the labpqr color space

The LabPQR Color Space

Giordano B. Beretta

Print Production Automation LabHewlett-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 asmall-field bipartite screen

Color-matching functions

Given a monochromatic stimulus Qλ of wavelength λ, it can be writtenas

Qλ = RλR + GλG + BλBwhere Rλ, Gλ, and Bλ are the spectral tristimulus values of Qλ

Assume an equal-energy stimulus E whose mono-chromaticconstituents are Eλ (equal-energy means Eλ ≡ 1)The equation for a color match involving a mono-chromatic constituentEλ of E is

Eλ = r(λ)R + g(λ)G + b(λ)Bwhere r(λ), g(λ), and b(λ), are the spectral tristimulus values of Eλ

Definition (color-matching functions)

The sets of such values r(λ), g(λ), and b(λ) are called color-matchingfunctions (CMF)

Color-matching functions

700 600 500 400

Stiles-Burch (1955;1959)

CIE 1931 standard colorimetric observer

We want to build an instrument delivering results valid for the group ofnormal trichromats (95% of population); since

R = k∫

Pλr(λ)dλ

G = k∫

Pλg(λ)dλ

B = k∫

Pλb(λ)dλ

an ideal observer can be defined by specifying values for thecolor-matching functions

Definition (CIE 1931 standard colorimetric observer)

The Commission Internationale de l’Éclairage (CIE) has recommendedsuch tables containing x(λ), y(λ), z(λ) for λ ∈ [360nm,830nm] in 1nmsteps

Illumination

The spectral power distribution of the light reflected to the eye by anobject is the product, at each wavelength, of the object’s spectralreflectance value by the spectral power distribution of the light source

500 700600400 500 700600400 500 700600400

500 700600400500 700600400 500 700600400

Incident SPD Reflected SPDReflectance curvex =

DeluxeCWF

Complexion

Mathematical interpretation

R = k∫

Pλ · r(λ)dλ

means that the red color coordinate is obtained by integrating the SPDusing 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 afixed illuminant for a standard observer, so we reorder to

R = k∫

r(λ)E(λ) · S(λ)dλ

Discretization

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

R = k∫

r(λ)E(λ)S(λ)dλ ≈ k∑

r(λi)E(λi)S(λi)∆λ

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 itwith simple linear algebraIn 1991 H. Joel Trussell has made available a comprehensiveMatLab library and several key papers for color scientistsSince then, spectral color science is mostly done with linearalgebra

Formalism

We use the vector-space notationWLOG, let k = 1R = (R>E)S, G = (G>E)S, B = (B>E)SInstead of doing this for each of R,G,B or X ,Y ,Z , using linearalgebra we can write it as a single equation by combining the CMFin 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 underdifferent illuminants, then we write

Υ = A>(ES) = A>η

η corresponds to the Pλ from earlier

Metameric stimuli

Consider two color stimuli

Q1 = R1R + G1G + B1BQ2 = R2R + G2G + B2B

Definition (metameric stimuli)If Q1 and Q2 have different spectral radiant power distributions, butR1 = R2 and G1 = G2 and B1 = B2, the two stimuli are calledmetameric stimuli

FactColor reproduction works because of metamerism

Fundamental and residue

How can we reconcile metamerism and color reproductiontechnology?In 1953 Günter Wyszecki pointed out that the SPD of stimuliconsists of a fundamental color-stimulus function η(λ) intrinsicallyassociated with the tristimulus values, and a metameric blackfunction κ(λ) called the residueκ(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 RtheoryUse an orthogonal projector to decompose stimuli in fundamentaland residueThe fundamental is a linear combination of the CMF AThe metameric black is the difference between the stimulus andthe fundamentalFor 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)−1A>

Matrix R is an orthogonal projectionA(A>A)−1 =: Mf , so R = Mf A> (remember: Υ = A>η)Because A has 3 independent columns, R has rank 3It decomposes the stimulus spectrum into fundamental η(λ) andthe 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 commonfundamental η, but different residues κInversely, a stimulus spectrum can be expressed as

ηi = η + κ = Rηi + (I − R)ηi

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

Spectral color reproduction

Sometimes colorimetry is insufficientSpectral printer modelsMapping from one device to anotherFluorescent inks and/or mediaPhysical media modelsInk-media interactionsSecurity printingMore than 3 colorant hues (e.g., CMYKOGV)Multiple illuminants (metamerism index minimization)Mapping K generation between two different CMYK printersScanner and camera characterization. . .

