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Optimization ofOptimization ofSensor Response FunctionsSensor Response Functions
for Colorimetry offor Colorimetry ofReflective and Emissive ObjectsReflective and Emissive Objects
Mark Wolski*, Charles A. Bouman, Jan P. AllebachPurdue University, School of Electrical and
Computer Engineering, West Lafayette, IN 47907
Eric WalowitColor Savvy Systems Inc., Springboro, OH 45066
*now with General Motors Research and Development Center, Warren, MI 48090-9055.
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Overall GoalOverall GoalDesign components (color filters) for an
inexpensive device to perform colorimetric measurements from surfaces of different types
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Device Operation HighlightsDevice Operation Highlights
Output: XYZ tristimulus values
3 modes of operation
D65
Reflective/D65
EE
Reflective/EEEmissive
n
n n
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Computation of Computation of Tristimulus ValuesTristimulus Values
Stimulus Vector – n
Emissive Mode
Reflective Mode
31 samples taken at 10 nm intervals
400 700
n
n =e
n=diag[iEE/D65] r
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Tristimulus VectorTristimulus Vector
Tristimulus vector
Color matching matrix – Am (3x31)
Effective stimulus
tm =[Xm,Ym,Zm]T =Am˜ n, m = EM, EE, 65D
˜ n =
e, m=EM
r, m= 65EE or D ⎧ ⎨ ⎩
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Color Matching MatrixColor Matching Matrix
400 450 500 550 600 650 7000
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
xy
z
3x31 matrix of color matching functions
AEM =xT
yT
zT
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
AEE =AEMdiag[iEE]
AD65 =AEMdiag[i65]
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Device ArchitectureDevice Architecture
LED’s LED’s
Detectors
Filters
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Computational ModelComputational Model
d d d d
r r r
...
...
X Y
l 5 l1 l5 l 1
f1 1f 2f4f
matrix multiply
...
Z
d
...
...
l5
2f
... ...
...
...
d
l 1
4f
...
...
...
r r r
Tm
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Estimate of Tristimulus VectorEstimate of Tristimulus Vector Estimate
Channel matrix emissive mode
reflective modes
t =TmFmn, m=EM, EE, 65D
FEM = diag[f1]d L diag[f 4 ]d[ ]T
FEE/D65 = diag[l1]diag[f1]d L diag[ l5]diag[f1]d[diag[l1]diag[f 2]d L diag[ l5]diag[f 2 ]dL
diag[l1]diag[f 4]d L diag[l5]diag[f 4]d]T,
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Error MetricError Metric
Tristimulus error
CIE uniform color space
u =L *a*b*
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥=
116(Y /Yw)1/3 −16
500[(X / Xw)1/3 −(Y / Yw)
1/3]200[(Y / Yw)
1/3 −(Z / Zw)1/3]
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Δt = t − ˆ t
= (Am − TmFm )˜ n
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Error Metric (cont.)Error Metric (cont.) Linearize about nominal tristimulus value t = t0
Linearized error norm
Δu0 ≈ J0Δt0
J0 =13
0 116 0500 −500 00 200 −200
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
Xw-1/3X0
-2/3 0 00 Yw
-1/3Y0-2/3 0
0 0 Zw-1/3Z0
-2/3
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
ΔE0 =Δu0 2 ≈ J0 (Am − TmFm)n0 2
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Error Metric (cont.)Error Metric (cont.)
Consider ensemble of 752 real stimuli nk
Rearrange and sum over k
ΔEEM
2 = 1N Jk (Am − TmFm)nk 2
2
k∑
ΔEEM
2 = Beqvec(Am ) − Beqvec(TmFm )2
2
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RegularizationRegularization Filter feasbility
Roughness cost
Design robustnessEffect of noise and/or component variations
Augment error metric
εs =Ks D2f k 2
2
k=1
4
∑
F m =Fm+ ΔFm
εm= Beq vec(Am) − Beq Fm
T ⊗I3( )vec(Tm )2
2+ Kr
2 vec(Tm ) 22
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Design ProblemDesign Problem Overall cost function
Solution procedure
For any fixed F = [f1, f2, f3, f4]T determine optimal
coefficient matrices TEM, TEE, and TD65 as solution to
least-squares problem
Minimize partially optimized cost via gradient
search
h*(F;Kr ,Ks) = min
TEM ,TEE,T 65D
h(F,TEM ,TEE,TD65;Kr,Ks)
h(f1, f2 ,f3 ,f 4 ,TEM ,TEE ,TD65;Kr ,Ks) = εEM , εEE , εD65[ ] 20 +εs
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Experimental ResultsExperimental Results
Optimal filter set for Kr = 0.1 and Ks = 1.0
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Experimental Results (cont.)Experimental Results (cont.)
Effect of system tolerance on mean-squared error
ΔE
0
0.5
1.0
10-4 10-3 10-2
Ω (log scale)
1.5
10 -5
avg
3σ limitsrms
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Experimental Results (cont.)Experimental Results (cont.)
Error performance in true L*a*b* for set of
752 spectral samples
modeavg
L*a*b* ΔE max
* * * L a b ΔE % Δ > 1E's
emissive
/ reflective EE
/ 65reflective D
0.27
0.18
0.15
1.56
1.00
0.91
1.5
0.2
0
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Experimental Results (cont.)Experimental Results (cont.)
Emissive mode L*a*b* error surface
-60-40
-200
2040
60
-50
0
50
1000
0.5
1
1.2311
a*b*
ΔE
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Approximation of Approximation of Color Matching MatrixColor Matching Matrix
450 500 550 600 650 700
emissive mode
0.5
1.0
1.5
4000
450 500 550 600 650 700
EE reflective mode
0.5
1.0
1.5
0400 450 500 550 600 650 700
D65 reflective mode
0.5
1.0
1.5
0400
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ConclusionsConclusions For given device architecture, it is possible to design
components that will yield satisfactory performance filters are quite smooth
device is robust to noise
excellent overall accuracy
Solution method is quite flexible independent of size of sample ensemble
Vector space methods provide a powerful tool for solving problems in color imaging