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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