data-based
sharpen-ing for each illuminant, closely resemble one another
andfurthermore that they resemble the unique set of sensor-based
sharpened sensors as well. Although all the sen-sors are similar,
the question remains regarding whetherthere might be an optimal
sharpening transform for theentire illuminant set.
In Ref. 23 we derive conditions for perfect DMT colorconstancy,
using sharpened sensors. Because the sharp-ening transform applies
to a whole space of illuminants,it is in essence a type of global
data-based sharpening.
The theoretical result is based on
finite-dimensionalapproximations of surface reflectance and
illumination,and what is shown is -that, if surface reflectances-
arewell modeled by three basis functions and illuminantsby two
basis functions, then there exists a set of new
-
1560 J. Opt. Soc. Am. A/Vol. 11, No. 5/May 1994
tions of Ac and N AC, as a result of Eq. (24). Conse-quently an
observation vector obtained for any surfaceunder an illuminant
EL(A) = aEC(A) + ,BE2(A) can be ex-pressed as a linear transform of
its descriptor vector:
pe = [al + I3.M]ACo = [al + /3IpC, (25)
where I is the identity matrix. Calculating the eigen-vector
decomposition of N,
(26)a M = aix ie f
and expressing the identity matrix in terms of C,
100
80
_60
40
20
100
80
; 602 0
40
20
100
80o
-60
40
0. it I 0 o -J . I o i
0 2 4 6 810 0 2 4 6 a 10 0 2 4 6 810%error %error %error
Fig. 6. Cumulative NFD histogram obtained with each
testilluminant (CIE A, D48, D65, D75, and D100) for optimal
fittingof VW observations (solid curves) and transformed
white-patchnormalization of VW observations (dotted curves). The
sixthcumulative NFD histogram shows average color
constancyperformance.
sensors for which a DMT can yield perfect color con-stancy.
These restrictions are quite strong; nonetheless,statistical
studies have shown that a three-dimensionalbasis set provides a
fair approximation to real surfacereflectance2 ' and that a
two-dimensional basis set de-scribes daylight illumination 9
reasonably well. More-over, Marimont and Wandell2 4 have developed
a methodfor deriving basis functions that is dependent on how
re-flectance, illuminant, and sensors interact to form sen-sor
responses [i.e., Eq. (2) above is at the heart of theirmethod]. A
three-dimensional model of reflectance anda two-dimensional model
of illumination is shown to pro-vide a good model of actual
response vectors.
Given these dimensionality restrictions on reflectanceand
illumination, cone response vectors of surfaces viewedunder a
canonical illuminant, that is descriptors, can bewritten as
pC = A r, (22)
(27)
enables us to rewrite the relationship between an obser-vation
and a descriptor, Eq. (25), as a diagonal transform:
T'pe = [a I + D ]Tpc. (28)
Writing pC in terms of pe leads directly to
tpc = [al +D] *Tpe. (29)
The import of Eq. (29) is that when the appropriateinitial
sharpening transformation T is applied, a diago-nal transform
supports perfect color constancy, subject ofcourse to the
restrictions imposed on illumination andreflectance.
These restrictions compare favorably with thoseemployed by
D'Zmura,2 5 who showed that, given three-dimensional reflectances
and three-dimensional illumi-nants, one can obtain perfect color
constancy, given twoimages of three color patches under two
different illumi-nants and using a nondiagonal transform.
