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O-24 A Reexamination of SRM as a
Means of Beer Color Specification
A.J. deLange
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….