The LabPQR Color Space

The LabPQR Color Space

Giordano B. Beretta

Print Production Automation Lab
Hewlett-Packard Laboratories
Palo Alto, California

8 April 2010

G. Beretta (HP Labs) LabPQR Overview 8 April 2010 1 / 35

Color matching

Colors are assessed by matching them with reference colors on a
small-ﬁeld bipartite screen


Color-matching functions
Given a monochromatic stimulus Qλ of wavelength λ, it can be written
as

Qλ = Rλ R + Gλ G + Bλ B
where Rλ , Gλ , and Bλ are the spectral tristimulus values of Qλ
Assume an equal-energy stimulus E whose mono-chromatic
constituents are Eλ (equal-energy means Eλ ≡ 1)
The equation for a color match involving a mono-chromatic constituent
Eλ of E is

r ¯ ¯
Eλ = ¯(λ)R + g (λ)G + b(λ)B
r ¯ ¯
where ¯(λ), g (λ), and b(λ), are the spectral tristimulus values of Eλ
Deﬁnition (color-matching functions)
r ¯ ¯
The sets of such values ¯(λ), g (λ), and b(λ) are called color-matching
functions (CMF)

Color-matching functions
3.0
Stiles-Burch (1955;1959)
2.5
2.0 b(λ)
1.5 g(λ)
1.0 r(λ)
0.5
0.0
nm
-0.5
400 500 600 700


CIE 1931 standard colorimetric observer
We want to build an instrument delivering results valid for the group of
normal trichromats (95% of population); since

R=k Pλ¯(λ)dλ
r

G=k ¯
Pλ g (λ)dλ

B=k ¯
Pλ b(λ)dλ

an ideal observer can be deﬁned by specifying values for the
color-matching functions
Deﬁnition (CIE 1931 standard colorimetric observer)
The Commission Internationale de l’Éclairage (CIE) has recommended
¯ ¯ ¯
such tables containing x (λ), y (λ), z (λ) for λ ∈ [360nm, 830nm] in 1nm
steps

Illumination
The spectral power distribution of the light reﬂected to the eye by an
object is the product, at each wavelength, of the object’s spectral
reﬂectance value by the spectral power distribution of the light source
CWF Complexion

400 500 600 700 400 500 600 700 400 500 600 700

Incident SPD x Reflectance curve = Reflected SPD

Deluxe Complexion
CWF

400 500 600 700 400 500 600 700 400 500 600 700

Mathematical interpretation

R=k Pλ · ¯(λ)dλ
r

means that the red color coordinate is obtained by integrating the SPD
using the red CMF for the measure, where

Pλ = E(λ) · S(λ)

is the product of the SPD of an illuminant E with the object spectrum S.
Usually we are interested in the coordinates of various objects under a
ﬁxed illuminant for a standard observer, so we reorder to

R=k ¯(λ)E(λ) · S(λ)dλ
r


Discretization

In practice, the CMF are given as a table with 1nm steps, and
instruments measure at steps of 1, 4, 10, 20nm etc., so in reality this is
a summation [for red R]:

R=k ¯(λ)E(λ)S(λ)dλ ≈ k
r ¯(λi )E(λi )S(λi )∆λ
r

The integration resp. summation is over the visible range [380, 780]nm,
but in practice it is often over [380, 730]nm for n = 36 samples
Instead of doing color science with measure theory, we can do it
with simple linear algebra
In 1991 H. Joel Trussell has made available a comprehensive
MatLab library and several key papers for color scientists
Since then, spectral color science is mostly done with linear
algebra


Formalism

We use the vector-space notation
WLOG, let k = 1
¯ ¯
R = (R E)S, G = (G E)S, ¯
B = (B E)S
Instead of doing this for each of R, G, B or X , Y , Z , using linear
algebra we can write it as a single equation by combining the CMF
in an n × 3 matrix A with the CMFs data in the columns:
Υ = (A E)S

Sometimes we are interested in the color of a ﬁxed object under
different illuminants, then we write
Υ = A (ES) = A η

η corresponds to the Pλ from earlier

Metameric stimuli

Consider two color stimuli

Q1 = R1 R + G1 G + B1 B
Q2 = R2 R + G2 G + B2 B

Deﬁnition (metameric stimuli)
If Q1 and Q2 have different spectral radiant power distributions, but
R1 = R2 and G1 = G2 and B1 = B2 , the two stimuli are called
metameric stimuli