Reducing the data

Storing a multidimensional vector for each pixel is expensiveCan we project on a lower-dimensional vector space?Yes, because the spectra are relatively smoothPopular technique: principal component analysisDue to the usually smooth spectra, the dimension can be quitelow: between 5 and 8

We have known how to deal with this for decades, it just requireslinearly more processing

The hard problem

We would like to use an ICC type workflow also for spectralimagingColorimetric workflow:

profile connection space 3-hue printerimage

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 · 1027G

Interim Connection Space

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

PCS to ICS

multi-hue printerscene

ICS to counts via low-dim. LUT

Choosing the basis vectors

Can we deviate from the usual PCA method of choosing thelargest eigenvectors and build on some other useful basis?When defining the basis vectors for XYZ, the new basis waschosen so that one vector coincides with luminous efficiency V (λ) compatibility of colorimetry with photometry1995 proposal by Bernhard Hill et al. at RWTH Aachen:incorporate three colorimetric dimensions compatibility of spectral technology with colorimetry

X Y Z V 4 ............. V 16

S1 S2 S3 .............. S64

L* a* b*

L* a* b* V* 4 ............ V* 16

L bit abit b bit V 4,bit .......... V 16,bit

nonlinear transform

spectral scan values

conventionalthree channel

display or printersystem interface

S1 S2 S3 .............. S16

multichanneldisplay orprinting

smoothing inverse

basis functions

quantization

spectral reconstruction

multispectral values

nonlinear representationencoding

decoding

LabPQR Approach

Mitchell Rosen et al. at RIT1 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 analysis3 Calculate tristimulus values XYZ4 Calculate PQR from residue (3 largest EV)5 Calculate LabPQR from XYZPQR using CIE equations

LabPQR notation

Reconstructed spectrum (LabPQR transform): P = TNc + VNpT : colorimetric transformationNc : tristimulus vector ΥV : basis vectors in PQRNp: residual

Constrained: Tck = A(A>A)−1 = MfRemember: matrix R = Mf A>

Unconstrained: Tu = RN>c (NcN>c )−1 via least squares analysisover a number of tristimulus vectors for spectra R = ηi

Calculation of V : first 3 eigenvectors in metameric black κ viaprincipal 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 isverifiedNote in particular the meaning of gamut mapping in PQR

The usage is to print a color chart and measure it spectrallyThe resulting table from device coordinates to spectra is then

1 converted to LabPQR2 inverted

The inverted table is used to interpolate LabPQR values to obtainthe device coordinates to reproduce a requested spectrum

Canon i9900 dye-based inks

Caveats

Green and black dyes tend to have an increasing reflectance inthe far redPaper brighteners act in the blue rangeRIT work: [400,700]nm for n = 31 samplesMost real world data: [380,730]nm for n = 36 samplesVisible range: [380,780]nm

The range has a strong effect on the principal components

Quality metric

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

Accuracy of Matrix R vs. unconstrained

What price in loss of accuracy do we pay for compatibilityconventional metamerism theory?

Constrained model depends only on CMFUnconstrained model additionally depends on device

Based on simulations (no LUT),the constrained model is more accurate in generalfor a single fixed printer, the unconstrained method allows the useof less principal components: LabPQ

Short spectral range [400,700]nm caused problems with green ink

Summary

1 Conventional ICC workflow is based on colorimetry2 A spectral workflow can can solve many more problems

proof printingfluorescencemetamerism. . .

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 ofTechnology.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 withthe 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 connectionspace.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 onMultispectral Color Science, volume 6062, pages60620K–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.

Shohei Tsutsumi, Mitchell R. Rosen, and Roy S. Berns.Spectral color reproduction using an interim connectionspace-based lookup table.Journal of Imaging Science, 52(4):040201–040201–13, 2008.

Bibliography IV

Shohei Tsutsumi, Mitchell R. Rosen, and Roy S. Berns.Spectral color management using interim connection spacesbased 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

interim connection space

interim connection space

spectral power distribution

spectral gamut mapping

spectral tristimulus values

imaging science

metameric black

matching functions

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