From the Munsell reflectance spectra and our six
testilluminants, we used principal-component analysis to de-rive
the basis vectors for reflectance and illumination.Using these
vectors, we constructed lighting matrices andthen calculated the
sharpening transform by means ofEq. (26). The formulas for the
perfect sharpened sensorsare given in Eqs. (30)-(32), where they
are contrastedwith the corresponding formulas obtained with respect
tosensor-based and data-based sharpening. The symbolsR, G, and B
denote the VW red, green, and blue conemechanisms, respectively,
scaled to have unit norms, andthe superscripts p, s, and d denote
perfect, sensor-based,and data-based sharpening, respectively:
RP = 2.44R - 1.93G + 0.110B,
where the superscript c denotes the canonical illuminant.Since
illumination is two dimensional, there is necessarilya second
illuminant E2 (A) linearly independent with EC(A)(together they
form the span). Associated with this sec-ond illuminant is a second
linearly independent lightingmatrix A2. It follows immediately that
A2 is some lineartransform NM away from AC:
A2 = MAC,
NM =A2[AC]-
(23)
(24)
Since every illuminant is a linear combination of EC(A)and
E2(A), lighting matrices in turn' are linear combina-
Rs = 2.46R - 1.97G + 0.075B,
Rd = 2.46R - 1.98G + 0.100B,
GP = 1.55G - 0.63R - 0.16B,
Gs = 1.58G - 0.66R - 0.12B,
Gd = 1.52G - 0.58R - 0.14B,
BP 1.OB - 0.13G + 0.081,
Bs = 1.OB - 0.14G + 0.09R,
Bd = 1.OB - 0.13G + 0.07R. (32)
CIE A D48
100
80
.- 60
N40
20
100
80
fl 60
aC 40
20
100
80
60
a 40
20
D65
0 2 4 6 8 10%error
Average
0 2 4 6 8 10%error
D75
0 2 4 6 8 10%error
D100
(30)
(31)
Finlayson et al.
= TgriTe
-
Vol. 11, No. 5/May 1994/J. Opt. Soc. Am. A 1561
It is reassuring that the perfect sharpened sensors arealmost
identical to those derived through sensor-basedand data-based
sharpening. Therefore, even though nei-ther the sensor-based
sharpened sensors nor the data-based sharpened ones are optimized
relative to a wholeset of illuminants, sharpening in all cases
generates sen-sors that are similar to those that work perfectly
for alarge, although restricted, class of illuminants. This
the-oretical result provides strong support for the hypothe-sis,
already confirmed in part by the consistency of thedata-based
sharpening results, that a single sharpeningtransformation will
work well for a reasonable range ofilluminants.
7. SPECTRAL SHARPENING AN])THE HUMAN VISUAL SYSTEM
If the human visual system employs a DMT for colorconstancy, our
results show that we should expect it touse sharpened sensors,
since doing so would optimize itsperformance. In this section we
briefly examine some ofthe psychophysical evidence for sharpened
sensitivities.
Sharpened sensitivities have been detected in both fieldand test
spectral sensitivity experiments (for a reviewof these terms see
Foster2 6). Sperling and Harwerth' 3
measured the test spectral sensitivities of human
subjectsconditioned to a large white background and found,
con-sistent with our findings for sharpened sensors, sharp-ened
peaks at 530 and 610 nm, with no sharpening of theblue
mechanism.
These authors propose that a linear combination ofthe cone
responses accounts for the sharpening. Theyfound that the sharpened
red sensor can be modeled asthe red cone minus a fraction of the
green and that thesharpened green sensor can be modeled as the
green coneminus a fraction of the red. This finding correspondswell
with our theoretical results in that our sharpeningtransformations,
Eqs. (30) and (31), basically involve redminus green and green
minus red, with only a slightcontribution from the blue.
More recently, Foster' 2 observed that field and testspectral
sensitivities show sharpened peaks when theyare derived in the
presence of a small monochromaticauxiliary field coincident with
the test field. Foster andSnelgar27 extended this work by
performing a hybrid ex-periment with a white, spatially coincident
auxiliary field,and sharpened sensitivities again were found. In
bothcases these experimentally determined, sharpened sen-sitivities
agree with our theoretical results. Foster andSnelgar28 verified,
as did Sperling and Harwerth,13 thatthe sharpened sensitivities
were a linear combination ofthe cone sensitivities.
Krastel and Braun2 9 measured spectral field sensitivi-ties
under changing illumination when, as in the experi-ments by
Sperling and Harwerth,' 3 a white conditioningfield is employed.
Illumination color was changed byplacement of colored filters in
front of the eye. Thesame test spectral-sensitivity curve results
under botha reddish and a bluish illuminant, suggesting that
theeye's sharpened mechanisms are unaffected by illumi-nation. More
recently Kalloniatis and Harwerth14 mea-sured cone spectral
sensitivities under white adaptingfields of different intensity and
found the sharpened sensi-
tivities to be independent of the intensity of the
adaptingfield.
