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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. 12.7. X. Compute X, Y, Z; Map to any coord. E 308. Compute X, Y, Z; Map to L*, a*, b* E 308.

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O 24 a reexamination of srm as a means of beer color specification

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

12.7

X

Compute

X, Y, Z;

Map to

any coord.

E 308

Compute

X, Y, Z;

Map to

L*, a*, b*

E 308

Current and Proposed Methods of Beer Color Specification

Beer-10A report

A430

1 cm

Absorption

Spectrum

SRM

/

< .039?

O.K.

Illuminant C

A700

(3) 10° CMFs

White Point

Beer-10C report

A380

1 cm

Absorption

Spectrum

Convert to

Transmission

Spectrum

L*

A385

Proposed report

a*

b*

A780

Any Illuminant

Any (3) CMFs

Any White Point

X

12.7

Average Normalized Spectrum

Avg. Norm. Spec.

(3) Eigenvectors

(3) Eigenvectors

A380

SRM

Normalize

by A430;

Convert to

Transmission

Spectrum

1 cm

Absorption

Spectrum

Compute

Spectrum

Deviation;

Encode into

SDCs

Reconstruct

Spectrum

Scale to any

Path;

Convert to

Transmission

A385

L*

SDC1

a* or u

SDC2

A430

SDC3

b* or v

A780


Beer s law
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


Deviation from average
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


O 24 a reexamination of srm as a means of beer color specification

Spectrum Compression: 59 Beer Transmission Spectra (1 cm). Ensemble variance (sum of squares of difference between spectrum and average spectrum) s2 = 6.48

Blue spectra are fruit beers


Normalize absorption spectra by a 430 convert to transmission s 2 0 29 4 4 of original
Normalize absorption spectra by A430; convert to Transmission: s2 = 0.29 (4.4% of original)

Conventional Beers

Fruit Beers

Normalization: Convert transmission to absorption (take -log10), divide by 430 nm value

and convert back to transmission (antilog[-A])


Transmission spectra normalized deviation from average s 2 0 29 i e 4 4 of original
Transmission Spectra (normalized) deviation from average (s2 = 0.29 i.e. 4.4% of original)

Singular value decomposition (SVD) of matrix of these data (eigen analysis of covariance

matrix) yield eigen vectors used to compute Principal Components of individual spectra


Variation from 1 st 2 pc s taken out average added back in s 2 00165 0 025 of original
Variation from 1st 2 PC’s taken out, average added back in: s2 = .00165 (0.025% of original)

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


Summary of last few slides
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(l)

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

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


Cielab color difference d e
CIELAB Color Difference, DE

  • 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

    • DE = [(L1-L2)2+ (a1-a2)2 + (b1-b2)2]1/2 (i.e. Euclidean Distance)

    • DE < 3 considered a “good match”

      • General accuracy of press reproduction: > 2


Example color differences center patch 16 srm 1 cm illum c
Example Color DifferencesCenter patch: ~16 SRM, 1 cm, Illum. C

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

DE this patch to

lower right corner:

20.8

Db*

+6

+3

0

-3

-6

Da* -6 -3 0 +3 +6

DE’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


Ensemble error in l ab color calculated from average spectrum unnormalized by srm no pc correction
Ensemble Error in L*ab color calculated from average spectrum unnormalized by SRM (no PC correction)

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


Ensemble error in l ab color calculated from srm 2 sdcs
Ensemble error in L*ab color calculated from SRM + 2 SDCs spectrum unnormalized by SRM (no PC correction)


Beer 10c l ab computation

1 cm Transmission Spectrum, 81 pts spectrum unnormalized by SRM (no PC correction)

Illum. C Distribution+, 81 pts

Accum,

Scale+

Accum,

Scale+

Accum,

Scale+

(X/Xr)1/3

(Z/Zr)1/3

S

S

S

Beer-10C L*ab Computation

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

take antilog

81 ~ 780nm

1 ~ 380nm

Point wise Multiply

x matching function+, 81 pts

Point wise Multiply

y matching function+, 81 pts

x data

y data

z data

z matching function+, 81 pts

Zr

Z

Y

Yr

X

Xr

Reference White+

(Y/Yr)1/3

+

-

-

+

116

+ = Tabulated in MOA

Other illuminants, matching functions,

reference whites allowed by E 308

-

16

200

500

b*

L*

a*


Beer 10c illustrated
Beer-10C Illustrated spectrum unnormalized by SRM (no PC correction)


Beer 10c word chart
Beer -10C Word Chart spectrum unnormalized by SRM (no PC correction)

  • 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.)


O 24 a reexamination of srm as a means of beer color specification

Accum spectrum unnormalized by SRM (no PC correction)

Accum

Accum

Proposed MOA SDC Computation

1 cm Absorption Spectrum, 81 pts

A430

1 ~ 380nm

81 ~ 780nm

Normalize (point wise divide)

Convert to transmission (10-A)

Point wise Subtract

Average Spectrum+, 81 pts

1st Eigenfunction+, 81 pts

Point wise Multiply

2nd Eigenfunction+, 81 pts

1st data

2nd data

3rd data

12.7

3rd Eigenfunction+, 81 pts

Reported Parameters:

SRM

1st SDC > 2nd SDC > 3rd SDC

+ = Tabulated in proposed MOA

Eigenfunctions are those of covariance matrix of normalized, de-meaned spectrum ensemble

“SDC” is, thus, a Principal Component of the input spectrum.


Proposed method illustrated
Proposed Method Illustrated spectrum unnormalized by SRM (no PC correction)

Note: Before application of matching function the tabulated average function

is subtracted from normalized function. This is not shown on this chart.


New method word chart
New Method Word Chart spectrum unnormalized by SRM (no PC correction)

  • 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.


O 24 a reexamination of srm as a means of beer color specification

Color Calculation from New Parameters spectrum unnormalized by SRM (no PC correction)

1st Eigenfunction+, 81 pts

2nd Eigenfunction+, 81 pts

3rd Eigenfunction+, 81 pts

Lab

E 308

XYZ

1 cm Absorption Spectrum, 81 pts

10-A

Luv

etc

1 ~ 380nm

81 ~ 780nm

A430

Path, cm

Un-normalize (point wise multiply)

Illuminant

Ref. XYZ

Convert to absorption (-log10)

Observer (CIE matching functions)

Point wise Add --> Aprox Norm. Spec.

Average Spectrum+, 81 pts

Sum scaled eigenfunctions = deviation

81

81

81

1/12.7

+ = Tabulated in proposed MOA

Input Parameters:

3rd SDC

1st SDC

2nd SDC

SRM

500


Color computation word chart
Color Computation Word Chart spectrum unnormalized by SRM (no PC correction)

  • 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
59 Beers in CIELAB Coordinates spectrum unnormalized by SRM (no PC correction)

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

SDC’s model deviation from corkscrew

Raspberry

Ale

Kriek


Summary
Summary spectrum unnormalized by SRM (no PC correction)

  • 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….