O 24 a reexamination of srm as a means of beer color specification
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O-24 A Reexamination of SRM as a Means of Beer Color Specification. A.J. deLange [email protected] 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

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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

[email protected]

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


Beer 10c l ab computation

1 cm Transmission Spectrum, 81 pts

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


Beer 10c word chart

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.)


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

Accum

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

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

  • 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

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

  • 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

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

SDC’s model deviation from corkscrew

Raspberry

Ale

Kriek


Summary

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….


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