Optimization of
Sensor Response Functions
for Colorimetry of
Reflective and Emissive Objects
Mark Wolski*, Charles A. Bouman, Jan P. Allebach
Purdue University, School of Electrical and
Computer Engineering, West Lafayette, IN 47907
Eric Walowit
Color Savvy Systems Inc., Springboro, OH 45066
*now with General Motors Research and Development
Center, Warren, MI 48090-9055.
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Overall Goal
Design components (color filters) for an
inexpensive device to perform colorimetric
measurements from surfaces of different types
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Device Operation Highlights


Output: XYZ tristimulus values
3 modes of operation
Emissive
Reflective/EE
EE
n
Reflective/D65
D65
n
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n
Computation of
Tristimulus Values

Stimulus Vector – n
n
31 samples taken at 10
nm intervals
400
l
ne

Emissive Mode

Reflective Mode n  diag[i EE/D65 ] r
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700
Tristimulus Vector

Tristimulus vector
t m  [Xm ,Ym , Zm ]T  A mn˜ , m = EM, EE, D65

Color matching matrix – Am (3x31)

Effective stimulus
m = EM
e,
n˜  
r, m = EE or D65
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Color Matching Matrix
z
1.8
1.6
x 
 y T 
 T 
z 
T
A EM
3x31 matrix
of color
matching
functions
1.4
1.2
x
y
1.0
0.8
0.6
0.4
0.2
0
400
450
500
550
l
600
A EE  A EMdiag[i EE ]
A D65  A EMdiag[i65 ]
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650
700
Device Architecture
Detectors
LED’s
LED’s
Filters
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Computational Model
X
Z
Y
matrix multiply
d ...
d
d ...
d
...
d ...
d
...
f1
f2
...
f2
...
f 4 ...
f4
l5
l1
f1
l1
...
r
Tm
r
...
r
...
l5
r
l1
...
l5
r
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r
Estimate of Tristimulus Vector

Estimate
ˆt  TmF mn,

m = EM, EE, D65
Channel matrix
 emissive mode
F EM  diag[f 1 ]d
diag[f 4 ]d
T
 reflective modes
F EE/D65  diag[l1 ]diag[f 1 ]d
diag[l1 ]diag[f 2 ]d
diag[l1 ]diag[f 4 ]d
diag[l5 ]diag[f 1 ]d
diag[l5 ]diag[f 2 ]d
diag[l5 ]diag[f 4 ]d ,
T
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Error Metric

Tristimulus error
t  t  ˆt
 (A m  T mF m )˜n

CIE uniform color space
1/3


L
*
116(Y
/
Y
)
 16
 
w
u  a *  500[(X / Xw )1/3  (Y / Yw )1/3 ]
b *  200[(Y / Y )1/3  (Z / Z )1/3 ] 
  
w
w

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Error Metric (cont.)

Linearize about nominal tristimulus value t = t0
u0  J0 t 0
116
 0
J 0  13 500 500
 0
200


0 Xw-1/3 X0-2/3
0
0  0
Y w-1/3Y0-2/3

200 
0
 0


0

Zw-1/3Z0-2/3 
0
Linearized error norm
E 0  u0
2
 J 0 (A m  T mF m )n0
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2
Error Metric (cont.)

Consider ensemble of 752 real stimuli nk
E 2EM

1
N

2
Jk (A m  T mF m )nk 2
k

Rearrange and sum over k
E 2EM
 Beq vec(Am )  Beq vec(T mF m )
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2
2
Regularization

Filter feasbility
 Roughness cost
 s  Ks
4
Df
2
k 1

2
k 2
Design robustness
 Effect of noise and/or component variations
˜  F  F
F
m
m
m
 Augment error metric
m 

B eq vec(A m )  Beq FTm  I3
vec(Tm ) 2 
2
K r2
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2
vec(T m ) 2
Design Problem
Overall cost function
h(f 1 , f 2 ,f 3 ,f 4 ,T EM ,T EE ,T D65 ; Kr , Ks ) 


Solution procedure
 EM ,  EE ,  D65  20   s
 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 ,TD65
h(F,TEM , TEE ,TD65 ;Kr , Ks )
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Experimental Results
Optimal filter set for Kr = 0.1 and Ks = 1.0
1
1
transmittance

400
1
400
700
400
700
1
700 400
wav elength
700
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Experimental Results (cont.)

Effect of system tolerance Won meansquared error
1.5
avg
1.0
E rms
3 limits
0.5
0
10 -5
10 -4
10 -3
10 -2
W (log s cale)
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Experimental Results (cont.)

Error performance in true L*a*b* for set of
752 spectral samples
mode
avg
L*a*b* E
max
L*a*b* E
% E's > 1
emissive
0.27
1.56
1.5
reflective / EE
0.18
1.00
0.2
reflective / D65
0.15
0.91
0
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Experimental Results (cont.)

Emissive mode L*a*b* error surface
1.2311
1
E
0.5
0
100
50
b*
0
-50 -60
-40
-20
0
20
40
a*
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60
Approximation of
Color Matching Matrix
emissive mode
1.5
1.0
0.5
0
400 450
1.5
500
550
600
700
D65 reflective mode
EE reflective mode
1.5
1.0
1.0
0.5
0.5
0
400 450
650
500
550
600
650
700
0
400 450
500
550
600
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650
700
Conclusions

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
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