O-24 A Reexamination of SRM as a

Means of Beer Color Specification

A.J. deLange

ajdel@cox.net

ASBC 2007 Annual Meeting

June 19, 2007

Current and Proposed Methods of Beer Color Specification

1 cmAbsorptionSpectrum

Normalizeby A430;

Convert toTransmission

Spectrum

A380

A385

A780

A430

X 12.7

ComputeSpectrumDeviation;

Encode intoSDCs

1 cmAbsorptionSpectrum

SRMX

12.7

A430

1 cmAbsorptionSpectrum

Convert toTransmission

Spectrum

A380

A385

A780

Illuminant C(3) 10° CMFs

White Point

Compute X, Y, Z;Map to

L*, a*, b*E 308

L*

a*b*

(3) EigenvectorsAverage Normalized Spectrum

ReconstructSpectrum

Scale to anyPath;

Convert toTransmission

SRM

SDC1

SDC2

SDC3

Compute X, Y, Z;Map to

any coord.E 308

Avg. Norm. Spec.(3) Eigenvectors

Any IlluminantAny (3) CMFs

Any White Point

L*

a* or u

b* or v

Beer-10C report

Proposed report

Beer-10A report

/A700

< .039?O.K.

Beer’s Law• Coloring matter in beer appears to follow Beer’s Law

– Absorption (log) is proportional to molar concentration

• Colorants are in fixed proportion in an ensemble of average beers

• If true, absorption spectra would be identical if normalized by absorption at one wavelength– Noted by Stone and Miller in 1949 when proposing SRM

4

3

2

1

0

Log Absorption

700600500400

Wavelength, nm

Ensemble of 60 Beer Absorp[tion (1cm) Spectra"Normalized" by SRM. Unusual curves are for fruit beers

1cmNormAbs60

Deviation From Average

• Miller and Stone studied 39 beers• Used deviation from average (A700/A430 ratio) to disqualify beers as

being suitable for SRM– Test still in MOA Beer-10A

• We propose to quantify deviation, encode it, and augment SRM report with this information– Encoding by spectral deviation Principal Components

• SRM plus encoded deviation permits reconstruction of spectrum– Spectrum inserted into ASTM E 308 for visible color calculation

under various conditions• Tested on an ensemble of 59 beers with good results• Worked with transmission spectra rather than absorption because they

give better computed color accuracy

Spectrum Compression: 59 Beer Transmission Spectra (1 cm). Ensemble variance (sum of squares of difference

between spectrum and average spectrum) 2 = 6.481.0

0.8

0.6

0.4

0.2

0.0

700600500400

Wavelength, nm

Ensemble of 59 Beer Transmission (1cm)Spectra. Average Variance: 6.48202

Blue spectra are fruit beers

Normalize absorption spectra by A430; convert to

Transmission: 2 = 0.29 (4.4% of original)

Normalization: Convert transmission to absorption (take -log10), divide by 430 nm valueand convert back to transmission (antilog[-A])

Conventional Beers

1.0

0.8

0.6

0.4

0.2

0.0

700600500400

Wavelength, nm

Ensemble of 59 Beer Transmission (1cm) Spectra"Normalized" by SRM. Average Variance: 0.28512(4.4% of Unnormalized Variance of 6.48202)

Fruit Beers

Transmission Spectra (normalized) deviation from average (2 = 0.29 i.e. 4.4% of original)

-0.3

-0.2

-0.1

0.0

0.1

Resudual Transmission Fraction

700600500400

Wavelength, nm

Normalized Transmission SpectraAfter Subtraction of Mean Spectrum

Singular value decomposition (SVD) of matrix of these data (eigen analysis of covariancematrix) yield eigen vectors used to compute Principal Components of individual spectra

Variation from 1st 2 PC’s taken out, average added back in: 2 = .00165 (0.025% of original)

0.8

0.6

0.4

0.2

0.0

700600500400

Wavelength, nm

Ensemble of 59 Beer Transmission (1cm) Spectra "Normalized" by SRM with First 2 Pricipal Components Taken Out. Average Variance: 0.001657(0.025% of Original Variance)

“Fuzziness” about average can be modeled by use of additional PC’s

Summary of Last Few Slides

• Normalizing by SRM removes 95% of variation (relative to average) in beer spectra

• First 2 Principal Components removes most of remainder (leaving but 0.025% of the original total)– As these PCs quantify deviation of individual beer spectrum from average

let’s call them“spectrum deviation coefficients” (SDC) • What’s left is the average plus 0.025% variation• Thus, if we take the average and add the 2 SDC’s worth of

variation back, then un-normalize by SRM we can reconstruct the transmission spectrum, T()

– T() ~ Log-1{(Log[Avg() + SDC1*E1() + SDC2*E2()])/(SRM/12.7)}

• From reconstructed spectrum we can calculate actual colors. Question: how accurately?

