Transcript SRM Color Revisited

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
A430
1 cm
Absorption
Spectrum
A700
/
X
< .039?
O.K.
A385
A780
SRM
Illuminant C
(3) 10° CMFs
White Point
A380
1 cm
Absorption
Spectrum
Beer-10A report
Convert to
Transmission
Spectrum
Compute
X, Y, Z;
Map to
L*, a*, b*
E 308
Average Normalized Spectrum
(3) Eigenvectors
A380
A385
1 cm
Absorption
Spectrum
A430
A780
X
L*
a*
b*
12.7
Compute
Spectrum
Deviation;
Encode into
SDCs
Proposed report
Any Illuminant
Any (3) CMFs
Any White Point
Avg. Norm. Spec.
(3) Eigenvectors
SRM
Normalize
by A430;
Convert to
Transmission
Spectrum
Beer-10C report
SDC1
SDC2
SDC3
Reconstruct
Spectrum
Scale to any
Path;
Convert to
Transmission
Compute
X, Y, Z;
Map to
any coord.
E 308
L*
a* or u
b* or v
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
Ensemble of 60 Beer Absorp[tion (1cm) Spectra
"Normalized" by SRM.
Unusual curves are for fruit beers
Log Absorption
3
2
1
1cmNormAbs60
0
400
500
600
Wavelength, nm
700
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) s2 = 6.48
Blue spectra are fruit beers
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 (s2 = 0.29 i.e. 4.4% of original)
0.1
Resudual Transmission Fraction
0.0
-0.1
Normalized Transmission Spectra
Af ter Subtraction of Mean Spectrum
-0.2
-0.3
400
500
600
700
Wavelength, nm
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 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
• 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, 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
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)
50
DE units
40
Kriek
Stout
40
Error in L* ab Color Reconstruct ed f rom
SRM alone
59 Beers, 5 cm Pat h, I lluminant C
2° Observ er
RMS DE: 14.2
Kriek
DE units
Error in L* ab Color Reconstruct ed f rom
SRM alone
59 Beers, 1 cm Pat h, I lluminant C
2° Observ er
RMS DE: 13.1
Raspberry Ale
30
Color Error, L*ab
Color Error, L*ab
30
20
20
Raspberry Ale
10
Stout
10
0
50
100
Beer SRM
50
150
30
Raspberry Ale
Kriek
40
DE units
Stout
Error in L* ab Color Reconstruct ed f rom
SRM alone
59 Beers, 1 cm Pat h, I lluminant A
2° Observ er
RMS DE: 11.8
Color Error, L*ab
Color Error, L*ab
DE units
40
20
10
100
Beer SRM
150
Error in L* ab Color Reconstruct ed f rom
SRM alone
59 Beers, 5 cm Pat h, I lluminant A
2° Observ er
RMS DE: 14.9
Kriek
30
20
Raspberry Ale
Stout
10
0
50
100
Beer SRM
150
50
100
Beer SRM
150
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
Beer-10C L*ab Computation
For different path (E 308) take log, scale,
take antilog
1 cm Transmission Spectrum, 81 pts
81 ~ 780nm
1 ~ 380nm
Point wise Multiply
Illum. C Distribution+, 81 pts
x matching function+, 81 pts
Point wise Multiply
y matching function+, 81 pts
x data
Xr
(X/Xr)1/3
S
Accum,
Scale+
Zr
Y
Yr
X
+
z data
Accum,
Scale+
Accum,
Scale+
z matching function+, 81 pts
Reference White+
y data
Z
(Z/Zr)1/3
(Y/Yr)1/3
-
+
-
S
116
+ = Tabulated in MOA
Other illuminants, matching functions,
reference whites allowed by E 308
500
16
a*
-
S
L*
200
b*
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
1 cm Absorption Spectrum, 81 pts
1 ~ 380nm
A430
81 ~ 780nm
Normalize (point wise divide)
Convert to transmission (10-A)
Point wise Subtract
Average Spectrum+, 81 pts
1st Eigenfunction+, 81 pts
2nd Eigenfunction+, 81 pts
3rd Eigenfunction+, 81 pts
Point wise Multiply
1st data
Accum
Reported Parameters: 1st SDC
2nd data
Accum
> 2nd SDC
3rd data
Accum
>
3rd SDC
12.7
SRM
+ = 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
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
• 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
Lab
XYZ
Luv
etc
E 308
1 cm Absorption Spectrum, 81 pts
10-A
1 ~ 380nm
Path, cm
Illuminant
Ref. XYZ
Observer (CIE matching functions)
Average Spectrum+, 81 pts
81 ~ 780nm
A430
Un-normalize (point wise multiply)
Convert to absorption (-log10)
Point wise Add --> Aprox Norm. Spec.
Sum scaled eigenfunctions = deviation
3rd Eigenfunction+, 81 pts
2nd Eigenfunction+, 81 pts
81
81
81
1/12.7
1st Eigenfunction+, 81 pts
+ = Tabulated in proposed MOA
Input Parameters:
1st SDC
2nd SDC
3rd SDC
SRM
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
Beer colors are restricted: generally follow “corkscrew” in (in ~ dark) to page
SDC’s model deviation from corkscrew
31.9
26.3
24.5
24.2
21.7
80
18.1
18.2
19.3
17.4
16 17.6
18.8
15.216.3
19.1
14.4
37
39.8
31.2
28.7
30.5
27.5
43.548.3
26.9
52.6
51.9
60
27.2
76.686.3
78.5
86.3
11.3
b*
99.69.8
9.9
40
20
17.4
Colors (chroma) for 59 Beers
1 cm Path; 2° Observer; Illuminant C
Line Connects Points in Order of SRM
SRM numbers indicated each vertex
7.37.5
65.9
5.76.2
6
4.7
4.4
3.8
3.8
3.8
3.2
3.3
33.6
2.4
2.9
Raspberry
Ale
Kriek
84.5
115.2
191.8
0
10
20
30
a*
40
50
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….