1 / 23

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

larryhansen
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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….

More Related