CIELAB Color Difference, E

• CIELAB Tristimulus Color:– Brightness L* (0 - 100)– a*: green-red (~ -100 to 100)– b*: blue-yellow (~ -100 to 100)– Calculated from 81 spectral transmission measurements (380, 385, 390…

780nm per ASTM E 308)

• All L*ab colors relative to a reference “White Point”– White: L* = 100, a* = 0, b* = 0

• Supposed to be uniform perceptual space• Difference between 2 colors

– E = [(L1-L2)2+ (a1-a2)2 + (b1-b2)2]1/2 (i.e. Euclidean Distance) – E < 3 considered a “good match”

• General accuracy of press reproduction: > 2

Example Color DifferencesCenter patch: ~16 SRM, 1 cm, Illum. C

a* -6 -3 0 +3 +6

b*

+6

+3

0

-3

-6

E’s Adjacent in same row or column (excluding top row): 3; Adjacent diagonal (excluding top row): 4.2 Center to corner (excluding top row): 8.5

Top Row Only L* -6 -3 0 +3 +6

E this patch tolower right corner:20.8

Ensemble Error in L*ab color calculated from average spectrum unnormalized by SRM (no PC

correction)50

40

30

20

10

Color Error, L*ab

E units

15010050 Beer SRM

Raspberry Ale * Error in L ab Color Reconstructed from SRM alone

59 , 1 , Beers cm Path Illuminant C2° Observer

RMS : 13.1E

Kriek

Stout

40

30

20

10

Color Error, L*ab

E units

15010050 Beer SRM

Raspberry Ale

* Error in L ab Color Reconstructed from SRM alone

59 , 1 , Beers cm Path Illuminant A2° Observer

RMS : 11.8EKriek

Stout

40

30

20

10

0

Color Error, L*ab

E units

15010050 Beer SRM

Raspberry Ale

* Error in L ab Color Reconstructed from SRM alone

59 , 5 , Beers cm Path Illuminant A2° Observer

RMS : 14.9E

Kriek

Stout

Calculate L*ab color from full spectrum; calculate lab color from average spectrum and SRM; plot difference

40

30

20

10

0

Color Error, L*ab

E units

15010050 Beer SRM

Raspberry Ale

* Error in L ab Color Reconstructed from SRM alone

59 , 5 , Beers cm Path Illuminant C2° Observer

RMS : 14.2E

Kriek

Stout

Ensemble error in L*ab color calculated from SRM + 2 SDCs

5

4

3

2

1

0

15010050

Beer SRM

Raspberry AleError in L*ab Color Reconstructed fromSRM and 2 Principal Components59 Beers, 1 cm Path, Illuminant C2° ObserverRMS : 1.11E

Kriek

5

4

3

2

1

0

15010050

Beer SRM

Raspberry Ale

Error in L*ab Color Reconstructed fromSRM and 2 Principal Components59 Beers, 5 cm Path, Illuminant C2° ObserverRMS : 1.11E

Kriek

5

4

3

2

1

0

15010050

Beer SRM

Raspberry Ale

Error in L*ab Color Reconstructed fromSRM and 2 Principal Components59 Beers, 1 cm Path, Illuminant A2° ObserverRMS : 1.17E

Kriek

5

4

3

2

1

0

15010050

Beer SRM

Raspberry Ale

Error in L*ab Color Reconstructed fromSRM and 2 Principal Components59 Beers, 5 cm Path, Illuminant A2° ObserverRMS : 1.19E

Kriek

Beer-10C L*ab Computation1 cm Transmission Spectrum, 81 pts

Illum. C Distribution+, 81 pts Point wise Multiply

x matching function+, 81 pts

y matching function+, 81 pts

z matching function+, 81 pts Accum,Scale+

Accum,Scale+

Accum,Scale+

x data y data z data

Point wise Multiply

1 ~ 380nm 81 ~ 780nm

X Y Z

(X/Xr)1/3 (Y/Yr)1/3 (Z/Zr)1/3

ZrYr

XrReference White+

++ -

-

116

16 200500-

a* b*L*

+ = Tabulated in MOAOther illuminants, matching functions,reference whites allowed by E 308

For different path (E 308) take log, scale,take antilog

Beer-10C Illustrated

Beer -10C Word Chart• Basis: ASTM E308 - Defines color measurement in US• Take 81 spectrum measurements: 380 to 780 nm; 5 nm steps; 1 cm path or scale to 1

cm from any other path length (Lambert Law).• Convert to transmission. Weight by spectral distribution of Illuminant C (tabulated

values)• Multiply point wise by each (3) color matching functions (table values of CIE 10°

observer). Scale sums by 100/2439.6 to compute X, Y, Z• Compute fx(X/Xr), fy(Y/Yr), fz(Z/Zr)

– f(u) = u1/3 (in E 308 f(u) is an offset linear function for u< .008856)– Xr = 97.285, Yr = 100, Zr = 116.145 (in E 308 these are calculated from illuminant spectral

distribution function)

• Compute – L* = 116 fx(X/Xr) - 16 – a*= 500[fx(X/Xr)- fy(Y/Yr)]– b*= 200[fy(Y/Yr) - fz(Z/Zr)]

• Report L*, a* and b* (could report X, Y and Z or other tristim.)

