O-24 A Reexamination of SRM as a Means of Beer Color Specification

1. 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

2. 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

3. 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. 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

5. 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

6. 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])

7. 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

8. 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

9. 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?

10. 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

11. 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

12. 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

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

14. 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*

15. Beer-10C Illustrated

16. 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.)

17. 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.

18. 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.

19. 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.

20. 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

21. 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

22. 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

23. 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….