O24 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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A.J. deLange
ajdel@cox.net
ASBC 2007 Annual Meeting
June 19, 2007
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 SpecificationBeer10A report
A430
1 cm
Absorption
Spectrum
SRM
/
< .039?
O.K.
Illuminant C
A700
(3) 10° CMFs
White Point
Beer10C 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
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
Conventional Beers
Fruit Beers
Normalization: Convert transmission to absorption (take log10), divide by 430 nm value
and convert back to transmission (antilog[A])
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
“Fuzziness” about average can be modeled by use of additional PC’s
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
Calculate L*ab color from full spectrum; calculate lab color from average spectrum and SRM; plot difference
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
Beer10C L*ab ComputationFor 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*
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 (10A)
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, demeaned spectrum ensemble
“SDC” is, thus, a Principal Component of the input spectrum.
Note: Before application of matching function the tabulated average function
is subtracted from normalized function. This is not shown on this chart.
Notes: 1. Table values would be published as part of a new MOA
2. Matching functions are eigenfunctions of covariance matrix
of “normalized”, demeaned transmission spectra thus coefficients
(SDC’s) are “Principal Components” of the beer’s spectrum.
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
10A
Luv
etc
1 ~ 380nm
81 ~ 780nm
A430
Path, cm
Unnormalize (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
Beer colors are restricted: generally follow “corkscrew” in (in ~ dark) to page
SDC’s model deviation from corkscrew
Raspberry
Ale
Kriek