Proposed MOA SDC Computation

Average Spectrum+, 81 pts Point wise Subtract

1st Eigenfunction+, 81 pts

2nd Eigenfunction+, 81 pts

3rd Eigenfunction+, 81 pts Accum Accum Accum

1st data 2nd data 3rd data

Point wise Multiply

+ = Tabulated in proposed MOAEigenfunctions are those of covariance matrix of normalized, de-meaned spectrum ensemble“SDC” is, thus, a Principal Component of the input spectrum.

1 cm Absorption Spectrum, 81 pts1 ~ 380nm 81 ~ 780nm

Normalize (point wise divide)

A430

12.7

1st SDC > 2nd SDC > 3rd SDC SRM

Convert to transmission (10-A)

Reported Parameters:

Proposed Method Illustrated

Note: Before application of matching function the tabulated average functionis subtracted from normalized function. This is not shown on this chart.

New Method Word Chart• Take 81 absorption (log) measurements: 380 to 780 nm, 5 nm steps, 1

cm path or scale (Lambert law) to 1 cm from any other path

• Compute SRM = 10*A430*2.54/2 = 12.7*A430

• Divide each point in spectrum by A430 (absorption at 430 nm)

• Convert to transmission (change sign and take antilog)

• Subtract average transmission spectrum (from published table values)

• Multiply point wise by each of 2 - 4 “matching functions” (published table values of ensemble eigenfunctions) and accumulate

• Report SRM and accumulated sums (SDC1, SDC2, )Notes: 1. Table values would be published as part of a new MOA

2. Matching functions are eigenfunctions of covariance matrix of “normalized”, de-meaned transmission spectra thus coefficients(SDC’s) are “Principal Components” of the beer’s spectrum.

Color Calculation from New Parameters

500

Average Spectrum+, 81 pts Point wise Add --> Aprox Norm. Spec.

1st Eigenfunction+, 81 pts

2nd Eigenfunction+, 81 pts

3rd Eigenfunction+, 81 pts

Sum scaled eigenfunctions = deviation

3rd SDC

+ = Tabulated in proposed MOA

1 cm Absorption Spectrum, 81 pts1 ~ 380nm 81 ~ 780nm

Un-normalize (point wise multiply)A430

1/12.7

2nd SDC1st SDC SRM

Convert to absorption (-log10)

818181

Path, cm

E 308 10-AXYZLuv

Lab

Illuminant

Observer (CIE matching functions)

etc

Ref. XYZ

Input Parameters:

Color Computation Word Chart

• Add point wise SDC1 times first matching function + SDC2 times second matching function (table values)… to average (tabulated values) spectrum– If no SDC values (i.e. SRM only) then just use average spectrum

• Convert to absorption (log) spectrum• Compute A430 = SRM/12.7• Multiply each point in spectrum by A430

– This is the reconstructed 1 cm absorption spectrum

• Compute color per ASTM E 308 (or Beer 10C)– Scale to any path length– Weight by any illuminant– Use either 10° or 2° color matching functions– Relative to any white point

59 Beers in CIELAB Coordinates

80

60

40

20

b*

50403020100

a*

2.42.933.23.33.6

3.83.83.8

4.44.7

5.75.9

6

6 6.2

7.37.5

99.69.89.9

11.3

14.4

15.216

16.3

17.4

17.417.618.1

18.218.8

19.1

19.3

21.724.224.5

26.3

26.9

27.2

27.5

28.730.5

31.2

31.937

39.8

43.548.3

51.952.6

76.678.5

84.5

86.3

86.3

115.2

191.8

Colors (chroma) for 59 Beers1 cm Path; 2° Observer; Illuminant CLine Connects Points in Order of SRMSRM numbers indicated each vertex

RaspberryAle

Kriek

Beer colors are restricted: generally follow “corkscrew” in (in ~ dark) to page

SDC’s model deviation from corkscrew

Summary• Beer colors are a subspace of all colors; spectra are similar

– This makes data compression possible

• SRM + 2 - 3 SDC’s (PCs) gives spectrum reconstruction sufficiently close for accurate tristimulus color calculation

• Calculation of SDC’s is as simple as calculation of tristim.– Can all be done in a spreadsheet like that for Beer 10C

• SRM + SDC’s is a candidate for new color reporting method• Plenty to be done before a new MOA could be promulgated

– Acceptance of concept– Verification of claim– Definition of ensemble and measurements for determination of average

spectrum, eigen functions– Trials, collaborative testing….