Poirson and Wandell'6 developed techniques for mea-suring the
spectral sensitivity of the eye with respect tothe task of color
discrimination. For color discriminationamong briefly presented
targets, the spectral sensitivitycurve has relatively sharp peaks
at 530 and 610 nm.
Although the general correspondence between oursharpened sensors
and the above psychophysical resultsdoes not imply that sharpening
in humans exists for thepurpose of color constancy, at least the
evidence thata linear combination of the cone responses is
employedsomewhere in the visual system lends plausibility to
theidea that sharpening might be used in human color-constancy
processing. On the other hand, our findingthat sharpening could
improve the performance of somecolor-constancy methods suggests an
explanation for theexistence of spectral sharpening in humans.
8. CONCLUSION
Spectral sharpening generates sensors that improve
theperformance of methods based on color constancy theo-ries (e.g.,
von Kries adaptation, Land's retinex) that use
0.4
D D 0.3N>0 0.2E c° u 0.1
0.0
-0.1
0.4
. > 0.3N>Ad 0.2E c8 XD 0.1
0.0
-0.1
0.4
, >, 0.3
0.2Ec°e 0.1
0.0
400 450 500 550Wavelength
400 450 500 550Wavelength
600 650
600 650
400 450 500 550 600 650Wavelength
Fig. 7. Solid curves show results for the Kodak Wratten
filters#66, #52, and #38; dotted curves show the results of
sensor-basedsharpening; dashed curves show the mean of the
data-basedsharpened sensors obtained for the five test illuminants
(CIEA, D48, D65, D75, and D100). Top, long-wavelength mecha-nism;
middle, medium-wavelength mechanism; bottom, short-wavelength
mechanism.
Finlayson et al.
-
1562 J. Opt. Soc.Am. A/Vol. 11, No. 5/May 1994
diagonal-matrix transformations. Data-based sharpen-ing finds
sensors that are optimal with respect to a givenset of surface
reflectances and illuminants. Sensor-based sharpening finds the
most-narrow-band sensorsthat can be created as a linear combination
of a givenset of sensors. Finally, for restricted classes of
illumi-nants and reflectance (they are constrained to be two
andthree dimensional, respectively) we have shown that thereexists
a sharpening transform with respect to which a di-agonal matrix
will support perfect color constancy. Thesharpening transform
derived by means of this analysis isin close agreement with the
sensor-based sharpening anddata-based sharpening transforms. In all
three casessharpened sensors substantially improve the accuracywith
which a DMT can model changes in illumination.The sensor-based and
the data-based sharpening tech-niques are quite general and can be
applied to visualsystems that have more than three sensors.
As with the cone sensitivities, sharpening of a colorcamera's
sensitivities can also have a significant effect.Use of
overlapping, broadband filters such as Wratten#66, #52, and #38
(Ref. 30) could be advantageous, sincefrom an exposure standpoint
they filter out less light, buttheir use could be disadvantageous
from a color-constancystandpoint. Sharpening such filters, as shown
in Fig. 7,can provide a good compromise among the
competingrequirements.
Spectral sharpening is not, in itself, a theory of
colorconstancy, in that it makes no statement about how tochoose
the coefficients of the diagonal matrix. Instead,we propose
sharpening as a mechanism for improving thetheoretical performance
of DMT algorithms of color con-stancy regardless of how any
particular algorithm mightcalculate the diagonal-matrix
coefficients to be used inadjustment for an illumination
change.
Since the performance of DMT algorithms improvessignificantly
when sharpened sensitivities are employed,and, furthermore, since
it then compares favorably withthe performance of the best possible
nondiagonal trans-form, our results suggest that if a linear
transform is acentral mechanism of human color constancy, then
afteran appropriate, fixed sharpening transformation of thesensors
there is little to be gained through the use of any-thing more
general than a DMT.
In general, our results lend support to DMT-basedtheories of
color constancy. In addition, since spectralsharpening aids color
constancy, we have advanced thehypothesis that sharpening might
provide an explanationfor the psychophysical finding of sharpening
in the hu-man visual system.
ACKNOWLEDGMENTS
This research has been supported by the Simon FraserUniversity
Center for Systems Science and the NaturalSciences and Engineering
Research Council of Canadaunder grant 4322.
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