Fact
Color reproduction works because of metamerism


Fundamental and residue

How can we reconcile metamerism and color reproduction
technology?
In 1953 Günter Wyszecki pointed out that the SPD of stimuli
consists of a fundamental color-stimulus function η (λ) intrinsically
´
associated with the tristimulus values, and a metameric black
function κ(λ) called the residue
κ(λ) is orthogonal to the space of the CMF


Matrix R theory

How does this translate to the discrete case?
In 1982 Jozef Cohen with William Kappauf developed the matrix R
theory
Use an orthogonal projector to decompose stimuli in fundamental
and residue
The fundamental is a linear combination of the CMF A
The metameric black is the difference between the stimulus and
the fundamental
For a set of metamers η1 (λ), η2 (λ), . . . , ηm (λ):

A η1 = A η2 = · · · = A ηm = Υ


Development of matrix R

R is deﬁned as the symmetric n × n matrix

Deﬁnition (matrix R)
R := A(A A)−1 A

Matrix R is an orthogonal projection
A(A A)−1 =: Mf , so R = Mf A (remember: Υ = A η)
Because A has 3 independent columns, R has rank 3
It decomposes the stimulus spectrum into fundamental η (λ) and
´
the metameric black κ:

η = Rηi
´
κ = ηi − η = ηi − Rηi = (I − R)ηi
´


Corollaries

Metameric black has tristimulus value zero

A κ = [0, 0, 0]

η = Rηi means that any group of metamers has a common
´
fundamental η , but different residues κ
´
Inversely, a stimulus spectrum can be expressed as

ηi = η + κ = Rηi + (I − R)ηi
´

i.e., the stimulus spectrum can be reconstructed if the
fundamental metamer and metameric black are known
Why is this useful?


Spectral color reproduction

Sometimes colorimetry is insufﬁcient
Spectral printer models
Mapping from one device to another
Fluorescent inks and/or media
Physical media models
Ink-media interactions
Security printing
More than 3 colorant hues (e.g., CMYKOGV)
Multiple illuminants (metamerism index minimization)
Mapping K generation between two different CMYK printers
Scanner and camera characterization
...


Reducing the data

Storing a multidimensional vector for each pixel is expensive
Can we project on a lower-dimensional vector space?
Yes, because the spectra are relatively smooth
Popular technique: principal component analysis
Due to the usually smooth spectra, the dimension can be quite
low: between 5 and 8
We have known how to deal with this for decades, it just requires
linearly more processing


The hard problem

We would like to use an ICC type workflow also for spectral
imaging
Colorimetric workflow:
profile connection
image 3-hue printer
space

The killer is the LUT used in the PCS:
bands in bands out levels per band size [bytes]
3 6 17 30K
6 6 17 145M
9 6 17 700G
31 6 17 8 · 1027 G


Interim Connection Space

Proposal by Mitchell Rosen et al. at RIT
Introduce a lower-dimensional Interim Connection Space ICS

PCS to ICS

scene multi-hue printer

ICS to counts via
low-dim. LUT


Choosing the basis vectors
Can we deviate from the usual PCA method of choosing the
largest eigenvectors and build on some other useful basis?
When deﬁning the basis vectors for XYZ, the new basis was
chosen so that one vector coincides with luminous efﬁciency V (λ)
compatibility of colorimetry with photometry
1995 proposal by Bernhard Hill et al. at RWTH Aachen:
incorporate three colorimetric dimensions
compatibility of spectral technology with colorimetry
spectral scan values

S1 S2 S3 .............. S16
smoothing inverse
spectral reconstruction S1 S2 S3 .............. S64 multichannel
display or
basis functions printing
multispectral values X Y Z V 4 ............. V 16
decoding

L* a* b* nonlinear transform

nonlinear representation L* a* b* V* 4 ............ V* 16
encoding
quantization
conventional L bit a bit b bit V 4,bit .......... V 16,bit
three channel system interface
display or printer


LabPQR Approach

Mitchell Rosen et al. at RIT
1 Calculate operator similar to matrix R using regression analysis
on a speciﬁc printer (unconstrained), or matrix R directly
(constrained)
2 Calculate residue using principal component analysis
3 Calculate tristimulus values XYZ
4 Calculate PQR from residue (3 largest EV)
5 Calculate LabPQR from XYZPQR using CIE equations


LabPQR notation
ˆ
Reconstructed spectrum (LabPQR transform): P = TNc + VNp
T : colorimetric transformation
Nc : tristimulus vector Υ
V : basis vectors in PQR
Np : residual
Constrained: Tck = A(A A)−1 = Mf
Remember: matrix R = Mf A
Unconstrained: Tu = RNc (Nc Nc )−1 via least squares analysis
over a number of tristimulus vectors for spectra R = ηi
Calculation of V : ﬁrst 3 eigenvectors in metameric black κ via
principal component analysis
Conventional notation:

η = Rηi
´ (= Mf Υ)
κ = ηi − η = ηi − Rηi = (I − R)ηi
´


LabPQR gamut


Using LabPQR

The diagram in the previous slide indicates how the algorithm is
veriﬁed
Note in particular the meaning of gamut mapping in PQR

The usage is to print a color chart and measure it spectrally
The resulting table from device coordinates to spectra is then
1 converted to LabPQR
2 inverted
The inverted table is used to interpolate LabPQR values to obtain
the device coordinates to reproduce a requested spectrum


Canon i9900 dye-based inks

G

K


Caveats

Green and black dyes tend to have an increasing reﬂectance in
the far red
Paper brighteners act in the blue range
RIT work: [400, 700]nm for n = 31 samples
Most real world data: [380, 730]nm for n = 36 samples
Visible range: [380, 780]nm
The range has a strong effect on the principal components


PQR


Quality metric

objective function = minimize (CIEDE2000 + k · ∆PQR)


Accuracy of Matrix R vs. unconstrained

What price in loss of accuracy do we pay for compatibility
conventional metamerism theory?
Constrained model depends only on CMF
Unconstrained model additionally depends on device
Based on simulations (no LUT),
the constrained model is more accurate in general
for a single ﬁxed printer, the unconstrained method allows the use
of less principal components: LabPQ
Short spectral range [400, 700]nm caused problems with green ink


Summary

1 Conventional ICC workflow is based on colorimetry
2 A spectral workflow can can solve many more problems
proof printing
fluorescence
metamerism
...
3 LabPQR is low-dimensional and compatible with colorimetry


Bibliography I

Henry R. Kang.
Computational Color Technology.
SPIE, Bellingham, 2006.
Bernhard Hill.
The history of multispectral imaging at Aachen University of
Technology.
Web Document, May 2002.
http://www.ite.rwth-aachen.de/Inhalt/Documents/
Hill/AachenMultispecHistory.pdf.
Thomas Keusen and Werner Praefcke.
Multispectral color system with an encoding format compatible with
the conventional tristimulus model.
In IS&T/SID Third Color Imaging Conference, pages 112–114,
1995.


Bibliography II

Mitchell R. Rosen and Ohta Noboru.
Spectral color processing using an interim connection space.
In IS&T/SID Eleventh Color Imaging Conference, pages 187–192,
2003.
Maxim W. Derhak and Mitchell R. Rosen.
Spectral colorimetry using LabPQR —- an interim connection
space.
In IS&T/SID Twelfth Color Imaging Conference, pages 246–250,
2004.
Mitchell R. Rosen and Maxim W. Derhak.
Spectral gamuts and spectral gamut mapping.
In Mitchell R. Rosen, Francisco H. Imai, and Shoji Tominaga,
editors, Spectral Imaging: Eighth International Symposium on
Multispectral Color Science, volume 6062, pages
60620K–1–60620K–11, 2006.

Bibliography III

Maxim W. Derhak and Mitchell R. Rosen.
Spectral colorimetry using LabPQR: An interim connection space.
Journal of Imaging Science and Technology, 50(1):53—63, 2006.
Shohei Tsutsumi, Mitchell R. Rosen, and Roy S. Berns.
Spectral gamut mapping using LabPQR.
Journal of Imaging Science and Technology, 51(6):473—485,
2007.
Spectral color reproduction using an interim connection
space-based lookup table.
Journal of Imaging Science, 52(4):040201–040201–13, 2008.


Bibliography IV

Spectral color management using interim connection spaces
based on spectral decomposition.
Color Research & Application, 33(4):282–299, August 2008.


Discussion
http://www.hpl.hp.com/personal/Giordano_Beretta/


The LabPQR Color Space

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